Samuel J. Ling; William Moebs; Jeff Sanny

Problems

3.1 Thermodynamic Systems

19.

A gas follows $p V = b p + c_{T}$ on an isothermal curve, where p is the pressure, V is the volume, b is a constant, and c is a function of temperature. If c can be determined from the measurement of a single other thermodynamic parameter, it establishes a temperature scale for this gas. Show that a temperature scale under an isochoric process can be established with this gas, with pressure being the measured parameter, and is identical to that of an ideal gas.

20.

A mole of gas has isobaric expansion coefficient $d V / d T = R / p$ and isochoric pressure-temperature coefficient $d p / d T = p / T$ . Show that the equation of state $p V - R T = 0$ is consistent with these coefficients.

21.

Find the isobaric expansion coefficient $d V / d T$ and the isothermal pressure-volume coefficient $d V / d P$ of a solid that has equation of state $V + b p T - c T^{2} = 0$ .

3.2 Work, Heat, and Internal Energy

22.

A gas at a pressure of 2.00 atm undergoes a quasi-static isobaric expansion from 3.00 to 5.00 L. How much work is done by the gas?

23.

It takes 500 J of work to compress quasi-statically 0.50 mol of an ideal gas to one-fifth its original volume. Calculate the temperature of the gas, assuming it remains constant during the compression.

24.

It is found that, when a dilute gas expands quasi-statically from 0.50 to 4.0 L, it does 250 J of work. Assuming that the gas temperature remains constant at 300 K, how many moles of gas are present?

25.

In a quasi-static isobaric expansion, 500 J of work are done by the gas. If the gas pressure is 0.80 atm, what is the fractional increase in the volume of the gas, assuming it was originally at 20.0 L?

26.

When a gas undergoes a quasi-static isobaric change in volume from 10.0 to 2.0 L, 15 J of work from an external source are required. What is the pressure of the gas?

27.

An ideal gas expands quasi-statically and isothermally from a state with pressure p and volume V to a state with volume 4V. Show that the work done by the gas in the expansion is pV(ln 4).

28.

As shown below, calculate the work done by the gas in the quasi-static processes represented by the paths (a) AB; (b) ADB; (c) ACB; and (d) ADCB.

The figure is a plot of pressure, p, in atmospheres on the vertical axis as a function of volume, V, in Liters on the horizontal axis. The horizontal volume scale runs from 0 to 5.0 Liters, and the vertical pressure scale runs from 0 to 4.0 atmospheres. Four points, A, B, C, and D are labeled. Point A is at 1.0 L, 1.0 atmospheres. Point B is at 3.0 L, 1.0 atmospheres. Point C is at 3.0 L, 2.0 atmospheres. Point D is at 1.0 L, 3.0 atmospheres. A straight horizontal line connects A to B, with an arrow pointing to the right indicating the direction from A to B. A straight horizontal line connects D to C, with an arrow to the right indicating the direction from D to C. A straight vertical line connects A to D, with an arrow pointing upward indicating the direction from A to D. A straight vertical line connects C to B, with an arrow downward indicating the direction from C to B. Finally, a straight diagonal line connects D to B with an arrow pointing in the direction from D to B.

29.

(a) Calculate the work done on the gas along the closed path shown below. The curved section between R and S is semicircular. (b) If the process is carried out in the opposite direction, what is the work done on the gas?

30.

An ideal gas expands quasi-statically to three times its original volume. Which process requires more work from the gas, an isothermal process or an isobaric one? Determine the ratio of the work done in these processes.

31.

A dilute gas at a pressure of 2.0 atm and a volume of 4.0 L is taken through the following quasi-static steps: (a) an isobaric expansion to a volume of 10.0 L, (b) an isochoric change to a pressure of 0.50 atm, (c) an isobaric compression to a volume of 4.0 L, and (d) an isochoric change to a pressure of 2.0 atm. Show these steps on a pV diagram and determine from your graph the net work done by the gas.

32.

What is the average mechanical energy of the atoms of an ideal monatomic gas at 300 K?

33.

What is the internal energy of 6.00 mol of an ideal monatomic gas at $200 ° C$ ?

34.

