---
title: "A heat engine contains \\(n\\) moles of a monatomic ideal gas. The gas is taken through a thermodynamic cycle consisting of three processes:  – **Process 1 \\(\\rightarrow\\) 2**: Isothermal expansion at a constant temperature \\(T_H\\) from an initial volume \\(V_0\\) to a final volume \\(3V_0\\). (The work done on the gas during this process is \\(W_{12} = -nRT_H \\ln 3\\)). – **Process 2 \\(\\rightarrow\\) 3**: Isobaric compression at constant pressure until the gas returns to its initial volume \\(V_0\\). – **Process 3 \\(\\rightarrow\\) 1**: Isochoric heating at constant volume back to state 1."
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url: "https://nerd-notes.com/ubq/117739/"
date_modified: "2026-08-04T07:54:49+00:00"
---

# A heat engine contains \(n\) moles of a monatomic ideal gas. The gas is taken through a thermodynamic cycle consisting of three processes:

– **Process 1 \(\rightarrow\) 2**: Isothermal expansion at a constant temperature \(T_H\) from an initial volume \(V_0\) to a final volume \(3V_0\). (The work done on the gas during this process is \(W_{12} = -nRT_H \ln 3\)).
– **Process 2 \(\rightarrow\) 3**: Isobaric compression at constant pressure until the gas returns to its initial volume \(V_0\).
– **Process 3 \(\rightarrow\) 1**: Isochoric heating at constant volume back to state 1.

A heat engine contains \(n\) moles of a monatomic ideal gas. The gas is taken through a thermodynamic cycle consisting of three processes:

- **Process 1 \(\rightarrow\) 2**: Isothermal expansion at a constant temperature \(T_H\) from an initial volume \(V_0\) to a final volume \(3V_0\). (The work done on the gas during this process is \(W_{12} = -nRT_H \ln 3\)).
- **Process 2 \(\rightarrow\) 3**: Isobaric compression at constant pressure until the gas returns to its initial volume \(V_0\).
- **Process 3 \(\rightarrow\) 1**: Isochoric heating at constant volume back to state 1.

**Part a)** **Sketch** a graph of the cycle on the \(PV\) axes provided. **Label** states 1, 2, and 3, and include arrows on the lines to indicate the direction of the cycle.

**Part b)** Student A makes the following claim: "The net work done on the gas over one complete cycle must be positive, because an external force must do work to compress the gas back to its initial volume during Process 2 \(\rightarrow\) 3." **Indicate** whether Student A's claim is correct or incorrect. - [ ] Correct - [ ] Incorrect **Justify** your answer using physical principles and features of the \(PV\) diagram you sketched in part (a).

**Part c)** **Derive** a symbolic expression for the net work done on the gas, \(W_\text{net}\), during one complete cycle. Express your answer in terms of \(n\), \(T_H\), \(V_0\), and physical constants, as appropriate.

**Part d)** **Explain** whether your derived expression in part (c) supports or refutes your evaluation of Student A's claim in part (b). (Note: \(\ln 3 \approx 1.1\))

**Part e)** Student B analyzes the entropy changes during the cycle and states: "Because the gas returns to its initial state at the end of the cycle, the total entropy of the universe (the gas plus its surroundings) does not change over one complete cycle."


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117739/*