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Assignment 8: Temperature, Heat and Thermal Properties Due: 2:00am on Saturday, November 6, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy. [Switch to Standard Assignment View]

A Sliding Crate of Fruit A crate of fruit with a mass of 34.0

and a specific heat capacity of 3650

down a ramp inclined at an angle of 36.6

slides 9.00

below the horizontal.

Part A If the crate was at rest at the top of the incline and has a speed of 2.05 work

at the bottom, how much

was done on the crate by friction?

Hint A.1

How to approach the problem Hint not displayed

Hint A.2

Find the initial and final kinetic energies Hint not displayed

Hint A.3

Find the difference between initial and final potential energy Hint not displayed

Use 9.81 ANSWER:

for the acceleration due to gravity and express your answer in joules. = -1720 Correct

The frictional force opposes the motion of the crate, so the work done on the crate by friction must be a negative quantity. Part B If an amount of heat equal to the magnitude of the work done by friction is absorbed by the crate of fruit and the fruit reaches a uniform final temperature, what is its temperature change ? Hint B.1

Equation for temperature change Hint not displayed

ANSWER:

= 1.38×10−2 Correct

Of course, the assumptions of "total heat absorption" and "uniform temperature change" are not very realistic; still, this simplified model provides a useful reminder about the transformation of mechanical energy into thermal energy when nonconservative forces are present.

mechanical energy into thermal energy when nonconservative forces are present.

Steam vs. Hot-Water Burns Just about everyone at one time or another has been burned by hot water or steam. This problem compares the heat input to your skin from steam as opposed to hot water at the same temperature. Assume that water and steam, initially at 100 , are cooled down to skin temperature, 34 , when they come in contact with your skin. Assume that the steam condenses extremely fast. We will further for both liquid water and steam. assume a constant specific heat capacity Part A Under these conditions, which of the following statements is true? ANSWER:

Steam burns the skin worse than hot water because the thermal conductivity of steam is much higher than that of liquid water. Steam burns the skin worse than hot water because the latent heat of vaporization is released as well. Hot water burns the skin worse than steam because the thermal conductivity of hot water is much higher than that of steam. Hot water and steam both burn skin about equally badly. Correct

The key point is that the latent heat of vaporization has to be taken into account for the steam. Part B How much heat

is transferred to the skin by 25.0

vaporization for steam is Hint B.1

of steam onto the skin? The latent heat of

.

Determine the heat transferred from steam to skin Hint not displayed

Express the heat transferred, in kilojoules, to three significant figures. ANSWER:

= 63.3 Correct

Here we assumed that the skin continues to remain at 34

. Actually the local temperature in

the area where the steam condenses can be raised quite significantly. Part C How much heat

is transferred by 25.0

of water onto the skin? To compare this to the result in

the previous part, continue to assume that the skin temperature does not change. Hint C.1

Determine the heat transferred from water to skin Hint not displayed

Express the heat transferred, in kilojoules, to three significant figures. ANSWER:

= 6.91 Correct

The amount of heat transferred to your skin is almost 10 times greater when you are burned by steam versus hot water. The temperature of steam can also potentially be much greater than 100 . For these reasons, steam burns are often far more severe than hot-water burns.

Dust Equipartitions Small dust particles suspended in air seem to dance randomly about, a phenomenon called Brownian motion. For this problem you will need to know Boltzmann's constant: . Part A What would you expect the mean translational kinetic energy

of such particles to be

if they are in air at a temperature of 290 K? Hint A.1

Kinetic energy in terms of average velocity Hint not displayed

Hint A.2

Equipartition Theorem Hint not displayed

Hint A.3

Use the Equipartition Theorem Hint not displayed

Express the mean translational kinetic energy numerically, in joules, to two significant figures. Note that has been factored out already to make your answer simpler. ANSWER:

= 6.0 Correct

Part B Find an expression for the rms (root-mean-square) speed spheres of diameter Hint B.1

of these particles, assuming them to be

and density

Find the rms speed in terms of temperature Hint not displayed

Hint B.2

Find the mass of the particles Hint not displayed

Express the rms speed in terms of

,

,

,

, and

.

Express the rms speed in terms of

,

,

,

, and

.

ANSWER: =

Correct

Part C Now calculate the rms (root-mean-square) speed and density

of diameter

of these particles, assuming them to be spheres . The mass of such a dust particle is

.