Calculate the internal energy of 15 mg of helium at a temperature of $0 ° C .$

35.

Two monatomic ideal gases A and B are at the same temperature. If 1.0 g of gas A has the same internal energy as 0.10 g of gas B, what are (a) the ratio of the number of moles of each gas and (b) the ration of the atomic masses of the two gases?

36.

The van der Waals coefficients for oxygen are $a = 0.138 J \cdot m^{3} / {mol}^{2}$ and $b = 3.18 \times 10^{−5} m^{3} / mol$ . Use these values to draw a van der Waals isotherm of oxygen at 100 K. On the same graph, draw isotherms of one mole of an ideal gas.

37.

Find the work done in the quasi-static processes shown below. The states are given as (p, V) values for the points in the pV plane: 1 (3 atm, 4 L), 2 (3 atm, 6 L), 3 (5 atm, 4 L), 4 (2 atm, 6 L), 5 (4 atm, 2 L), 6 (5 atm, 5 L), and 7 (2 atm, 5 L).

Figures a through f are plots of p on the vertical as a function of V on the horizontal axis. Figure a has points 1 and 2 at the same pressure and with V 2 larger than V 1. A horizontal line with a rightward arrow goes from point 1 to point 2. Figure b has points 1 and 3 at the same volume and with p 3 larger than p 1. A vertical line with an upward arrow goes from point 1 to point 3. Figure c has points 1 and 4, where p 1 is larger than p 4 and V 1 is smaller than V 4. A diagonal line with an arrow pointing down and to the right goes from point 1 to point 4. Figure d has points 1, 3 and 5, where V 1 and V 3 are equal, and larger than V 5. P 1 is smaller than P 5 which is smaller than P 3. A diagonal line with an arrow pointing up and to the left goes from point 1 to point 5. A second diagonal line with an arrow pointing up and to the right goes from point 5 to point 3. Figure e has points 1, 2 and 6, where p 1 and p 2 are equal, and smaller than p 6. V 1 is smaller than V 6 which is smaller than V 2. A diagonal line with an arrow pointing up and to the right goes from point 1 to point 6. A second diagonal line with an arrow pointing down and to the right goes from point 6 to point 2. Figure f has points 1, 2 and 7, where p 1 and p 2 are equal, and larger than p 7. V 1 is smaller than V 6 which is smaller than V 2. A diagonal line with an arrow pointing down and to the right goes from point 1 to point 7. A second diagonal line with an arrow pointing up and to the right goes from point 7 to point 2.

3.3 First Law of Thermodynamics

38.

When a dilute gas expands quasi-statically from 0.50 to 4.0 L, it does 250 J of work. Assuming that the gas temperature remains constant at 300 K, (a) what is the change in the internal energy of the gas? (b) How much heat is absorbed by the gas in this process?

39.

In an expansion of gas, 500 J of work are done by the gas. If the internal energy of the gas increased by 80 J in the expansion, how much heat does the gas absorb?

40.

An ideal gas expands quasi-statically and isothermally from a state with pressure p and volume V to a state with volume 4V. How much heat is added to the expanding gas?

41.

As shown below, if the heat absorbed by the gas along AB is 400 J, determine the quantities of heat absorbed along (a) ADB; (b) ACB; and (c) ADCB.

42.

During the isobaric expansion from A to B represented below, 3,100 J of heat are added to the gas. What is the change in its internal energy?

The figure is a plot of pressure, p, in Newtons per square meter on the vertical axis as a function of volume, V, in cubic meters on the horizontal axis. The horizontal volume scale runs from 0 to 3.0 cubic meters, and the vertical pressure scale is labeled at only one pressure, 1.0 times 10 to the 4 Newtons per square meter. Two points, A and B, are labeled, both at the labeled pressure of 1.0 times 10 to the 4 Newtons per square meter. Point A is at 0.15 cubic meters. Point B is at 0.3 cubic meters. A horizontal line connects A to B, with an arrow pointing to the right, from A to B.

43.

(a) What is the change in internal energy for the process represented by the closed path shown below? (b) How much heat is exchanged? (c) If the path is traversed in the opposite direction, how much heat is exchanged?

44.

