Specific Heat Ratio Nigel Cabaluna, Ronald Dela Cruz and Tom Mari Jandel Rentoy* National Institute of Physics, University of the Philippines, Diliman, Quezon City 1101 *Corresponding author:
[email protected]
Abstract One common method of measuring the ratio of heat capacity of gases, γ = Cp/Cv, is Ruchardt’s method. This method uses the oscillation of a mass supported by the pressure gas. For the ideal diatomic gas has Cp = 7R/2, Cv = 5R/2 and γ=1.4. Percentage errors might have occurred because of infinitesimal errors. This is done with a steel bearing oscillating in a precisely fitted glass tube attached to a glass reservoir. Exact measurements of the frequency are difficult to obtain and thus the use of Vernier Lab Pro® is very convenient to obtain high accuracy data. The experimental specific heat ratio is calculated using the slope of the reciprocal of the square of period and the reciprocal of the heights of the piston. © 2013 Samahang Pisika ng Pilipinas Keywords: Heat capacity in chemical thermodynamics, 82.60.Fa; Specific heat, Specific heat ratio, Ruchardt’s Method, ideal diatomic gas. I.
Introduction The specific heat ratio γ is defined by the ratio γ=
(1)
where Cp and Cv are the specific heat of a gas at constant pressure and constant volume, respectively. For an ideal diatomic gas, the specific heats have the values: C p = 7R/2 and Cv = 5R/2, such that the ratio is γ=1.4. In this experiment we shall determine the value experimentally. The setup for the experiment employed in the method devised by Eduard Ruchardt, a German physicist, for determining the specific heat ratio is shown in. In Ruchardt’s method, the piston is given a small displacement in the vertical direction. The piston then oscillates about the original position, with a period that is dependent theoretically on the specific heat ratio. Theoretically, we can describe the height of the piston from the ground y(t) by using Newton’s second law: 1
⁄ 2) = PA-PatmA-mg,
m(
(2)
where m is the mass of the piston, P is the pressure of the gass, A is the cross-sectional area of the piston, g = 9.81m/s2 is the acceleration due to gravity and Patm = 10.1325kPa is the atmospheric pressure. When the gas is given quick and small changes in the volume, approximately no heat is transferred into or out of the gas. This is called an adiabatic process and characterized by P(Vγ)= constant = P0(V0γ)
(3)
where P0A=PatmA + mg is the initial pressure on the gas and V0 is the volume of the gas at the middle of oscillation. The equation of motion this simplifies to: m d2y(t)/dt2 = P0(V0/V)γA - PatmA – mg,
(4)
where V is the volume of the gas at pressure P. In terms of the small vertical displacement δy, V=A(y0 + δy).
(5)
The coefficient of the first term can be expanded to P0V0γ / Vγ = P0(Ay0) γ/ [A(y0 + δy)] γ ≈ P0 – γP0
(6)
using binomial expansion. The motion of the piston can then be expressed as d2(δy(t))/dt2 = -ω2δy,
(7)
were the angular frequency is defined as ω=√
.
(8)
Equation (7) is the differential equation for a (translated) simple harmonic motion with solution Y(t) = y0 –a cos(ωt)
(9)
where a is the amount of initial displacement (amplitude). In the reality, the piston is damped by friction due to its contact with the cylinder glass. Accounting for friction, Newton’s second law then becomes md2y/dt2 = PA – PatmA – mg - b
= -ω2y - b ,
(10)
where b is the damping parameter. For relatively small damping, the solution is y(t) = y0 + a exp(
) cos (ω’t),
(11)
2
where ω’= √
.
(12)
The above relation can be expressed as : 1/T2 =( γPatmA +mg / 4π2my0) – b2/16π2m2
(13)
Where T= 2π/ω’ is the period of the damped oscillation, or the time interval between two successive peaks. II. Methodology To begin the experiment, the materials and equipment needed were a Pasco heat engine apparatus and a Vernier LabPro with gas pressure sensor. The instruments were set up with care before experimental data were gathered. The diameter of the piston used was 32.5mm or 0.0325m and the mass of the platform of the piston was 35g or 0.035kg. When the instruments have been set up, the piston was put at a height of 0.075m and tightened. Then, the LabPro started collecting data as the piston was lightly tapped. By zooming in the graph of the pressure versus time graph in the LabPro, the period was recorded. To ensure that one whole period is measured, the peaks of the graphs were taken account. This process is done for heights of 0.080m, 0.085m, .0.090m and 0.095m of the piston. Data were then input on Microsoft Excel. The data was plotted by the computer program and the linearly fit. The equation of the best fit line was then recorded. The specific heat ratio was then calculated. III. Results and Discussion The measured periods can be found in Table W2 as the height of the piston varies. Height of Piston (m) Period (s) 0.075 0.032 0.080 0.034 0.085 0.036 0.090 0.038 0.095 0.04 Table W2. Measured Data The calculated reciprocal of the height of the piston y-1 and the calculated reciprocal of the square of the period T-2 can be found in Table W2.1.
3
y-1 T-2 13 976 12 865 11 771 11 692 10 625 Table W2.1. T-2 and y-1 Data The graph of the reciprocal of the height of the piston y-1 and the reciprocal of the square of the period T-2 can be found in Figure W2.
T-2 vs y-1 1200 1000 y = 119.31x - 574.31 800 T-2 600 400 200 0 0
5
10
15
y-1
Figure W2. T-2 vs y-1 Graph From the graph of the reciprocal of the height of the piston y-1 and the reciprocal of the square of the period T-2, the best fit line is found to be y = 119.31x - 574.31 From equation (11) the specific heat ratio is computed from the slope of the best fit line. γ = [ (slope)(4π2m) –mg ]/ PatmA
(12)
The experimental specific heat ratio and percent deviation from the theoretical heat ratio of 7/5 or 1.4 is given in Table W3. Experimental Specific Percent Deviation Heat Ratio 1.96 40% Table W3. Specific Heat Ratio 4
IV.
Conclusion
The specific heat ratio can be computed using the area of a piston, the mass of a piston, atmospheric pressure, varying heights of a piston and varying time periods. By using the slope of the best fit 1st degree line of the reciprocal of the square of the periods versus the reciprocal of the heights of the piston, the experimental specific heat ratio can be computed. As the height of the piston increases the period also increases thus having a directly proportional relationship. Aknowledgements This experiment was done with the help of lab instructor, Ms. Jen-Jen Manuel. Reference: National Institute of Physics University of the Philippines Diliman Lab Manual
5
