Real Gases

Real gases – as opposed to a perfect or ideal gas – exhibit properties that 

cannot be explained entirely using the ideal gas law. law. To understand the behaviour of real gases, the following must be taken into account: compressibility compressibility effects; variable specific heat capacity; capacity; van der Waals forces; forces; non-equilibrium non-equilibrium thermodynamic thermodynamic effects; issues with molecular dissociation and elementary reactions with variable composition. For most applications, such a detailed analysis is unnecessary, and the ideal gas approximation can be used with reasonable accuracy. On the other hand, realgas models have to be used near the condensation point of gases, near critical points,, at very high pressures, and in other less usual cases. points van der Waals model

Main article: article: van der Waals equation Real gases are often modeled by taking into account their molar weight and molar volume

The Virial equation derives from a perturbative treatment of treatment of statistical mechanics.

or alternatively

where A, B, C, A , B , and C are temperature dependent constants. ′

′

′

Peng– Peng –Robinson model

This two parameter equation (named after D.-Y. Peng and D. B. Robinson Robinson[3]) has the interesting property being useful in modeling some liquids as well as real gases.

Wohl model

The Wohl equation (named after A. Wohl Wohl[4]) is formulated in terms of c ritical values, making it useful when real gas constants are not a vailable. Where P is the pressure, T is the temperature, R the ideal gas constant, and V m the molar volume. volume . a and b are parameters that are determined empirically empirically for each gas, but are sometimes estimated from their critical temperature (Tc) and critical pressure (Pc) using these relations:

where

. Beattie– Beattie –Bridgeman model

Redlich– Redlich–Kwong model

The Redlich–Kwong equation is another two-parameters equation that is used to model real gases. It is almost always more accurate than the van der Waals equation,, and often more accurate than some equations with more than two equation parameters. The equation is

[5]

this equation is based on five experimentally determined constants. it is expressed as

where where a and b two empirical parameters that are not the same parameters as in the van der Waals equation. These parameters can be determined:

Berthelot and modified Berthelot model

The Berthelot equation (named after D. Berthelot  Berthelot [1] is very rarely used,

but the modified version is somewhat more accurate

Dieterici model

This model (named after C. Dieterici Dieterici[2]) fell out of usage in recent years

this equation is known to be reasonably accurate for densities up to about  0.8ρcr where ρcr is the density of the substance at the critical point. the constants appearing in the above equation are available in following table when P is in KPa, v is in \frac{m^3}{Kmol} , T is in K and R=8.314frac{kPa.m^3}{Kmol.K}}[6] R=8.314frac{kPa.m^3}{Kmol.K Gas

A_0

a

B_0

b

c

Air Argon,Ar Carbon Dioxide, Co_2 Helium,He Hydrogen,H_2 Nitrogen,N_2 Oxygen,O_2

131.8441 0.01931 0.04611 -0.001101 4.34×10^4 130.7802 0.02328 0.03931 0.0 5.99×10^4 507.2836 0.07132 0.10476 0.07235 6.60×10^5 2.1886 0.05984 0.01400 0.0 40 20.0117 -0.00506 0.02096 -0.04359 504 136.2315 0.02617 0.05046 -0.00691 4.20×10^4 151.0857 0.02562 0.04624 0.004208 4.80×10^4

Benedict –Webb– Webb–Rubin model

. Clausius model

Main article: Benedict –Webb–Rubin equation The BWR equation, sometimes referred to as the BWRS equation,

The Clausius equation (named after Rudolf Clausius) Clausius) is a very simple threeparameter equation used to model gases.

where

where Vc is critical volume. Virial model

where d is the molal density and where a, b, c, A, B, C, α, and γ are empirical constants. Note that the γ constant is a derivative of constant α and therefore almost identical to 1. The compressibility compressibility factor (Z), also known as the compression factor, is a useful thermodynamic property for modifying the ideal gas law to account for the real gas behavior. behavior.[1] In general, deviation from ideal behavior becomes more significant the closer a gas is to a phase c hange, the lower the temperature or the larger the pressure. Compressibility Compressibility factor values are usually obtained by calculation from equations of state (EOS), such as the virial equation which take compound specific empirical constants as input. For a gas that is a mixture of two or more pure gases (air or natural gas, for example), a gas composition is required before compressibility compressibility can be calculated. Alternatively, the compressibility factor for specific gases can be read from

generalized compressibility chart s[1] that plot as a function of pressure at  constant temperature. The compressibility factor is defined as

where is the molar volume, volume, is the molar volume of the corresponding corresponding ideal gas, is the pressure pressure,, is the temperature,, and temperature is the gas constant . For engineering applications, it is frequently expressed as

where is the density of the gas and is the specific gas constant  constant ,[2] being the molar mass. mass. For an ideal gas the compressibility compressibility factor is per definition. In many real world applications requirements for accuracy demand that deviations from ideal gas behaviour, i.e., i.e. , real gas behaviour, is taken into account. The value of  generally increases with pressure and decreases with temperature. At high pressures molecules are colliding more often. This allows repulsive forces between molecules to have a noticeable effect, making the molar volume of the real real gas gas (

) great greater er than than the molar molar volu volume me of of the the corre corresp spond onding ing ideal ideal gas gas (

), which causes to exceed one. one.[3] When pressures are lower, the molecules are free to move. In this case attractive forces dominate, making . The closer the gas is to its critical point or point or its boiling point, the more deviates from the ideal case. Generalized compressibility factor graphs for pure gases