When a gas expands along path AC shown below, it does 400 J of work and absorbs either 200 or 400 J of heat. (a) Suppose you are told that along path ABC, the gas absorbs either 200 or 400 J of heat. Which of these values is correct? (b) Give the correct answer from part (a), how much work is done by the gas along ABC? (c) Along CD, the internal energy of the gas decreases by 50 J. How much heat is exchanged by the gas along this path?

The figure is a plot of pressure, p on the vertical axis as a function of volume, V on the horizontal axis. Four points, A, B, C, and D are shown. B is directly above A, at the same volume but with p B greater than p A. Likewise, C is directly above D, at the same volume but with p C greater than p D. A and D are at the same pressure, with p D greater than p A. B and C are at the same pressure, with p C greater than p B. Four paths are shown. One path connects from A straight up to B. One path connects from B horizontally to the right to C. One path connects from C straight down to D. And the last path connects from A to C with a somewhat wavy curve that remains above the A D pressure and below the B C pressure.

45.

When a gas expands along AB (see below), it does 20 J of work and absorbs 30 J of heat. When the gas expands along AC, it does 40 J of work and absorbs 70 J of heat. (a) How much heat does the gas exchange along BC? (b) When the gas makes the transition from C to A along CDA, 60 J of work are done on it from C to D. How much heat does it exchange along CDA?

46.

A dilute gas is stored in the left chamber of a container whose walls are perfectly insulating (see below), and the right chamber is evacuated. When the partition is removed, the gas expands and fills the entire container. Calculate the work done by the gas. Does the internal energy of the gas change in this process?

The figure is an illustration of a container with a partition in the middle dividing it into two chambers. The outer walls are insulated. The chamber on the left is full of gas, indicated by blue shading and many small dots representing the gas molecules. The right chamber is empty.

47.

Ideal gases A and B are stored in the left and right chambers of an insulated container, as shown below. The partition is removed and the gases mix. Is any work done in this process? If the temperatures of A and B are initially equal, what happens to their common temperature after they are mixed?

48.

An ideal monatomic gas at a pressure of $2.0 \times 10^{5} {N/m}^{2}$ and a temperature of 300 K undergoes a quasi-static isobaric expansion from $2.0 \times 10^{3} to 4.0 \times 10^{3} {cm}^{3} .$ (a) What is the work done by the gas? (b) What is the temperature of the gas after the expansion? (c) How many moles of gas are there? (d) What is the change in internal energy of the gas? (e) How much heat is added to the gas?

49.

Consider the process for steam in a cylinder shown below. Suppose the change in the internal energy in this process is 30 kJ. Find the heat entering the system.

50.

The state of 30 moles of steam in a cylinder is changed in a cyclic manner from a-b-c-a, where the pressure and volume of the states are: a (30 atm, 20 L), b (50 atm, 20 L), and c (50 atm, 45 L). Assume each change takes place along the line connecting the initial and final states in the pV plane. (a) Display the cycle in the pV plane. (b) Find the net work done by the steam in one cycle. (c) Find the net amount of heat flow in the steam over the course of one cycle.

51.

A monatomic ideal gas undergoes a quasi-static process that is described by the function $p (V) = p_{1} + 3 (V - V_{1})$ , where the starting state is $(p_{1}, V_{1})$ and the final state $(p_{2}, V_{2})$ . Assume the system consists of n moles of the gas in a container that can exchange heat with the environment and whose volume can change freely. (a) Evaluate the work done by the gas during the change in the state. (b) Find the change in internal energy of the gas. (c) Find the heat input to the gas during the change. (d) What are initial and final temperatures?

52.

A metallic container of fixed volume of $2.5 \times 10^{−3} m^{3}$ immersed in a large tank of temperature $27 ° C$ contains two compartments separated by a freely movable wall. Initially, the wall is kept in place by a stopper so that there are 0.02 mol of the nitrogen gas on one side and 0.03 mol of the oxygen gas on the other side, each occupying half the volume. When the stopper is removed, the wall moves and comes to a final position. The movement of the wall is controlled so that the wall moves in infinitesimal quasi-static steps. (a) Find the final volumes of the two sides assuming the ideal gas behavior for the two gases. (b) How much work does each gas do on the other? (c) What is the change in the internal energy of each gas? (d) Find the amount of heat that enters or leaves each gas.