Express your answer in millimeters per second to one decimal place only. ANSWER:

= 0.3 Correct

This speed is several orders of magnitude smaller than the typical velocities of gas molecules at this temperature (which are of the order of hundreds of meters per second). This is simply because the mass of these particles is much larger than the mass of typical gas molecules. For particles larger than the ones described here, the weight can no longer be ignored. Such particles tend to settle quite quickly on account of their weight. Then such a calculation is no longer valid.

Heating a Room Imagine you've been walking outside on a cold winter's day. When you arrive home at your studio apartment, you realize that you left a window open and your room is only slightly warmer than the outside. You turn on your 1.0-

space heater right away and wait impatiently for the

room to warm up. In this problem, make the following assumptions: The entire output of the space heater goes into warming the air in the room. The air in the room is an ideal gas with five degrees of freedom per particle (three translational degrees of freedom and two rotational degrees of freedom—about right for nitrogen and oxygen). The air in the room is at a constant pressure of 1.00 . At room temperature and atmospheric pressure, 1

of air fills a volume of 23

. This is slightly

larger than the volume of air at standard temperature and pressure, because room temperature is hotter than 0 .

Part A How long will it be before the heater warms the air in the room by 10. Hint A.1

?

How to approach the problem

Since you know the power output of the heater, you know how much energy per unit time is being added to the air in the room. To determine how long it will take to warm up the room, then, you need to determine the total energy needed to raise the temperature of the air in the room by 10 . Once you have this value, simply divide by the power of the heater to determine the time:

Once you have this value, simply divide by the power of the heater to determine the time: .

Hint A.2

Find the energy needed to raise the temperature

How much energy (in joules) is needed to raise the temperature of the room by 10.

?

Hint A.2.1 How to approach the problem

Hint not displayed Hint A.2.2 Find the heat capacity

Hint not displayed Hint A.2.3 Find the amount of air in the room

Hint not displayed Express your answer in joules to two significant figures. ANSWER:

energy = Answer not displayed

Express your answer in minutes to two significant figures. ANSWER:

time =

16 Correct

In practice, it would probably take more than an hour to heat the room by 10.

because the

walls and any items in the room are in thermal contact with the air and would have to be warmed up also.

Melting Point of Platinum The ratio of the pressure of a gas at the melting point of platinum to its pressure at the triple point of water, when the gas is kept at constant volume, is found to be 7.476. Part A What is the Celsius temperature Hint A.1

of the melting point of platinum?

How to approach the problem. Hint not displayed

Hint A.2

Find an expression for the pressure ratio Hint not displayed

Hint A.3

Triple-point temperature of water Hint not displayed

Express your answer in degrees Celsius to four significant figures. ANSWER:

= Answer not displayed

Adding Ice to Water An insulated beaker with negligible mass contains liquid water with a mass of 0.330 temperature of 64.8

and a

.

Part A How much ice at a temperature of -14.0 temperature of the system will be 29.0 Hint A.1

must be dropped into the water so that the final ?

How to approach the problem Hint not displayed

Hint A.2

Calculate the heat lost by the water Hint not displayed

Hint A.3

How to calculate the heat gained by the ice Hint not displayed

Hint A.4

Heat gained by the ice Hint not displayed

Take the specific heat of liquid water to be 4190 , and the heat of fusion for water to be 334 ANSWER:

, the specific heat of ice to be 2100 .

= Answer not displayed

Thermal Energy from Friction on a Rope A capstan is a rotating drum or cylinder over which a rope or cord slides to provide a great amplification of the rope's tension while keeping both ends free . Since the added tension in the rope is due to friction, the capstan generates thermal energy.