Together they define the critical point of a fluid above which distinct liquid and gas phases of a given fluid do not exist. The pressure-volume-temperature pressure-volume-temperature (PVT) data for real gases varies from one pure gas to another. However, when the compressibility factors of various single-component gases are graphed versus pressure along with temperature isotherms many of the graphs exhibit similar isotherm shapes. In order to obtain a generalized graph that can be used for many different gases, the reduced pressure and temperature, and , are used to normalize the compressibility compressibility factor data. Figure 2 is an example of a generalized compressibility compressibility factor graph derived from hundreds of experimental PVT data points of 10 pure gases, namely methane, ethane, ethylene, propane, n-butane, ipentane, n-hexane, nitrogen, carbon dioxide and steam. There are more detailed generalized compressibility factor graphs based on as many as 25 or more different pure gases, such as the Nelson-Obert graphs. Such graphs are said to have an accuracy accuracy within 1-2 percent percent for values greater than 0.6 and within 4-6 percent for values of 0.3-0.6. The generalized compressibility factor graphs may be considerably in error for strongly polar gases which are gases for which the centers of positive and negative charge do not coincide. In such such cases cases the estimate estimate for may be in error by as much as 15-20 percent. The quantum gases hydrogen, helium, and neon do not conform to the corresponding-states corresponding-states behavior and t he reduced pressure and temperature for those three gases should be redefined in the following manner to improve the accuracy of predicting their compressibility factors when using the g eneralized graphs:

and where the temperatures are in kelvin and the pressures are in atmospheres. [4] Theoretical models

The virial equation is especially useful to describe the causes of non-ideality at a molecular level (very few gases are mono-atomic) as it is derived directly from statistical mechanics:

Where the coefficients in the numerator are known as virial coefficients and are functions of temperature. The virial coefficients account for interactions between successively larger groups of molecules. For example, accounts for interactions between pairs, for interactions between three gas molecules, and so on. Because interactions between large numbers of molecules are rare, the virial equation is usually truncated after the third term. term.[5] The Real gas article features more theoretical methods to compute compressibility compressibility factors Experimental values

Generalized compressibility factor diagram. The unique relationship between the compressibility factor and the reduced temperature, temperature, , and the reduced pressure, pressure, , was first recognized by Johannes Diderik van der Waals in 1873 and is known as t he two-parameter principle of corresponding states. states . The principle of corresponding states expresses the generalization that the properties of a gas which are dependent  on intermolecular forces are related to the critical properties of the gas in a universal way. That provides a most important basis for developing correlations of molecular properties. As for the compressibility compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature,

It is extremely difficult to generalize at what pressures or temperatures the deviation from the ideal gas becomes important. important. As a rule of thumb, the ideal gas law is reasonably accurate up to a pressure of about 2 atm atm,, and even higher for small non-associating molecules. For example methyl chloride, chloride, a highly polar molecule and therefore with significant intermolecular forces, the experimental value for the compressibility compressibility factor is at a pressure of 10 atm and temperature of 100 °C. °C.[6] For air (small non-polar molecules) at  approximately the same conditions, the compressibility compressibility factor is only (see table below for 10 bars bars,, 400 K). Compressibility Compressibility of air

Normal air comprises in crude numbers 80 percent nitrogen percent  nitrogen N2 and 20 percent  oxygen O2. Both molecules are small and non-polar (and therefore nonassociating). We can therefore expect that the behaviour of air within broad temperature and pressure ranges can be approximated as an ideal gas with reasonable accuracy. Experimental values for the compressibility factor confirm this. Z for air as function of pressure 1-500 bar

, and reduced

pressure, , should have the same compressibility compressibility factor. The reduced temperature and pressure are defined by

and Here

and

are known as the critical temperature and critical pressure of 

a gas. They are characteristics of each specific gas with

being the

temperature above which it is not possible to liquify a given gas and is the minimum pressure required required to liquify a given gas at its critical temperature.

75-200 K isotherms

Material constants that vary for each type of material are eliminated, in a recast  reduced form of a constitutive equation. equation . The reduced variables are defined in terms of  of critical critical variables. variables. It originated with the work of Johannes of  Johannes Diderik van der Waals in about 1873 1873[3] when he used the critical temperature and critical pressure to characterize a fluid. The most prominent example is the van der Waals equation of state, the reduced form of which applies to all fluids. Compressibility Compressibility factor at the critical point 