53.

A gas in a cylindrical closed container is adiabatically and quasi-statically expanded from a state A (3 MPa, 2 L) to a state B with volume of 6 L along the path $1.8 p V = constant .$ (a) Plot the path in the pV plane. (b) Find the amount of work done by the gas and the change in the internal energy of the gas during the process.

3.4 Thermodynamic Processes

54.

Two moles of a monatomic ideal gas at (5 MPa, 5 L) is expanded isothermally until the volume is doubled (step 1). Then it is cooled isochorically until the pressure is 1 MPa (step 2). The temperature drops in this process. The gas is now compressed isothermally until its volume is back to 5 L, but its pressure is now 2 MPa (step 3). Finally, the gas is heated isochorically to return to the initial state (step 4). (a) Draw the four processes in the pV plane. (b) Find the total work done by the gas.

55.

Consider a transformation from point A to B in a two-step process. First, the pressure is lowered from 3 MPa at point A to a pressure of 1 MPa, while keeping the volume at 2 L by cooling the system. The state reached is labeled C. Then the system is heated at a constant pressure to reach a volume of 6 L in the state B. (a) Find the amount of work done on the ACB path. (b) Find the amount of heat exchanged by the system when it goes from A to B on the ACB path. (c) Compare the change in the internal energy when the AB process occurs isothermally with the AB change through the two-step process on the ACB path.

56.

Consider a cylinder with a movable piston containing n moles of an ideal gas. The entire apparatus is immersed in a constant temperature bath of temperature T kelvin. The piston is then pushed slowly so that the pressure of the gas changes quasi-statically from $p_{1}$ to $p_{2}$ at constant temperature T. Find the work done by the gas in terms of n, R, T, $p_{1},$ and $p_{2} .$

57.

An ideal gas expands isothermally along AB and does 700 J of work (see below). (a) How much heat does the gas exchange along AB? (b) The gas then expands adiabatically along BC and does 400 J of work. When the gas returns to A along CA, it exhausts 100 J of heat to its surroundings. How much work is done on the gas along this path?

The figure is a plot of pressure, p, on the vertical axis as a function of volume, V, on the horizontal axis. Three Points, A, B, C, and D are labeled. Point A is at the smallest volume and highest pressure. Point C is at the largest volume and lowest pressure. Point B is at an intermediate pressure and volume, but above the A C line. A path from A to B, to C, and back to A is shown. The path leaves A, goes down but with decreasing slope to reach B. It leaves B and descends steeply to C. It then curves back up to A. All the curves are concave up.

58.

Consider the processes shown below for a monatomic gas. (a) Find the work done in each of the processes AB, BC, AD, and DC. (b) Find the internal energy change in processes AB and BC. (c) Find the internal energy difference between states C and A. (d) Find the total heat added in the ADC process. (e) From the information given, can you find the heat added in process AD? Why or why not?

59.

Two moles of helium gas are placed in a cylindrical container with a piston. The gas is at room temperature $25 ° C$ and under a pressure of $3.0 \times 10^{5} Pa .$ When the pressure from the outside is decreased while keeping the temperature the same as the room temperature, the volume of the gas doubles. (a) Find the work the external agent does on the gas in the process. (b) Find the heat exchanged by the gas and indicate whether the gas takes in or gives up heat. Assume ideal gas behavior.

60.

An amount of n moles of a monatomic ideal gas in a conducting container with a movable piston is placed in a large thermal heat bath at temperature $T_{1}$ and the gas is allowed to come to equilibrium. After the equilibrium is reached, the pressure on the piston is lowered so that the gas expands at constant temperature. The process is continued quasi-statically until the final pressure is 4/3 of the initial pressure $p_{1} .$ (a) Find the change in the internal energy of the gas. (b) Find the work done by the gas. (c) Find the heat exchanged by the gas, and indicate, whether the gas takes in or gives up heat.

3.5 Heat Capacities of an Ideal Gas

61.

The temperature of an ideal monatomic gas rises by 8.0 K. What is the change in the internal energy of 1 mol of the gas at constant volume?

62.