Part A If the difference in tension (

) between the two ends of the rope is

and turns once in

a diameter of

, find the rate

and the capstan has

at which thermal energy is

being generated. Hint A.1

Torque applied to the rope Hint not displayed

Hint A.2

Power due to the torque Hint not displayed

Hint A.3

Power from force Hint not displayed

Hint A.4

How both methods work equally well Hint not displayed

Give a numerical answer, in watts, rounded to the nearest 10 W. ANSWER:

= Answer not displayed W

Part B If the capstan is made of iron (with a specific heat capacity , at what rate

) and has a mass of

does its temperature rise? Assume that the temperature in the capstan

is uniform and that all the thermal energy generated flows into it. Note that is a temperature. Hint B.1

Use the chain rule Hint not displayed

Give a numerical answer, in degrees Celsius per second, rounded to two significant figures. ANSWER:

= Answer not displayed

Particle Gas Review A particle gas consists of

monatomic particles each of mass

all contained in a volume

at

A particle gas consists of

monatomic particles each of mass

. Your answers should be written in terms of the Boltzmann constant

temperature

rather than

number

all contained in a volume

at

and Avagadro's

.

Part A Find

, the average speed squared for each particle.

Hint A.1

how to approach the problem Hint not displayed

Hint A.2

Find

for each particle Hint not displayed

Hint A.3

Relating the

,

, and

velocities Hint not displayed

Express the average speed squared in terms of the gas temperature

and any other given

quantities. ANSWER:

= Answer not displayed

Part B Find

, the internal energy of the gas.

Hint B.1

How to approach the problem Hint not displayed

Hint B.2

Kinetic energy of a single gas particle Hint not displayed

Express the internal energy in terms of the gas temperature ANSWER:

and any other given quantities.

= Answer not displayed

Part C

Part not displayed Part D

Part not displayed Now imagine that the mass of each gas particle is increased by a factor of 3. All other information given in the problem introduction remains the same.

given in the problem introduction remains the same. Part E What will be the ratio of the new molar mass ANSWER:

to the old molar mass

?

to the old rms speed

?

= Answer not displayed

Part F What will be the ratio of the new rms speed Hint F.1

Definition of rms speed Hint not displayed

ANSWER:

= Answer not displayed

Part G What will be the ratio of the new molar heat capacity

to the old molar heat capacity

?

Hint G.1 How to approach the problem

Hint not displayed ANSWER:

= Answer not displayed

pV Diagram for a Piston A container holds a sample of ideal gas in thermal equilibrium, as shown in the figure. One end of the container is sealed with a piston whose head is perfectly free to move, unless it is locked in place. The walls of the container readily allow the transfer of energy via heat, unless the piston is wrapped in insulation.

Refer to the pV diagram presented to answer the questions below. In each case, the piston head is initially unlocked and the gas is in equilibrium at the pressure and volume indicated by point 0 on the diagram.

Part A Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process? "Lock the piston head in place, and hold the container above a very hot flame." Hint A.1

Understand the graph Hint not displayed

Hint A.2

Find the change in volume Hint not displayed

ANSWER:

point point point point point point point point

1 2 3 4 5 6 7 8

Answer not displayed

Part B Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process? "Immerse the container into a large water bath at the same temperature, and very slowly push the piston head further into the container." Hint B.1

Find the change in volume Hint not displayed

Hint B.2

Find the change in temperature Hint not displayed

ANSWER:

ANSWER:

point point point point point point point point

1 2 3 4 5 6 7 8

Answer not displayed

Part C Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process? "Lock the piston head in place and plunge the piston into water that is colder than the gas." ANSWER:

point point point point point point point point

1 2 3 4 5 6 7 8

Answer not displayed

Part D Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process? "Wrap the piston in insulation. Pull the piston head further out of the container." Hint D.1

Find the change in volume Hint not displayed

Hint D.2

Find the change in temperature Hint not displayed

ANSWER:

point point point point point point point point

1 2 3 4 5 6 7 8

Answer not displayed

Average Spacing of Gas Molecules Consider an ideal gas at 27.0 degrees Celsius and 1.00 atmosphere pressure. Imagine the molecules to be uniformly spaced, with each molecule at the center of a small cube.

be uniformly spaced, with each molecule at the center of a small cube. Part A What is the length Hint A.1

of an edge of each small cube if adjacent cubes touch but don't overlap?