The compressibility factor at the critical point, which is defined as

, where the subscript indicates the critical point is point is predicted to be a constant independent of substance by many equations of state; the Van der Waals equation e.g. predicts a value of 250-1000 K isotherms Compressibility Compressibility factor for air (experimental values) Pressure, bar (absolute) Te mp, K 1 5 10 20 40 60 80 100 150 200 250 300 400 500 0.00 0.02 0.05 0.10 0.20 0.30 0.40 0.50 0.75 1.01 75 52 60 19 36 63 82 94 99 81 25 0.02 0.04 0.09 0.19 0.29 0.39 0.48 0.72 0.95 1.19 1.41 80 50 99 95 81 58 27 87 58 88 31 39 0.97 0.02 0.04 0.09 0.18 0.27 0.36 0.46 0.67 0.89 1.10 1.31 1.71 2.11 90 64 36 53 40 66 81 86 81 79 29 98 10 61 05 0.97 0.88 0.04 0.09 0.17 0.26 0.34 0.43 0.63 0.83 1.03 1.22 1.59 1.95 100 97 72 53 00 82 35 98 37 86 77 95 27 37 36 0.98 0.93 0.88 0.67 0.17 0.25 0.33 0.41 0.59 0.77 0.95 1.10 1.50 1.73 120 80 73 60 30 78 57 71 32 64 20 30 76 91 66 0.99 0.96 0.92 0.82 0.58 0.33 0.37 0.43 0.59 0.76 0.91 1.03 1.32 1.59 140 27 14 05 97 56 13 37 40 09 99 14 93 02 03 0.99 0.97 0.94 0.89 0.78 0.66 0.56 0.54 0.63 0.75 0.88 1.01 1.25 1.49 160 51 48 89 54 03 03 96 89 40 64 40 05 85 70 0.99 0.98 0.96 0.93 0.86 0.79 0.74 0.70 0.71 0.79 0.90 1.00 1.22 1.43 180 67 32 60 14 25 77 32 84 80 86 00 68 32 61 0.99 0.98 0.97 0.95 0.91 0.87 0.83 0.81 0.80 0.85 0.93 1.01 1.20 1.39 200 78 86 67 39 00 01 74 42 61 49 11 85 54 44 0.99 0.99 0.99 0.98 0.96 0.95 0.94 0.94 0.94 0.97 1.01 1.07 1.19 1.33 250 92 57 11 22 71 49 63 11 50 13 52 02 90 92 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.03 1.06 1.10 1.20 1.31 300 99 87 74 50 17 01 03 30 74 26 69 89 73 63 1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.01 1.03 1.06 1.09 1.13 1.21 1.30 350 00 02 04 14 38 75 21 83 77 35 47 03 16 15 1.00 1.00 1.00 1.00 1.01 1.01 1.02 1.03 1.05 1.07 1.10 1.14 1.21 1.28 400 02 12 25 46 00 59 29 12 33 95 87 11 17 90 1.00 1.00 1.00 1.00 1.01 1.02 1.02 1.03 1.06 1.09 1.11 1.14 1.20 1.27 450 03 16 34 63 33 10 87 74 14 13 83 63 90 78 1.00 1.00 1.00 1.00 1.01 1.02 1.03 1.04 1.06 1.09 1.11 1.14 1.20 1.26 500 03 20 34 74 51 34 23 10 50 13 83 63 51 67 1.00 1.00 1.00 1.00 1.01 1.02 1.03 1.04 1.06 1.09 1.11 1.14 1.19 1.24 600 04 22 39 81 64 53 40 34 78 20 72 27 47 75 1.00 1.00 1.00 1.00 1.01 1.02 1.03 1.04 1.06 1.08 1.10 1.12 1.17 1.21 800 04 20 38 77 57 40 21 08 21 44 61 83 20 50 100 1.00 1.00 1.00 1.00 1.01 1.02 1.02 1.03 1.05 1.07 1.09 1.11 1.15 1.18 0 04 18 37 68 42 15 90 65 56 44 48 31 15 89 Source: Perry's chemical engineers' handbook (6ed ed.). MCGraw-Hill. 1984. ISBN 0-07-049479-7 0-07-049479-7.. (table 3-162). values are calculated from values of  pressure, volume (or density), and temperature in Vassernan, Kazavchinskii, and Rabinovich, "Thermophysical Properties of Air and Air Components;' Moscow, Nauka, 1966, and NBS-NSF Trans. TT 70-50095, 1971: and Vassernan and Rabinovich, "Thermophysical Properties Properties of Liquid Air and Its Component, "Moscow, 1968, and NBS-NSF Trans. 69-55092, 1970.

.

Substance Value

H2O

0.23 0.23[4]

4He

0.31 0.31[4]

He

0.30 0.30[5]

H2

0.30 0.30[5]

Ne

0.29 0.29[5]

N2

0.29 0.29[5]

Ar 0.29 0.29[5] In physics and thermodynamics , an equation of state is a relation between state variables .[1] More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions. It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature , pressure , volume, or internal energy . Equations of state are useful in describing the properties of  fluids, mixtures of fluids, fluids, solids , and even the interior of  stars. Overview The most prominent use of an equation of state is to correlate densities of gases and liquids to temperatures and pressures. One of the simplest equations of  state for this purpose is the ideal gas law, law, which is roughly accurate for weakly polar gases at low pressures and moderate temperatures. However, this equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid. Therefore, a number of more accurate equations of state have been developed for gases and liquids. At present, there is no single equation of state that  accurately predicts the properties of all substances under all conditions. In addition, there are also equations of state describing solids solids,, including the transition of solids from one crystalline state to another. There are equations that model the interior of  of stars stars,, including neutron stars, stars, dense matter (quarkgluon plasmas) plasmas) and radiation fields. A related concept is the perfect fluid equation of state used in cosmology cosmology.. Historical Boyle's law (1662) Boyle's Law was perhaps the first expression of an equation of state. In 1662, the noted Irish physicist and chemist Robert chemist Robert Boyle performed a series of  experiments employing employing a J-shaped glass tube, which was sealed on one end. Mercury was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was carefully measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as: The above relationship has also been attributed to Edme Mariotte and is sometimes referred to as Mariotte's Mariotte's law. However, Mariotte's work was not  published until 1676. Charles's law or Law of Charles and Gay-Lussac (1787) In 1787 the French physicist Jacques physicist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 kelvin interval. Later, in 1802, 1802, Joseph Louis Gay-Lussac published results of  similar experiments, indicating a linear relationship between volume and temperature:

Compressibility Compressibility of ammonia gas

Ammonia is small but highly polar molecule with significant interactions. Values can be obtained from Perry 4th ed (awaits future library visit) According to van der Waals, the theorem of corresponding corresponding states (or all fluids fluids,, when compared at  principle of corresponding states ) indicates that all the same reduced temperature and reduced pressure, pressure , have approximately the same compressib compressibility ility factor and all deviate from ideal gas behavior to about the same degree. degree.[1][2]

Dalton's law of partial pressures (1801) Dalton's Law of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone. Mathematically, this can be represented for n species as:

where

is molar volume, and

and

are substance-specific substance-specific constants.

