### Problems

## 3.1 Thermodynamic Systems

A gas follows $pV=bp+{c}_{T}$ 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.

A mole of gas has isobaric expansion coefficient $dV\text{/}dT=R\text{/}p$ and isochoric pressure-temperature coefficient $dp\text{/}dT=p\text{/}T$. Show that the equation of state $pV\u2013RT=0$is consistent with these coefficients.

Find the isobaric expansion coefficient $dV\text{/}dT$ and the isothermal pressure-volume coefficient $dV\text{/}dP$ of a solid that has equation of state $V+bpT\u2013c{T}^{2}=0$.

## 3.2 Work, Heat, and Internal Energy

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?

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.

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?

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?

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?

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).

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.

(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?

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.

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.

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

What is the internal energy of 6.00 mol of an ideal monatomic gas at $200\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ ?

Calculate the internal energy of 15 mg of helium at a temperature of $0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}.$

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?

The van der Waals coefficients for oxygen are $a=0.138\phantom{\rule{0.2em}{0ex}}\text{J}\xb7{\text{m}}^{3}\text{/}{\text{mol}}^{2}$ and $b=3.18\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}{\phantom{\rule{0.2em}{0ex}}\text{m}}^{3}\text{/}\text{mol}$. 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.

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).

## 3.3 First Law of Thermodynamics

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?

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?

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?

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.

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?

(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?

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?

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*?

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?

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?

An ideal monatomic gas at a pressure of $2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}{\phantom{\rule{0.2em}{0ex}}\text{N/m}}^{2}$ and a temperature of 300 K undergoes a quasi-static isobaric expansion from $2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}4.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}{\phantom{\rule{0.2em}{0ex}}\text{cm}}^{3}.$ (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?

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.

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.

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 $\left({p}_{1},{V}_{1}\right)$ and the final state $\left({p}_{2},{V}_{2}\right)$. 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?

A metallic container of fixed volume of $2.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}{\phantom{\rule{0.2em}{0ex}}\text{m}}^{3}$ immersed in a large tank of temperature $27\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ 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.

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 $1.8\phantom{\rule{0.2em}{0ex}}pV=\text{constant}\text{.}$ (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

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.

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.

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 ${p}_{2}.$

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?

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?

Two moles of helium gas are placed in a cylindrical container with a piston. The gas is at room temperature $25\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ and under a pressure of $3.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{Pa}\text{.}$ 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.

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

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?

For a temperature increase of $10\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ 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?

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

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?

When 400 J of heat are slowly added to 10 mol of an ideal monatomic gas, its temperature rises by $10\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. What is the work done on the gas?

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

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

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?

Pressure and volume measurements of a dilute gas undergoing a quasi-static adiabatic expansion are shown below. Plot ln p vs. V and determine $\text{\gamma}$ 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 |

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

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

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

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?

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

$W=\frac{nR}{\gamma -1}({T}_{1}-{T}_{2}).$

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?

(a) An ideal gas expands adiabatically from a volume of $2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}{\phantom{\rule{0.2em}{0ex}}\text{m}}^{3}$ to $2.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}{\phantom{\rule{0.2em}{0ex}}\text{m}}^{3}$. If the initial pressure and temperature were $5.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{Pa}$ and 300 K, respectively, what are the final pressure and temperature of the gas? Use $\gamma =5\text{/}3$ for the gas. (b) In an isothermal process, an ideal gas expands from a volume of $2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}{\phantom{\rule{0.2em}{0ex}}\text{m}}^{3}$ to $2.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}{\phantom{\rule{0.2em}{0ex}}\text{m}}^{3}$. If the initial pressure and temperature were $5.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{Pa}$ and 300 K, respectively, what are the final pressure and temperature of the gas?

On an adiabatic process of an ideal gas pressure, volume and temperature change such that $p{V}^{\gamma}$ is constant with $\gamma =5\text{/}3$ for monatomic gas such as helium and $\gamma =7\text{/}5$ 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 ${T}_{1}=500\phantom{\rule{0.2em}{0ex}}\text{K},\phantom{\rule{0.2em}{0ex}}{V}_{1}=1\phantom{\rule{0.2em}{0ex}}\text{L},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{T}_{2}=300\phantom{\rule{0.2em}{0ex}}\text{K}$ for your plot.

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.