How to approach the problem Hint not displayed

Hint A.2

Calculate the volume per mole Hint not displayed

Hint A.3

Calculate the volume per molecule Hint not displayed

Hint A.4

The edge length of a cube Hint not displayed

Express your answer numerically in meters. ANSWER:

= Answer not displayed

Velocity and Energy Scaling Hydrogen molecules have a mass of

and oxygen molecules have a mass of

, where

is defined

). Compare a gas of hydrogen molecules to a gas of

as an atomic mass unit ( oxygen molecules. Part A At what gas temperature

would the average translational kinetic energy of a hydrogen molecule be

equal to that of an oxygen molecule in a gas of temperature 300 K? Hint A.1

Find the energy associated with one degree of freedom Hint not displayed

Hint A.2

Total translational kinetic energy Hint not displayed

Express the temperature numerically in kelvins. ANSWER:

= Answer not displayed K

Part B At what gas temperature

would the root-mean-square (rms) speed of a hydrogen molecule be

At what gas temperature

would the root-mean-square (rms) speed of a hydrogen molecule be

equal to that of an oxygen molecule in a gas at 300 K? Hint B.1

Find the rms speed Hint not displayed

State your answer numerically, in kelvins, to the nearest integer. ANSWER:

= Answer not displayed K

The Speed of Nitrogen Molecules The kinetic theory of gases states that the kinetic energy of a gas is directly proportional to the temperature of the gas. A relationship between the microscopic properties of the gas molecules and the macroscopic properties of the gas can be derived using the following assumptions: The gas is composed of pointlike particles separated by comparatively large distances. The gas molecules are in continual random motion with collisions being perfectly elastic. The gas molecules exert no long-range forces on each other. One of the most important microscopic properties of gas molecules is velocity. There are several different ways to describe statistically the average velocity of a molecule in a gas. The most obvious measure is the average velocity . However, since the molecules in a gas are moving in random directions, the average velocity is approximately zero. Another measure of velocity is

, the

average squared velocity. Since the square of velocity is always positive, this measure does not average to zero over the entire gas. A third measure is the root-mean-square (rms) speed, , equal to the square root of

. The rms speed is a good approximation of the the typical speed of the molecules

in a gas. This histogram shows a theoretical distribution of speeds of molecules in a sample of nitrogen ( ) gas. In this problem, you'll use the histogram to compute properties of the gas.

Part A What is the average speed Hint A.1

of the molecules in the gas?

How to use the histogram Hint not displayed

Hint A.2

More on computing the average Hint not displayed

Express your answer numerically to three significant digits. ANSWER:

= Answer not displayed

Part B Because the kinetic energy of a single molecule is related to its velocity squared, the best measure of the kinetic energy of the entire gas is obtained by computing the mean squared velocity, , or its square root

. The quantity

is more common than

because it has the dimensions of

velocity instead of the less-familiar velocity-squared. What is the rms speed of the molecules in the nitrogen gas? Hint B.1

How to approach the problem Hint not displayed

Hint B.2

Find the mean square velocity Hint not displayed

Express your answer numerically to three significant digits. ANSWER:

= Answer not displayed

Part C What is the temperature Hint C.1

of the sample of

gas described in the histogram?

How to approach the problem Hint not displayed

Hint C.2

Find the molar mass of N2 Hint not displayed

Express your answer in degrees Celsius to three significant figues. ANSWER:

= Answer not displayed

Part D

Part not displayed

Pressure Cooker A pressure cooker is a pot whose lid can be tightly sealed to prevent gas from entering or escaping. Part A

Part A If an otherwise empty pressure cooker is filled with air of room temperature and then placed on a hot stove, what would be the magnitude of the net force on the lid when the air inside the cooker had been heated to

? Assume that the temperature of the air outside the pressure cooker is

(room temperature) and that the area of the pressure cooker lid is

. Take atmospheric pressure to be

. Treat the air, both inside and outside the pressure cooker, as an ideal gas obeying Hint A.1

.

Calculate the pressure inside Hint not displayed

Hint A.2

Relating pressure and force Hint not displayed

Hint A.3

Determine the role of the outside pressure Hint not displayed

Express the force in terms of given variables. ANSWER:

= Answer not displayed

Part B The pressure relief valve on the lid is now opened, allowing hot air to escape until the pressure inside the cooker becomes equal to the outside pressure . The pot is then sealed again and removed from the stove. Assume that when the cooker is removed from the stove, the air inside it is still at What is the magnitude of the net force down to Hint B.1

on the lid when the air inside the cooker has cooled back

?

How to approach the problem Hint not displayed

Hint B.2

What stays constant when the cooker is opened? Hint not displayed

Hint B.3

Calculate the pressure inside Hint not displayed

Express the magnitude of the net force in terms of given variables. ANSWER:

Score Summary:

.