They can be calculated from the critical properties The ideal gas law (1834) In 1834 Émile Clapeyron combined Boyle's Law and Charles' law into the first  statement of the ideal gas law . Initially the law was formulated as  pV m = R(T C  C  + 267) (with temperature expressed in degrees Celsius), where R is the gas constant . However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with 0 °C = 273.15 K, giving: Van der Waals equation of state (1873) In 1873, J. D. van der Waals introduced the first  first equation equation of state derived by the assumption of a finite volume occupied by the constituent molecules. molecules.[2] His new formula revolutionized the study of equations of state, and was most famously continued via the Redlich–Kwong equation of state and the Soave modification of Redlich–Kwong Kwong.. Equation of state of an Ideal gas : Boyle’s law, Charles’ law and Avogadro’s law could be combined to give a general relation between the volume, pressure, temperature and the number of  moles of a particular gas. The equation, PV = constant , describes the variation of P with V at constant T and the equation V/T , represents the variation of V with T at constant P. On combining these equations, we get: PV/T = constant and this relating the variables P. V and T of an ideal gas is known as the equation of state. The product ‘ PV’ over T is always constant for all specified states of the gas. Hence, if we know these values for any one state the constant can be calculated. In standard state or STP, with the pressure at 1 atm and temperature at being 273.16 K , t he volume occupied by a mole of an ideal gas would be equal to 22.414 L. According to Avogadro’s law this volume is same for all ideal gases and if we consider ‘ n’ moles of an ideal gas at STP, then the equation becomes PV/T = PoVo/To = nR PV = nRT, Where, R is a universal gas constant per mole. The above equation is

known as the ideal gas equation and it connects directly all the components and permits all kinds of c alculations. Major equations of state For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form: In the following equations the variables are defined as follows. Any consistent  set of units may be used, although SI units are preferred. preferred. Absolute temperature refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero. = pressure (absolute) = volume = number of moles of a substance

= = molar volume, the volume of 1 mole of gas or liquid = absolute temperature = ideal gas constant (8.314472 constant (8.314472 J/(mol·K)) = pressure at the critical point 

that

and

(noting

is the molar volume at the critical point) as:

Also written as

Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this landmark equation equation is called the the attraction attraction parameter parameter and the repulsion repulsion parameter or the effective molecular volume. While the equation is definitely superior superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate. The van der Waals equation may be considered as the ideal gas law, “improved” due to two independent reasons: Molecules are thought as particles with volume, not material points. Thus cann cannoot be be to too lilittle ttle,, les lesss tha than n so some con consstan tant. So we we get get ( ) ins inste tead ad of  . While ideal gas molecules do not interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the ideal ideal gas law), law), so we write write ( someth something ing)) instea instead d of . To evalua evaluate te this this ‘something’, let's exa mine an additional force acting on an element of gas surface. surface. While While the force force acting acting on each surface surface molecul moleculee is ~ , the force force acting

on the whole element is ~ ~ . With the reduced state variables, i.e. Vr=Vm/Vc, Pr=P/Pc and Tr=T/Tc, the reduced form of the Van der Waals equation can be formulated:

The benefit of this form is that for given T r and Pr, the reduced volume of the liquid and gas can be calculated directly using Cardano's method for the reduced cubic form:

For Pr Videal * The repulsion forces become more significant than the attractive forces. * The gas cannot be compressed easily. * Usually the Z > 1 for so called permanent gases like He, H 2. Case-II: If Z < 1

Where is defined in the Peng-Robinson equation of state, T r r  is the reduced temperature,, P r r  is the reduced pressure, temperature pressure, Z is the compressib compressibility ility factor factor,, and

* Vreal < Videal * The attractive forces are more significant than the repulsive forces. * The gas can be liquefied easily. * Usually the Z < 1 for gases like NH3, CO2, SO2. The isotherms for one mole of different gases, plotted against the Z value and pressure, P at 0oC are shown below:

Typically, one knows two of the three state properties ( P , V m, T ), ), and must  compute the third directly from the equation of state under consideration. To calculate the third state property, it is necessary to know three constants for the species at hand: the critical temperature T c, critical pressure P c, and the acentric factor ω. But once these constants are known, it is possible to evaluate all of the above expressions and hence determine the enthalpy and entropy departures. In thermodynamics a residual property is defined as the difference between a real gas property and an ideal gas property, both considered at the same pressure,, temperature pressure temperature,, and composition composition.. BEHAVIOUR OF REAL GASES: DEVIATIONS FROM IDEAL GAS BEHAVIOUR AND VAN DER WAALS EQUATION FOR REAL GASES Real gases do not obey ideal gas equation under all c onditions. They nearly obey ideal gas equation at higher temperatures and very low pressures. However they show deviations from ideality at low temperatures and high pressures. The deviations from ideal gas behaviour can be illustrated as follows: The isotherms obtained by plotting pressure, P against volume, V for real g ases do not coincide with that of ideal gas, as shown below. * For gases like He, H2 the Z value increases with increase in pressure (positive deviation). It is because, the repulsive forces become more significant and the attractive forces become less dominant. Hence these gases are difficult to be condensed. * For gases like CH 4, CO2, NH3 etc., the Z value decreases initially (negative deviation) but increases at higher pressures. It is because: at low pressures, the attraction forces are more dominant over the repulsion forces, whereas at higher pressures the repulsion forces become significant as the molecules approach closer to each other. * But for all the gases, the Z value approaches one at very low pressures, indicating the ideal behaviour. Also consider the following graphs of Z vs P for a particular gas, N 2 at different  temperatures.