For a temperature increase of $10 ° C$ at constant volume, what is the heat absorbed by (a) 3.0 mol of a dilute monatomic gas; (b) 0.50 mol of a dilute diatomic gas; and (c) 15 mol of a dilute polyatomic gas?

63.

If the gases of the preceding problem are initially at 300 K, what are their internal energies after they absorb the heat?

64.

Consider 0.40 mol of dilute carbon dioxide at a pressure of 0.50 atm and a volume of 50 L. What is the internal energy of the gas?

65.

When 400 J of heat are slowly added to 10 mol of an ideal monatomic gas, its temperature rises by $10 ° C$ . What is the work done on the gas?

66.

One mole of a dilute diatomic gas occupying a volume of 10.00 L expands against a constant pressure of 2.000 atm when it is slowly heated. If 400.0 J of heat are added in the process, what is its final volume?

3.6 Adiabatic Processes for an Ideal Gas

67.

A monatomic ideal gas undergoes a quasi-static adiabatic expansion in which its volume is doubled. How is the pressure of the gas changed?

68.

An ideal gas has a pressure of 0.50 atm and a volume of 10 L. It is compressed adiabatically and quasi-statically until its pressure is 3.0 atm and its volume is 2.8 L. Is the gas monatomic, diatomic, or polyatomic?

69.

Pressure and volume measurements of a dilute gas undergoing a quasi-static adiabatic expansion are shown below. Plot ln p vs. V and determine $γ$ for this gas from your graph.

P (atm)	V (L)
20.0	1.0
17.0	1.1
14.0	1.3
11.0	1.5
8.0	2.0
5.0	2.6
2.0	5.2
1.0	8.4

70.

An ideal monatomic gas at 300 K expands adiabatically and reversibly to twice its volume. What is its final temperature?

71.

An ideal diatomic gas at 80 K is slowly compressed adiabatically and reversibly to half its volume. What is its final temperature?

72.

An ideal diatomic gas at 80 K is slowly compressed adiabatically to one-third its original volume. What is its final temperature?

73.

Compare the charge in internal energy of an ideal gas for a quasi-static adiabatic expansion with that for a quasi-static isothermal expansion. What happens to the temperature of an ideal gas in an adiabatic expansion?

74.

The temperature of n moles of an ideal gas changes from $T_{1}$ to $T_{2}$ in a quasi-static adiabatic transition. Show that the work done by the gas is given by

$W = \frac{n R}{γ - 1} (T_{1} - T_{2}) .$

75.

A dilute gas expands quasi-statically to three times its initial volume. Is the final gas pressure greater for an isothermal or an adiabatic expansion? Does your answer depend on whether the gas is monatomic, diatomic, or polyatomic?

76.

(a) An ideal gas expands adiabatically from a volume of $2.0 \times 10^{−3} m^{3}$ to $2.5 \times 10^{−3} m^{3}$ . If the initial pressure and temperature were $5.0 \times 10^{5} Pa$ and 300 K, respectively, what are the final pressure and temperature of the gas? Use $γ = 5 / 3$ for the gas. (b) In an isothermal process, an ideal gas expands from a volume of $2.0 \times 10^{−3} m^{3}$ to $2.5 \times 10^{−3} m^{3}$ . If the initial pressure and temperature were $5.0 \times 10^{5} Pa$ and 300 K, respectively, what are the final pressure and temperature of the gas?

77.

On an adiabatic process of an ideal gas pressure, volume and temperature change such that $p V^{γ}$ is constant with $γ = 5 / 3$ for monatomic gas such as helium and $γ = 7 / 5$ for diatomic gas such as hydrogen at room temperature. Use numerical values to plot two isotherms of 1 mol of helium gas using ideal gas law and two adiabatic processes mediating between them. Use $T_{1} = 500 K, V_{1} = 1 L, and T_{2} = 300 K$ for your plot.

78.

Two moles of a monatomic ideal gas such as helium is compressed adiabatically and reversibly from a state (3 atm, 5 L) to a state with pressure 4 atm. (a) Find the volume and temperature of the final state. (b) Find the temperature of the initial state of the gas. (c) Find the work done by the gas in the process. (d) Find the change in internal energy of the gas in the process.