= Answer not displayed

Your score on this assignment is 99.6%. You received 39.85 out of a possible total of 40 points.

View more...
A Sliding Crate of Fruit A crate of fruit with a mass of 34.0

and a specific heat capacity of 3650

down a ramp inclined at an angle of 36.6

slides 9.00

below the horizontal.

Part A If the crate was at rest at the top of the incline and has a speed of 2.05 work

at the bottom, how much

was done on the crate by friction?

Hint A.1

How to approach the problem Hint not displayed

Hint A.2

Find the initial and final kinetic energies Hint not displayed

Hint A.3

Find the difference between initial and final potential energy Hint not displayed

Use 9.81 ANSWER:

for the acceleration due to gravity and express your answer in joules. = -1720 Correct

The frictional force opposes the motion of the crate, so the work done on the crate by friction must be a negative quantity. Part B If an amount of heat equal to the magnitude of the work done by friction is absorbed by the crate of fruit and the fruit reaches a uniform final temperature, what is its temperature change ? Hint B.1

Equation for temperature change Hint not displayed

ANSWER:

= 1.38×10−2 Correct

Of course, the assumptions of "total heat absorption" and "uniform temperature change" are not very realistic; still, this simplified model provides a useful reminder about the transformation of mechanical energy into thermal energy when nonconservative forces are present.

mechanical energy into thermal energy when nonconservative forces are present.

Steam vs. Hot-Water Burns Just about everyone at one time or another has been burned by hot water or steam. This problem compares the heat input to your skin from steam as opposed to hot water at the same temperature. Assume that water and steam, initially at 100 , are cooled down to skin temperature, 34 , when they come in contact with your skin. Assume that the steam condenses extremely fast. We will further for both liquid water and steam. assume a constant specific heat capacity Part A Under these conditions, which of the following statements is true? ANSWER:

Steam burns the skin worse than hot water because the thermal conductivity of steam is much higher than that of liquid water. Steam burns the skin worse than hot water because the latent heat of vaporization is released as well. Hot water burns the skin worse than steam because the thermal conductivity of hot water is much higher than that of steam. Hot water and steam both burn skin about equally badly. Correct

The key point is that the latent heat of vaporization has to be taken into account for the steam. Part B How much heat

is transferred to the skin by 25.0

vaporization for steam is Hint B.1

of steam onto the skin? The latent heat of

.

Determine the heat transferred from steam to skin Hint not displayed

Express the heat transferred, in kilojoules, to three significant figures. ANSWER:

= 63.3 Correct

Here we assumed that the skin continues to remain at 34

. Actually the local temperature in

the area where the steam condenses can be raised quite significantly. Part C How much heat

is transferred by 25.0

of water onto the skin? To compare this to the result in

the previous part, continue to assume that the skin temperature does not change. Hint C.1

Determine the heat transferred from water to skin Hint not displayed

Express the heat transferred, in kilojoules, to three significant figures. ANSWER:

= 6.91 Correct

The amount of heat transferred to your skin is almost 10 times greater when you are burned by steam versus hot water. The temperature of steam can also potentially be much greater than 100 . For these reasons, steam burns are often far more severe than hot-water burns.

Dust Equipartitions Small dust particles suspended in air seem to dance randomly about, a phenomenon called Brownian motion. For this problem you will need to know Boltzmann's constant: . Part A What would you expect the mean translational kinetic energy

of such particles to be

if they are in air at a temperature of 290 K? Hint A.1

Kinetic energy in terms of average velocity Hint not displayed

Hint A.2

Equipartition Theorem Hint not displayed

Hint A.3

Use the Equipartition Theorem Hint not displayed

Express the mean translational kinetic energy numerically, in joules, to two significant figures. Note that has been factored out already to make your answer simpler. ANSWER:

= 6.0 Correct

Part B Find an expression for the rms (root-mean-square) speed spheres of diameter Hint B.1

of these particles, assuming them to be

and density

Find the rms speed in terms of temperature Hint not displayed

Hint B.2

Find the mass of the particles Hint not displayed

Express the rms speed in terms of

,

,

,

, and

.

Express the rms speed in terms of

,

,

,

, and

.

ANSWER: =

Correct

Part C Now calculate the rms (root-mean-square) speed and density

of diameter

of these particles, assuming them to be spheres . The mass of such a dust particle is

.