It is clear from above graphs that the volume of real gas is more than or less than expected in certain cases. The deviation from ideal gas behaviour can also be expressed by compressibility compressibility factor, Z. Compressibility Compressibilit y factor (Z): The ratio of PV to nRT is known as compressibility factor.

(or) The ratio of volume of real gas, V real  real  to the ideal volume of that gas, V  perfect   perfect  calculated by ideal gas equation is known as compressibility factor.

But from ideal gas equation: PVperfect  = nRT or

In above graphs, the curves are approaching the horizontal line with increase in the temperature i.e., the gases approach ideal behaviour at higher temperatures. VAN DER WAALS EQUATION : DERIVATION

Therefore

The deviations from ideal gas behaviour can be ascertained to the following faulty assumptions by kinetic theory of gases. * The real volume of the gas molecules is negligible when compared to the volume of the container. * There are no forces of attraction or repulsion between the gas molecules. Hence van der Waals suggested the following corrections: * The gas molecules possess finite volume and hence should not be neglected. It  is especially true at high pressures and low temperatures and should be accounted for. * In case of real gases, both the forces of attraction as well as repulsion operate between gas molecules. Note: If the gases obey the kinetic theory of gases, then they cannot be compressed since the attractions between the gas molecules is negligible. Also the following corrections are applied by van der Waals to the ideal gas equation: Volume correction:

The volume available for the gas molecules is less than the volume of the container, V. The available volume is obtained by subtracting excluded volume of ‘n’ moles of  gas, nb from the volume of the container. Available volume = V - nb Where 'b' is a constant characteristic of a gas. The ideal gas equation can be written after correcting for this as: P(V-nb) = nRT Pressure correction:

CHARACTERISTICS OF VAN DER WAALS CONSTANTS: a & b * ‘a’ is called van der Waals constant of attraction. Higher values of ‘a’ indicate greater attraction between gas molecules. The easily compressible gases like ammonia, HCl possess higher ‘a’ values. Greater the value of ‘a’ for a gas easier is the liquefaction. * ‘b’ is called excluded volume of the gas.

Where NA = Avogadro number and r = radius of molecule * For a given gas, the numerical value of ‘a’ is greater than that of ‘b’. * The Boyle's temperature (TB) and inversion temperature (T i) are related to the two van der Waals constants as:

Note: Boyle’s temperature is the temperature at which the gases obey the ideal gas

laws over a wide range of pressures. Inversion temperature is the temperature at which the Joule Thompson coefficient changes its sign i.e., from positive to negative or vice versa. What is excluded volume?

Van der Waals considered the gas molecules are hard spheres which are not  penetrated by other molecules. Consider two hard spheres of radius ‘r’ in closest contact are enveloped by a sphere of radius ‘d’ that is forbidden for the centers of the other particles.

The pressure of the real gas is less than the expected pressure due to attractions between the molecules. These attractions slow down the motion of gas molecules and result in: i) reduction of frequency of collissions over the walls and ii) reduction in the force with which the molecules strike the walls. Hence Pideal = Preal + p Where p = reduction in pressure However the reduction in pressure is proportional proportional to the t he square of molar concentration, n/V.

One factor for reduction in frequency of collisions and the second factor for reduction in strength of their impulses on the walls. or

Hence the volume excluded for the third sphere is given by:

However the volume excluded by one particle is:

where ‘a’ is a proportionality constant characteristic of a g as. Therefore:

Now the ideal gas equation can be modified once again by introducing this pressure correction as:

This is known as van der Waals equation of state. For one mole of a gas, the equation can be written as:

Therefore the volume excluded by N A particles, b is given by:

Note: This factor four is considered erroneous by Maxwell and proposed that  the excluded volume is 16 times of the volume of NA molecules. EXPLANATION OF DEVIATIONS UNDER DIFFERENT CONDITIONS USING VAN DER WAALS EQUATION  At low pressures: pressures: Since V is large and theref ore ore ‘b’ is negligible compared to V, the van der Waals equation for one mole of a gas is reduced to:

Where Vm = volume occupied by one mole of a real gas Remember that:

P = observed pressure of the real gas

whereas, V = volume occupied by the real gas and is equal to the volume of the container. whereas, (V - nb) = available volume for gas molecules Units of ‘a’ and ‘b’:

----------- atm L 2 mol-2 For ‘a’ ----------For ‘b’ ----------- L. mol -1

* Note that V = Vm = molar volume. Therefore: PV > RT at lower pressures. This is the reason for decrease in the Z value at low pressures. Note: By dividing with RT on each side, the above equation can be written as:

 At very low pressures: Since V is very large and therefore both ‘b’ and ‘a/V 2’

values become negligible. Hence the van der Waals equation for one mole of gas is reduced to: PV = RT Therefore, at very low pressures all the gases obey the ideal gas equation.

value of ‘b’ cannot  be neglected. Although a/V2 is also large its value may be neglected in comparison with very high value of P. Hence the van der Waals equation is reduced to: P(V - b) = RT PV - Pb = RT PV = RT + Pb Therefore: at high pressures, PV > RT. This explains the raising parts of the isotherms, at high pressures, plotted between Z vs P. Note: By dividing with RT on each side, the above equation can be written as:

