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title: "A sample of an ideal gas is initially in state \\(i\\) with pressure \\(P_0\\) and volume \\(V_0\\). The gas expands to a final volume of \\(2V_0\\) along three different thermodynamic paths shown in the \\(PV\\) diagram: Path 1 is an isobaric expansion, Path 2 is an isothermal expansion, and Path 3 is an adiabatic expansion. Which of the following statements correctly compares the work done by the gas along these paths and provides the correct physical justification?"
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url: "https://nerd-notes.com/ubq/116238/"
date_modified: "2026-08-03T11:46:25+00:00"
---

# A sample of an ideal gas is initially in state \(i\) with pressure \(P_0\) and volume \(V_0\). The gas expands to a final volume of \(2V_0\) along three different thermodynamic paths shown in the \(PV\) diagram: Path 1 is an isobaric expansion, Path 2 is an isothermal expansion, and Path 3 is an adiabatic expansion. Which of the following statements correctly compares the work done by the gas along these paths and provides the correct physical justification?

A sample of an ideal gas is initially in state \(i\) with pressure \(P_0\) and volume \(V_0\). The gas expands to a final volume of \(2V_0\) along three different thermodynamic paths shown in the \(PV\) diagram: Path 1 is an isobaric expansion, Path 2 is an isothermal expansion, and Path 3 is an adiabatic expansion. Which of the following statements correctly compares the work done by the gas along these paths and provides the correct physical justification?

![A PV diagram with pressure P on the vertical axis and volume V on the horizontal axis. The origin represents zero volume and pressure. On the horizontal axis, two volume values are labeled: V_0 and 2V_0. On the vertical axis, one pressure value is labeled: P_0. An initial state point labeled i is located at coordinates (V_0, P_0). Three paths originate from state point i and extend to a final volume of 2V_0. Path 1 is a horizontal line segment from (V_0, P_0) to (2V_0, P_0) with an arrow pointing right, labeled Path 1. Path 2 is a smooth curve bending downward from (V_0, P_0) to a point at volume 2V_0 with pressure equal to 0.5 P_0, with an arrow pointing down and right, labeled Path 2. Path 3 is a steeper curve bending downward below Path 2 from (V_0, P_0) to a point at volume 2V_0 with pressure less than 0.5 P_0, with an arrow pointing down and right, labeled Path 3. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785757585-oGOQHD.jpg)

- **A.** The work done by the gas is greatest for Path 3 because no heat enters or leaves the system, allowing all thermal energy to be converted directly into mechanical work.
- **B.** The heat added to the gas is greatest for Path 2 because the gas must absorb heat continuously to keep its temperature constant during expansion.
- **C.** The change in internal energy of the gas is zero for all three paths because the gas undergoes the exact same volume expansion from \(V_0\) to \(2V_0\).
- **D.** The work done by the gas is greatest for Path 1 and least for Path 3 because Path 1 maintains the highest pressure throughout the expansion, resulting in the greatest area under the \(PV\) curve.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/116238/*