Express your answer in millimeters per second to one decimal place only. ANSWER:

= 0.3 Correct

This speed is several orders of magnitude smaller than the typical velocities of gas molecules at this temperature (which are of the order of hundreds of meters per second). This is simply because the mass of these particles is much larger than the mass of typical gas molecules. For particles larger than the ones described here, the weight can no longer be ignored. Such particles tend to settle quite quickly on account of their weight. Then such a calculation is no longer valid.

Heating a Room Imagine you've been walking outside on a cold winter's day. When you arrive home at your studio apartment, you realize that you left a window open and your room is only slightly warmer than the outside. You turn on your 1.0-

space heater right away and wait impatiently for the

room to warm up. In this problem, make the following assumptions: The entire output of the space heater goes into warming the air in the room. The air in the room is an ideal gas with five degrees of freedom per particle (three translational degrees of freedom and two rotational degrees of freedom—about right for nitrogen and oxygen). The air in the room is at a constant pressure of 1.00 . At room temperature and atmospheric pressure, 1

of air fills a volume of 23

. This is slightly

larger than the volume of air at standard temperature and pressure, because room temperature is hotter than 0 .

Part A How long will it be before the heater warms the air in the room by 10. Hint A.1

?

How to approach the problem

Since you know the power output of the heater, you know how much energy per unit time is being added to the air in the room. To determine how long it will take to warm up the room, then, you need to determine the total energy needed to raise the temperature of the air in the room by 10 . Once you have this value, simply divide by the power of the heater to determine the time:

Once you have this value, simply divide by the power of the heater to determine the time: .

Hint A.2

Find the energy needed to raise the temperature

How much energy (in joules) is needed to raise the temperature of the room by 10.

?

Hint A.2.1 How to approach the problem

Hint not displayed Hint A.2.2 Find the heat capacity

Hint not displayed Hint A.2.3 Find the amount of air in the room

Hint not displayed Express your answer in joules to two significant figures. ANSWER:

energy = Answer not displayed

Express your answer in minutes to two significant figures. ANSWER:

time =

16 Correct

In practice, it would probably take more than an hour to heat the room by 10.

because the

walls and any items in the room are in thermal contact with the air and would have to be warmed up also.

Melting Point of Platinum The ratio of the pressure of a gas at the melting point of platinum to its pressure at the triple point of water, when the gas is kept at constant volume, is found to be 7.476. Part A What is the Celsius temperature Hint A.1

of the melting point of platinum?

How to approach the problem. Hint not displayed

Hint A.2

Find an expression for the pressure ratio Hint not displayed

Hint A.3

Triple-point temperature of water Hint not displayed

Express your answer in degrees Celsius to four significant figures. ANSWER:

= Answer not displayed

Adding Ice to Water An insulated beaker with negligible mass contains liquid water with a mass of 0.330 temperature of 64.8

and a

.

Part A How much ice at a temperature of -14.0 temperature of the system will be 29.0 Hint A.1

must be dropped into the water so that the final ?

How to approach the problem Hint not displayed

Hint A.2

Calculate the heat lost by the water Hint not displayed

Hint A.3

How to calculate the heat gained by the ice Hint not displayed

Hint A.4

Heat gained by the ice Hint not displayed

Take the specific heat of liquid water to be 4190 , and the heat of fusion for water to be 334 ANSWER:

, the specific heat of ice to be 2100 .

= Answer not displayed

Thermal Energy from Friction on a Rope A capstan is a rotating drum or cylinder over which a rope or cord slides to provide a great amplification of the rope's tension while keeping both ends free . Since the added tension in the rope is due to friction, the capstan generates thermal energy.