* As the temperature is lowered, the isotherms show deviation from ideal behaviour. * At 30.98 oC, carbon dioxide remains as gas up to 73 atm. But liquid appears for the first time at 73 atm (represented by point ‘O’). Hence 30.98 oC is called critical temperature for CO 2. And above 73 atm. there is a steep rise in the pressure. This steep portion of the curve represents the isotherm of liquid state for which small decrease in volume results in steep rise in the pressure. * At even lower temperature, 20 oC, the liquid appears at point ‘A’. Further compression does not change the pressure up to point ‘B’. After point, B the curve again becomes steep representing the isotherm for liquid CO 2. CRITICAL CONSTANTS Critical Temperature (T c): It is the temperature above which a gas cannot be liquefied by applying pressure.

 At high pressures: pressures: Since the volume of the gas is small, the

pressure required to cause Critical Pressure (P c): It is the minimum pressure liquefaction at critical temperature, T c.

Critical Volume (Vc): It is the volume occupied by one mole of a gas at T c and

Pc. V c = 3b Super critical fluid: The dense fluid obtained by compressing a gas above its

 At high temperatures: temperatures: In this case, V is very large and attractions are

negligible. Hence both ‘b’ and ‘a/V 2’ are negligible. This reduces the van der Waals equation to: PV = RT for one mole For H2 and He gases: Since the actual volume of these gas molecules is very

small, the intermolecular forces of attractions are very small. i.e., a/V 2 can be ignored. Thus the van der Waals equation is reduced to: P(V - b) = RT PV - Pb = RT PV = RT + Pb Therefore, for H2 and He gases, PV > RT. Hence for these gases, the Z value is always greater than one as evident from the isotherms plotted between Z vs P. MERITS & APPLICATIONS OF VAN DER WAAL'S EQUATION * The Vander Waal's equation holds good for real gases up to moderately high pressures. * It explains the isotherms of PV/RT vs P for various gases. * From this equation it is possible to obtain expressions for Boyle's temperature, critical constants and inversion temperature in terms of the Vander Waal's constants 'a' and 'b'. Liquefaction of gases: The isotherms plotted between P vs V at different 

temperatures for one mole of CO 2 gas are shown below.

Following conclusions can be drawn from these graphs: * At higher temperatures, say 50 oC, the isotherms show ideal behaviour.

critical temperature is called super critical fluid. * It is not a liquid though its density is similar to that of liquid. * It is not a gas due to high density and no distinct surface that separates it from the vapour phase. * It can be used as a solvent. E.g. The super critical fluid of CO 2 is used in the extraction of caffeine from coffee beans. Van der Waals Equation

The behavior of real gases usually agrees with the predictions of the ideal gas equation to within within 5% at at normal normal temperatures and pressures. At low low temperatures or high pressures, real gases deviate significantly from ideal gas behavior. In 1873, while searching for a way to link the behavior of liquids and gases, the Dutch physicist Johannes van der Waals developed an explanation for these deviations and an equation that was able to fit the behavior of real ga ses over a much wider range of pressures. Van der Waals realized that two of the assumptions of the kinetic molecular theory were questionable. The kinetic theory assumes that gas particles occupy a negligible fraction of the total volume of the g as. It also assumes that the force of attraction between gas molecules is zero. The first assumption works at pressures close to 1 atm. But something happens to the validity of this assumption as the gas is compressed. Imagine for the moment that the atoms or molecules in a gas were all clustered in one corner of  a cylinder, as shown in the figure below. At normal pressures, the volume occupied by these particles is a negligibly small fraction of the total volume of  the gas. But at high pressures, this is no longer true. As a result, real gases are not as compressible at high pressures as an ideal gas. The volume of a real gas is therefore larger than expected from the ideal gas equation at high pressures. Van der Waals proposed that we correct for the fact that the volume of a real gas is too large at high pressures by subtracting a term from the volume of the real gas before we substitute it into the ideal gas equation. He therefore introduced a constant  constant (b) into the ideal gas equation that was equal to the volume actually occupied by a mole of gas particles. Because the volume of the gas particles depends on the number of moles of gas in t he container, the term that is subtracted from the real volume of the gas is equal to the number of moles of  gas times b. P (V - nb) = nRT  When the pressure is relatively small, and the volume is reasonably large, the nb term is too small to make any difference in the calculation. But at high pressures, when the volume of the gas is small, the nb term corrects for the fact  that the volume of a real g as is larger than expected from the ideal gas equation. The assumption that there is no force of attraction between gas particles cannot  be true. If it was, gases would never condense to form liquids. In reality, there is a small force of attraction between gas molecules that tends to hold the molecules together. This force of attraction has two consequences: (1) gases condense to form liquids at low temperatures and (2) the pressure of a real gas is sometimes smaller than expected for an ideal gas. To correct for the fact that the pressure of a real gas is smaller than expected from the ideal gas equation, van der Waals added a term to the pressure in this equation. This term contained a second constant (a) and has the form: an2/V 2. The complete van der Waals equation is therefore written as follows.

P = 52.6 atm As the pressure of CO 2 increases the van der Waals equation initially gives pressures that are smaller than the ideal gas equation, as shown in the figure below, because of the strong force of attraction between CO 2 molecules.

