---
title: "A sample of a monatomic ideal gas undergoes the cyclic process \\(1 \\to 2 \\to 3 \\to 4 \\to 1\\) shown in the \\(PV\\) diagram. An ideal Carnot heat engine operates between the minimum and maximum absolute temperatures reached by the gas during this cycle. What is the ratio of the thermal efficiency of the cyclic engine to the thermal efficiency of the Carnot engine, \\(\\dfrac{\\eta_{\\text{cycle}}}{\\eta_{\\text{Carnot}}}\\)?"
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url: "https://nerd-notes.com/ubq/123444/"
date_modified: "2026-09-28T11:59:22+00:00"
---

# A sample of a monatomic ideal gas undergoes the cyclic process \(1 \to 2 \to 3 \to 4 \to 1\) shown in the \(PV\) diagram. An ideal Carnot heat engine operates between the minimum and maximum absolute temperatures reached by the gas during this cycle. What is the ratio of the thermal efficiency of the cyclic engine to the thermal efficiency of the Carnot engine, \(\dfrac{\eta_{\text{cycle}}}{\eta_{\text{Carnot}}}\)?

A sample of a monatomic ideal gas undergoes the cyclic process \(1 \to 2 \to 3 \to 4 \to 1\) shown in the \(PV\) diagram. An ideal Carnot heat engine operates between the minimum and maximum absolute temperatures reached by the gas during this cycle. What is the ratio of the thermal efficiency of the cyclic engine to the thermal efficiency of the Carnot engine, \(\dfrac{\eta_{\text{cycle}}}{\eta_{\text{Carnot}}}\)?

![A Cartesian coordinate graph with a horizontal axis labeled V and a vertical axis labeled P. The horizontal axis has two labeled tick marks: V_0 and 2V_0. The vertical axis has two labeled tick marks: P_0 and 2P_0. A solid rectangular closed path connects four vertices in the first quadrant: state 1 at (V_0, P_0), state 2 at (V_0, 2P_0), state 3 at (2V_0, 2P_0), and state 4 at (2V_0, P_0). Clockwise arrows sit on the four segments: pointing upward from state 1 to state 2, pointing rightward from state 2 to state 3, pointing downward from state 3 to state 4, and pointing leftward from state 4 to state 1. Thin dashed vertical reference lines extend downward from 2V_0 to the horizontal axis, and thin dashed horizontal reference lines extend leftward from 2P_0 to the vertical axis. No other labels, curves, shading, or text appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790596761-gkh6vo.jpg)

- **A.** \(\dfrac{8}{39}\)
- **B.** \(\dfrac{4}{15}\)
- **C.** \(\dfrac{8}{27}\)
- **D.** \(\dfrac{4}{13}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123444/*