Part A If the difference in tension (

) between the two ends of the rope is

and turns once in

a diameter of

, find the rate

and the capstan has

at which thermal energy is

being generated. Hint A.1

Torque applied to the rope Hint not displayed

Hint A.2

Power due to the torque Hint not displayed

Hint A.3

Power from force Hint not displayed

Hint A.4

How both methods work equally well Hint not displayed

Give a numerical answer, in watts, rounded to the nearest 10 W. ANSWER:

= Answer not displayed W

Part B If the capstan is made of iron (with a specific heat capacity , at what rate

) and has a mass of

does its temperature rise? Assume that the temperature in the capstan

is uniform and that all the thermal energy generated flows into it. Note that is a temperature. Hint B.1

Use the chain rule Hint not displayed

Give a numerical answer, in degrees Celsius per second, rounded to two significant figures. ANSWER:

= Answer not displayed

Particle Gas Review A particle gas consists of

monatomic particles each of mass

all contained in a volume

at

A particle gas consists of

monatomic particles each of mass

. Your answers should be written in terms of the Boltzmann constant

temperature

rather than

number

all contained in a volume

at

and Avagadro's

.

Part A Find

, the average speed squared for each particle.

Hint A.1

how to approach the problem Hint not displayed

Hint A.2

Find

for each particle Hint not displayed

Hint A.3

Relating the

,

, and

velocities Hint not displayed

Express the average speed squared in terms of the gas temperature

and any other given

quantities. ANSWER:

= Answer not displayed

Part B Find

, the internal energy of the gas.

Hint B.1

How to approach the problem Hint not displayed

Hint B.2

Kinetic energy of a single gas particle Hint not displayed

Express the internal energy in terms of the gas temperature ANSWER:

and any other given quantities.

= Answer not displayed

Part C

Part not displayed Part D

Part not displayed Now imagine that the mass of each gas particle is increased by a factor of 3. All other information given in the problem introduction remains the same.

given in the problem introduction remains the same. Part E What will be the ratio of the new molar mass ANSWER:

to the old molar mass

?

to the old rms speed

?

= Answer not displayed

Part F What will be the ratio of the new rms speed Hint F.1

Definition of rms speed Hint not displayed

ANSWER:

= Answer not displayed

Part G What will be the ratio of the new molar heat capacity

to the old molar heat capacity

?

Hint G.1 How to approach the problem

Hint not displayed ANSWER:

= Answer not displayed

pV Diagram for a Piston A container holds a sample of ideal gas in thermal equilibrium, as shown in the figure. One end of the container is sealed with a piston whose head is perfectly free to move, unless it is locked in place. The walls of the container readily allow the transfer of energy via heat, unless the piston is wrapped in insulation.

Refer to the pV diagram presented to answer the questions below. In each case, the piston head is initially unlocked and the gas is in equilibrium at the pressure and volume indicated by point 0 on the diagram.

Part A Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process? "Lock the piston head in place, and hold the container above a very hot flame." Hint A.1

Understand the graph Hint not displayed

Hint A.2

Find the change in volume Hint not displayed

ANSWER:

point point point point point point point point

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Answer not displayed

Part B Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process? "Immerse the container into a large water bath at the same temperature, and very slowly push the piston head further into the container." Hint B.1

Find the change in volume Hint not displayed

Hint B.2

Find the change in temperature Hint not displayed

ANSWER:

ANSWER:

point point point point point point point point

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Answer not displayed

Part C Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process? "Lock the piston head in place and plunge the piston into water that is colder than the gas." ANSWER:

point point point point point point point point

1 2 3 4 5 6 7 8

Answer not displayed

Part D Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process? "Wrap the piston in insulation. Pull the piston head further out of the container." Hint D.1

Find the change in volume Hint not displayed

Hint D.2

Find the change in temperature Hint not displayed

ANSWER:

point point point point point point point point

1 2 3 4 5 6 7 8

Answer not displayed

Average Spacing of Gas Molecules Consider an ideal gas at 27.0 degrees Celsius and 1.00 atmosphere pressure. Imagine the molecules to be uniformly spaced, with each molecule at the center of a small cube.

be uniformly spaced, with each molecule at the center of a small cube. Part A What is the length Hint A.1

of an edge of each small cube if adjacent cubes touch but don't overlap?