This equation is something of a mixed blessing. It provides a much better fit  A plot of the product of the pressure times the volume for samples of H 2, N2, CO2 gases versus the p with the behavior of a real gas than the ideal gas equation. But it does this at the cost of a loss in generality. The ideal g as equation is equally valid for any gas, these gases. Let's now compress the gas even further, raising the pressure until the volume whereas the van der Waals equation contains a pair of constants ( a and b) that  of the gas is only 0.0500 liters. The ideal gas equation predicts that the pressure change from gas to gas. would have to increase to 448 atm to co ndense 1.00 mole of CO2 at 0oC to a The ideal gas equation predicts that a plot of  PV versus P for a gas would be a volume of 0.0500 L. horizontal line because PV should be a constant. Experimental data for PV  versus P for H2 and N2 gas at 0C and CO2 at 40C are given in the figure below. Values of the van der Waals constants for these and other gases are given in the table below. van der Waals Constants for Various Gases Compound  a (L 2-atm/mol  2 ) b (L/mol)

He 0.03412 0.02370 Ne 0.2107 0.01709 H2 0.2444 0.02661 Ar 1.345 0.03219 O2 1.360 0.03803 N2 1.390 0.03913 CO 1.485 0.03985 CH4 2.253 0.04278 CO2 3.592 0.04267 NH3 4.170 0.03707 The magnitude of the deviations from ideal gas behavior can be illustrated by comparing the results of calculations using the ideal gas equation and the van der Waals equation for 1.00 mole of CO 2 at 0oC in containers of different  volumes. Let's start with a 22.4 L container. According to the ideal gas equation, the pressure of this gas should be 1.00 atm.

Substituting what we know about CO 2 into the van der Waals equation gives a much more complex equation.

The van der Waals equation predicts that the pressure will have to reach 1620 atm to achieve the same results.

P = 1620 atm

The van der Waals equation gives results that are larger than the ideal gas equation at very high pressures, as shown in the figure above, because of the volume occupied by the CO2 molecules.  Analysis of the van van der Waals Constants Constants

The van der Waals equation contains two constants, a and b, that are characteristic properties of a particular gas. The first of these constants corrects for the force of attraction between gas particles. Compounds for which the force of attraction between particles is strong have large values for a. If you think about what happens when a liquid boils, you might expect that compounds with large values of a would have higher boiling points. (As the force of attraction between gas particles becomes stronger, we have to go to higher temperatures before we can break the bonds between the molecules in the liquid to form a gas.) It isn't surprising to find a correlation between the value of the a constant  in the van der Waals equation and the boiling points of a number of simple compounds, as shown in the fugure below. Gases with very small values of  a, such as H2 and He, must be cooled to almost absolute zero before they condense to form a liquid. The other van der Waals constant, b, is a rough measure of the size of a gas particle. According to the table of van der Waals constants, constants, the volume of a mole of argon atoms is 0.03219 liters. This number can be used to estimate the volume of an individual argon atom.

The volume of an argon atom can then be converted into cubic centimeters using the appropriate unit factors. This equation can be solved, however, for the pressure of the gas. P = 0.995 atm

At normal temperatures and pressures, the ideal gas and van der Waals equations give essentially the same results. Let's now repeat this calculation, assuming that the gas is compressed so that it  fills a container that has a volume of only 0.200 liters. According to the ideal gas equation, the pressure would have to be increased to 112 atm to compress 1.00 mol of CO 2 at 0C to a volume of 0.200 L.

If we assume that argon atoms are spherical, we can estimate the radius of these atoms. We start by noting that the volume of a sphere is related to its radius by the following formula. V = 4/3 r 3 We then assume that the volume of an argon atom is 5.345 x 10 -23 cm and calculate the radius of the atom. r = 2.3 x 10-8 cm According to this calculation, an argon atom has a radius of about 2 x 10-8 cm. Deviations from Ideal Behavior Real Gases fail to obey the Ideal Gas equation of state exactly. Why? 

For exactly one mole of an ideal gas: The van der Waals equation, however, predicts that the pressure will only have to increase to 52.6 atm to achieve the same results. Plotting the experimentally determined value of ( pV/RT) for exactly one mole of various REAL GASes as a function of pressure, p, shows a deviation from ideality (The quantity (pV)/(nRT) = Z is called the COMPRESSIBILITY FACTOR and should be unity for an Ideal Gas):

The deviation from ideal behavior is large at high pressure and low temperatue At lower pressures and high temperatures, the deviation from ideal behavior is typically small, and the ideal gas law can be used to predict behavior with little error. Deviation from ideal behavior can also be shown for a given gas (Nitrogen, in this example) as a function of temperature:

To put the observed (real) pressure into the Ideal Gas expression, we must  correct for the decrease in pressure due to molecular stickiness. For Stickiness to be a factor, the two gas molecules must have a collision. The probability of a collision is the probability of two molecules being in the same place at the same time. The probability of the first molecule being at the place of the collision is proportional to the number density (n/V). The probability of the second one being in the same place is the same, (n/V). Thus the reduction in pressure due to stickiness should be proportional to (n/V)2. If the proportionality constant is called a, then the ideal pressure is

Thus, correcting for molecular stickiness alone, the I deal Gas equation would become:

As pressures and density increase, the volume of the molecules themselves becomes significant relative to the size of the container. As temperature is decreased below a critical value, the deviation from ideal gas behavior becomes severe, because the gas CONDENSES to become a LIQUID. The van der Waals Equation (A fix to the Ideal Gas Equation)

One of the most useful equations to predict the behavior of real gases was developed by Johannes van der Waals (1837-1923) It takes into account  Molecular Stickiness and Molecular Size The Ideal Gas Equation:

