4.3.3: Problems

Exercise \(\PageIndex{1}\): Three cycle engine

\(\color{red}{\text{There is a time delay-since the system must be in equilibrium-before the initial change of state occurs.}}\)

\(\color{red}{\text{You must go in order.}}\)

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). Restart.

1. Describe the types of expansions and compressions that are a part of this engine cycle.
2. For this ideal gas, what is \(\gamma\) (the adiabatic constant, or the ratio of the specific heat at constant pressure to the specific heat at constant volume)?
3. Find the work done in each step.
4. What is the net work done?

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{2}\): Otto engine

\(\color{red}{\text{There is a time delay-since the system must be in equilibrium-before the initial change of state occurs.}}\)

\(\color{red}{\text{You must go in order.}}\)

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). To run this Otto engine (cycles of adiabatic and isochoric expansions and contractions), you must go through the steps in order. The work shown in the data table is the work done during each step. Restart.

1. Is the gas a monatomic or diatomic ideal gas?
2. What is the net work and what is the heat absorbed?
3. Find the efficiency of this Otto engine.

Note

Note that the high temperature and low temperature reservoirs do not stay at constant temperatures.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{3}\): Brayton cycle

\(\color{red}{\text{There is a time delay-since the system must be in equilibrium-before the initial change of state occurs.}}\)

\(\color{red}{\text{You must go in order.}}\)

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). To run this Brayton cycle, go through the steps in order. Restart.

1. Describe the types of expansions and compressions that are a part of this engine cycle.
2. For this monatomic gas, find the work done in each step.
3. What is the net work done?
4. What is the efficiency of this cycle?

Note

Note that the high temperature and low temperature reservoirs do not stay at constant temperatures.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{4}\): Internal combustion engine

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). We will assume an ideal diatomic gas in the engine. Restart.

The Otto engine cycle is close to the cycle of an internal combustion engine and consists of adiabatic and isochoric processes.

1. Identify which parts of the engine cycle correspond to which processes.
2. Explain why, during the gas intake and smoke exhaust process, there is no net work done. Also explain how heat is released.
3. Calculate the work done during the adiabatic expansion and compression.
4. Find the heat exchanged during the isochoric processes.
5. Determine the efficiency of this engine.

Problem authored by Anne J. Cox.
Script authored by Anne J. Cox and Wolfgang Christian.
Applet authored by CoLoS and modified by Wolfgang Christian.

Exercise \(\PageIndex{5}\): Engine cycle and \(PV\) diagram

A mass is pushed downward on top of a container (the dimension of the container into the computer screen is \(30\text{ cm}\)) filled with an ideal monatomic gas that undergoes a thermodynamic process as shown in the animation (pressure is given in atmospheres, position is given in centimeters, temperature is given in degrees Celsius, and time is given in seconds). Both the dial and the relative height of the black box on the piston represent the changing pressure. Restart.

1. Determine how many moles of gas are in the container.
2. Find the following: 
   • The work done by or on the gas in each step.
   • The heat gained or lost by the gas in each step.
3. Draw a \(PV\) diagram for this engine cycle.
4. Calculate the efficiency of this engine.

Problem authored by Anne J. Cox.
Script authored by Mario Belloni and Anne J. Cox.

Exercise \(\PageIndex{6}\): Match the expansion to the \(TS\) graph

\(\color{red}{\text{There is a time delay-since the system must be in equilibrium-before the change of state occurs.}}\)

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). Assume an ideal monatomic gas. Restart.

Which temperature-entropy graph goes with the animation shown? Explain.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{7}\): Find the entropy of an expansion

\(\color{red}{\text{There is a time delay-since the system must be in equilibrium-before the change of state occurs.}}\)

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). Assume an ideal monatomic gas. Restart.

1. Identify the thermodynamic process.
2. Draw a \(TS\) diagram and calculate the change in entropy.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{8}\): Refrigerator and entropy

\(\color{red}{\text{There is a time delay-since the system must be in equilibrium-before the initial change of state occurs.}}\)

\(\color{red}{\text{You must go in order.}}\)

In this animation \(N = nR\) (i.e., \(k_{B} = 1\)). This, then, gives the ideal gas law as \(PV = NT\). Assume an ideal gas. Restart.

The reverse of an engine is a refrigerator. An engine uses heat to produce work, while a refrigerator does negative work on the gas (something else does work on the gas) to remove heat. The coefficient of performance is \(K = |Q_{C}|/|W|\) (heat transferred out of the cold reservoir divided by the work required)

1. In which step is the heat removed from the cold reservoir (i.e., heat removed from the refrigerator and absorbed by the gas)?
2. In which step is work done on the gas?
3. For the refrigerator, find the work done during each cycle, the heat transferred from the cold reservoir, and the coefficient of performance.
4. What is the change in entropy for the complete cycle?

Problem authored by Anne J. Cox.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.