How to approach the problem Hint not displayed

Hint A.2

Calculate the volume per mole Hint not displayed

Hint A.3

Calculate the volume per molecule Hint not displayed

Hint A.4

The edge length of a cube Hint not displayed

Express your answer numerically in meters. ANSWER:

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Velocity and Energy Scaling Hydrogen molecules have a mass of

and oxygen molecules have a mass of

, where

is defined

). Compare a gas of hydrogen molecules to a gas of

as an atomic mass unit ( oxygen molecules. Part A At what gas temperature

would the average translational kinetic energy of a hydrogen molecule be

equal to that of an oxygen molecule in a gas of temperature 300 K? Hint A.1

Find the energy associated with one degree of freedom Hint not displayed

Hint A.2

Total translational kinetic energy Hint not displayed

Express the temperature numerically in kelvins. ANSWER:

= Answer not displayed K

Part B At what gas temperature

would the root-mean-square (rms) speed of a hydrogen molecule be

At what gas temperature

would the root-mean-square (rms) speed of a hydrogen molecule be

equal to that of an oxygen molecule in a gas at 300 K? Hint B.1

Find the rms speed Hint not displayed

State your answer numerically, in kelvins, to the nearest integer. ANSWER:

= Answer not displayed K

The Speed of Nitrogen Molecules The kinetic theory of gases states that the kinetic energy of a gas is directly proportional to the temperature of the gas. A relationship between the microscopic properties of the gas molecules and the macroscopic properties of the gas can be derived using the following assumptions: The gas is composed of pointlike particles separated by comparatively large distances. The gas molecules are in continual random motion with collisions being perfectly elastic. The gas molecules exert no long-range forces on each other. One of the most important microscopic properties of gas molecules is velocity. There are several different ways to describe statistically the average velocity of a molecule in a gas. The most obvious measure is the average velocity . However, since the molecules in a gas are moving in random directions, the average velocity is approximately zero. Another measure of velocity is

, the

average squared velocity. Since the square of velocity is always positive, this measure does not average to zero over the entire gas. A third measure is the root-mean-square (rms) speed, , equal to the square root of

. The rms speed is a good approximation of the the typical speed of the molecules

in a gas. This histogram shows a theoretical distribution of speeds of molecules in a sample of nitrogen ( ) gas. In this problem, you'll use the histogram to compute properties of the gas.

Part A What is the average speed Hint A.1

of the molecules in the gas?

How to use the histogram Hint not displayed

Hint A.2

More on computing the average Hint not displayed

Express your answer numerically to three significant digits. ANSWER:

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Part B Because the kinetic energy of a single molecule is related to its velocity squared, the best measure of the kinetic energy of the entire gas is obtained by computing the mean squared velocity, , or its square root

. The quantity

is more common than

because it has the dimensions of

velocity instead of the less-familiar velocity-squared. What is the rms speed of the molecules in the nitrogen gas? Hint B.1

How to approach the problem Hint not displayed

Hint B.2

Find the mean square velocity Hint not displayed

Express your answer numerically to three significant digits. ANSWER:

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Part C What is the temperature Hint C.1

of the sample of

gas described in the histogram?

How to approach the problem Hint not displayed

Hint C.2

Find the molar mass of N2 Hint not displayed

Express your answer in degrees Celsius to three significant figues. ANSWER:

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Part D

Part not displayed

Pressure Cooker A pressure cooker is a pot whose lid can be tightly sealed to prevent gas from entering or escaping. Part A

Part A If an otherwise empty pressure cooker is filled with air of room temperature and then placed on a hot stove, what would be the magnitude of the net force on the lid when the air inside the cooker had been heated to

? Assume that the temperature of the air outside the pressure cooker is

(room temperature) and that the area of the pressure cooker lid is

. Take atmospheric pressure to be

. Treat the air, both inside and outside the pressure cooker, as an ideal gas obeying Hint A.1

.

Calculate the pressure inside Hint not displayed

Hint A.2

Relating pressure and force Hint not displayed

Hint A.3

Determine the role of the outside pressure Hint not displayed

Express the force in terms of given variables. ANSWER:

= Answer not displayed

Part B The pressure relief valve on the lid is now opened, allowing hot air to escape until the pressure inside the cooker becomes equal to the outside pressure . The pot is then sealed again and removed from the stove. Assume that when the cooker is removed from the stove, the air inside it is still at What is the magnitude of the net force down to Hint B.1

on the lid when the air inside the cooker has cooled back

?

How to approach the problem Hint not displayed

Hint B.2

What stays constant when the cooker is opened? Hint not displayed

Hint B.3

Calculate the pressure inside Hint not displayed

Express the magnitude of the net force in terms of given variables. ANSWER:

Score Summary:

.

= Answer not displayed

Your score on this assignment is 99.6%. You received 39.85 out of a possible total of 40 points.

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