Has been presented as an Empirical relation, but it doesn't work perfectly for all gases under all conditions because it is based on imperfect assumptions. Remember the conditions assumed for an Ideal Gas?: Molecules are perfectly elastic (no STICKINESS) Molecules are point masses (no SIZE) Molecules move at random

To correct for the effect of finite molecular volume, we must recognize that in the ideal gas equation the volume used is the "free volume" that the molecules find themselves in. The free volume is just the real (container) volume minus the volume that is taken up by the molecules of the gas itself.

where b is a constant representing the volume of a mole of gas molecules at  rest. Thus, the ideal gas equation, if corrected for molecular size and stickiness, looks like:

The first two of these assumptions is clearly wrong for all gases, because at low temperatures all gases CONDENSE, or form a liquid phase. This must happen because the molecules stick to one another, at least a little. We can correct the Ideal Gas equation for stickiness. Moreover, the liquid has measurable molar volume, and this volume is simply the size of the close-packed molecules of the liquid. At high pressures, and thus high densities, the intermolecular distances can become quite short, and attractive forces between molecules becomes significant. Neighboring molecules exert a relatively long-ranged attractive force on one another, which will reduce the momentum in which they tranfer to the container walls. The observed pressure exerted by the gas under these conditions will be less than that for an Ideal Gas

We can rearrange this expression slightly to give the familiar form of the van der Waals equation.

Unlike the universal gas constant, R, The van der Waals constants a and b are different for different gases.

Substance

a (L2 atm/mol2)

b (L/mol)

He

0.0341

0.0237

H2

0.244

0.0266

O2

1.36

0.0318

H2O

5.46

0.0305

CCl4 20.4 0.1383 Numerical Example: Use the van der Waals equation to calculate the pressure of a sample of 1.0000 mol of oxygen gas in a 22.415 L vessel at 0.0000°C (Note: For an ideal gas, this temperature and volume would lead to conditions of STP, i.e. a pressure of  exactly 1.0000 atm) V = 22.4 L T = (0.000 + 273.15) = 273.15K a (O2) = 1.36 L2 atm/mol2 (From Table) b (O2) = 0.0318 L /mol (From Table) (nRT)/(V-nb) = 1.0014 atm - a (n/V)2 = -0.0027 atm p = (nRT)/(V-nb) - a (n/V) 2 = 0.9987 atm If O2 was perfectly ideal, the pressure would be 1.0000 atm. The first term on the right-hand-side of the van der Waals equation, (nRT)/(Vnb), represents the ideal gas pressure corrected for finite molecular volume. In other words the volume is slightly less than 22.415 L due to the mole of O2 in it. The the molecules collide a bit more frequently with the walls of the container, because they have less room to fly around in in the middle of the container. The value of 1.0014 atm for this term represents this increase in pressure. The second term in the equation, - a (n/V)2, represents the reduction in pressure due to molecular stickiness. This correction to ideal gas behavior dominates the finite volume correction, leading to a compressibility compressibility factor, (0.9987, in fact). CalculatorZ=pV/nRT of less than 1 (0.9987, http://www.chem.ufl.edu/~itl/2045/lectures/glaw_calc.html

Enthalpy, denoted by H, is a convenient energy concept defined from properties of a system: (1)

where U is internal energy, p is pressure and V is volume, and is thus also a property of the system. The term U + d(pV) appears often in equations resulting from the First Law of Thermodynamics. Internal energy, and hence enthalpy, can only be quantified relative to an arbitrary reference value. In most applications, however, it is the changes in enthalpy that are important, so the values of the reference state chancel out. The convenience of enthalpy as a thermodynamic concept is illustrated by the definitions of the molar heat capacities of a closed system of constant volume, pressure,

, and constant

, in a reversible process:

(2)

(3)

Enthalpy and other thermodynamic properties of gases and vapors are usually expressed in terms of their deviation from ideal behavior.  Residual enthalpy, , is defined as the difference between the actual enthalpy and the ideal enthalpy. For an isothermal process, (4)

Accurate experimental data should be used in evaluating this expression, but approximate values (±2%) can be obtained by using the Compressibility Factor , which is plotted in compressibility charts as a function of reduced temperature and pressure. Residual enthalpy can be expressed in terms of the compressibility factor for a change at constant temperature and composition: (5)

It follows from the definition of enthalpy and the First Law of Thermodynamics that the change in enthalpy in a closed system at constant pressure in a reversible process is equal to the heat input. The change in enthalpy in a chemical reaction at constant pressure is then the heat of reaction , which in most reactions result mainly from changes in bond strength over the course of a reaction. As enthalpy is a state function, the change in enthalpy is independent of the path of change, so that

, for any reaction can be calculated from a

suitable combination of reference reactions.

values for standard

The heat of formation of a chemical compound is a special case of the heat of reaction, where the compound is the only product from reactants comprising its component elements. The heat of combustion is the enthalpy change when a substance is oxidized with molecular oxygen. Enthalpy change associated with a

where h, u and v are the molar enthalpy, molar internal energy and molar volume, respectively.

phase change in a pure substance, known as latent heat, is calculated from the Clapeyron Equation: Equation :

For solids and liquids, enthalpy change resulting from a change in pressure and temperature can be calculated from

(6)

where p* is the saturation pressure and is the change in molar volume between the two phases. In the special case of vaporization (or sublimation), where the vapor behaves as an ideal gas and the volume of liquid (or solid) is small compared to that of the vapor, (7)

Enthalpy changes associated with changes in concentration, such as the heat of dilution, mixing or crystallization, can be evaluated from enthalpy concentration charts.