In this worksheet, we will practice calculating the work done on or by gases that undergo a combination of thermodynamic processes and calculating the change in their internal energy.

**Q1: **

One mole of an ideal gas is initially in a chamber of volume m^{3} at a temperature of
.

How much heat is absorbed by the gas when it slowly expands isothermally to twice its initial volume?

Suppose the gas is slowly expanded to twice its initial volume by first decreasing the pressure at constant volume and then expanding the gas isobarically. How much heat is transferred in this case?

Calculate the heat transferred when the gas is expanded quasi-statically to twice its initial volume by expanding the gas isobarically, then decreasing its pressure at constant volume.

**Q2: **

An ideal gas expands isothermally along the path , as shown in the accompanying diagram. The gas does J of work along this path.

How much heat does the gas exchange along ?

The gas then expands adiabatically along and does 400 J of work. When the gas returns to along , it exhausts 100 J of heat to its surroundings. How much work is done on the gas along this path?

**Q3: **

Consider the process shown in the accompanying diagram, relating the pressure and volume of steam in a cylinder. The change in the internal energy in this process is J. Find the heat entering the system during the process.

**Q4: **

A gas undergoes the process shown in the accompanying diagram. When the gas expands along , it does 500 J of work and absorbs 250 J of heat. When the gas expands along , it does 700 J of work and absorbs 300 J of heat.

How much heat does the gas exchange along ?

When the gas makes the transition from to along , 800 J of work is done on it between and . How much heat does the gas exchange along ?

**Q5: **

Consider the closed path shown in the diagram.

What is the change in internal energy for the process represented by the path?

How much heat is exchanged in the process represented by the path?

If the direction in which the path is traversed is reversed, how much heat is exchanged in the process represented by the path?

**Q6: **

The accompanying diagram shows a process for which along the path the system is heated by 400 J.

By how much is the system heated along the path ?

By how much is the system heated along the path ?

By how much is the system heated along the path ?

**Q7: **

Several quasi-static processes are shown in the accompanying diagram. In each case, pressure is in atmospheres and volume is in liters. The values of pressure and volume corresponding to the points plotted on the graphs are: Point 1 ( atm, L), Point 2 ( atm, L), Point 3 ( atm, L), Point 4 ( atm, L), Point 5 ( atm, L), Point 6 ( atm, L), and Point 7 ( atm, L).

Find the work done during the process in case (a).

Find the work done during the process in case (b).

Find the work done during the process in case (c).

Find the work done during the process in case (d).

Find the work done during the process in case (e).

Find the work done during the process in case (f).

**Q8: **

Consider a transformation from point to point in a two-step process. First, the pressure is lowered from 3.00 MPa at point to a pressure of 1.00 MPa, while keeping the volume at 2.00 L by cooling the system. The state reached is labeled . The system is then heated at a constant pressure to reach a volume of 6.00 L in the state .

Find the amount of work done on the path.

Find the amount of heat exchanged by the system when it goes from to on the path.

**Q9: **

The state of 45 moles of steam in a cylinder is changed in a cyclic manner from , where the pressure and volume of the states are (35 atm, 18 L), (56 atm, 18 L), and (56 atm, 36 L). Assume each change takes place along the straight line connecting the initial and final states in the P-V plane.

Find the net work done by the steam in one cycle.

Find the net cooling of the steam over the course of one cycle.

**Q11: **

The temperature of 5.8 moles of helium gas is reduced from to by reducing the pressure of the gas from 11.0 atm to 2.5 atm.

What is the final volume of helium?

What is the change in helium’s internal energy?

**Q12: **

An ideal gas is at an initial pressure, volume, and temperature of
Pa,
m^{3}, and
320 K respectively. The gas
then expands to a volume m^{3}. Use a value of
for the ratio of the molar heat capacities of the gas.

What is the final temperature of the gas if it expands isothermally?

What is the final pressure of the gas if it expands isothermally?

- A Pa
- B Pa
- C Pa
- D Pa
- E Pa

What is the final temperature of the gas if it expands quasi-statically?

What is the final pressure of the gas if it expands adiabatically and quasi-statically?

- A Pa
- B Pa
- C Pa
- D Pa
- E Pa

**Q13: **

Three moles of a monatomic ideal gas are at a pressure of 6.8 MPa and occupy a volume of 4.0 L. The gas is expanded isothermally until the volume is doubled. The gas is then cooled isochorically until the gas pressure is 2.6 MPa, reducing the gas’s temperature. The gas is then compressed isothermally until its volume is again 4.0 L, but this time at a pressure of 3.5 MPa. Finally, the gas is heated isochorically to return to the initial state where its volume was 4.0 L and its pressure was 6.8 MPa.

What is the net work done by the gas?

What is the net heating of the gas?

**Q14: **

For an isentropic expansion of an ideal gas from state 1 to state 2, what is the relationship between the temperatures and pressures of the states?

- A
- B
- C
- D
- E

**Q15: **

A tank contains 105.0 g chlorine gas (), which is at a temperature of and an absolute pressure of Pa. The temperature of the air outside the tank is . In determining the properties of the gas, use a value of 70.9 g/mol for the molar mass of .

What is the volume of the tank?

What is the internal energy of the gas?

- A J
- B J
- C J
- D J
- E J

**Q16: **

Use the van der Waals equation to find the volume of a cloud of at a temperature of 373 K if the mass of the cloud is 1 kg.

**Q17: **

Dry steam with a mass of 1.0 kg is heated at a constant pressure of
200 kPa until its volume is
1.3 m^{3}. What is the net heating
required for the process?

**Q18: **

Calculate the work required to compress 1.5 cubic meters of air initially at a temperature of and a pressure of 102 kPa to a pressure of 600 kPa.

**Q19: **

Which power cycle used for internal combustion consists of two adiabatic processes, an isochoric process, and an isobaric process?

- ALenoir
- BRankine
- COtto
- DDiesel

**Q20: **

What is the power cycle that consists of two adiabatic processes and two isochoric processes?

- ABrayton
- BCarnot
- CDiesel
- DOtto

**Q21: **

A rigid vessel contains a 7.00 kg mass of an ideal gas at a pressure of 5.00 atm. A valve is opened, and half of the mass of the gas escapes. If the final pressure in the tank is 1.50 atm and the final temperature in the tank is , what is the initial temperature of the gas?

**Q22: **

What is a thermodynamic cycle?

- AA thermodynamic process
- BA series of identical thermodynamic processes
- CA system in equilibrium
- DA series of thermodynamic processes that return to the first state of the first process

**Q23: **

What restrictions exist on the transfer of mass and energy into and out of a closed system?

- AIn a closed system, no energy may be transferred in or out of the system boundaries.
- BIn a closed system, no mass may be transferred in or out of the system boundaries.
- CIn a closed system, no work may be transferred in or out of the system boundaries.
- DIn a closed system, both mass and energy cannot be transferred in or out of the system boundaries.

**Q24: **

What set of thermodynamic processes are used to form a Carnot cycle?

- ATwo isobaric processes and two adiabatic processes.
- BTwo irreversible isothermal processes and two adiabatic processes.
- CTwo reversible isentropic processes and two adiabatic processes.
- DTwo reversible isothermal processes and two adiabatic processes.

**Q25: **

What is the power cycle that consists of two adiabatic processes and two isobaric processes?

- ADiesel
- BCarnot
- COtto
- DBrayton