Ideal-Gas Internal Energy and Temperature

Introduction

For an ideal gas of fixed composition, internal energy depends on temperature rather than volume or pressure independently. Heating it raises U; cooling lowers U. This fact simplifies first-law problems and explains why isothermal ideal-gas expansions have zero internal-energy change even when heat and work are nonzero.

Core explanation

At constant volume, pressure-volume work is zero. If no other work occurs, the first law gives ΔU = q V. The molar constant-volume heat capacity C V,m describes how much internal energy changes per mole for a small temperature change: dU = n C V,m(T)dT for a fixed amount of ideal gas. If C V,m is approximately constant over the temperature interval, integrate to obtain ΔU ≈ nC V,mΔT. Temperature differences may be written in kelvin or degrees Celsius because the interval sizes are equal, but absolute-temperature formulas elsewhere require kelvin.

The heat capacity may depend on temperature, especially when molecular rotational or vibrational modes become active. For a larger temperature span, use the integral ΔU = n∫ from T i to T f C V,m(T)dT rather than treating C V,m as an exact constant. A problem usually supplies a heat-capacity value or states an approximation. Do not assume all ideal gases have the same C V,m; molecular structure affects accessible energy modes.

Because ideal-gas U depends only on T, ΔU = 0 for any isothermal change between states of the same fixed-composition ideal gas. Reversible isothermal expansion has nonzero negative work balanced by positive heat. Insulated free expansion has q = w = 0. Their common ΔU follows from the equal endpoint temperatures, not from identical paths.

The Joule relation is a property of the ideal-gas model. Real gases have intermolecular interactions, so their internal energy can depend on volume and temperature. A real gas expanding without heat or work may change temperature. Always check whether the problem explicitly uses an ideal gas and whether composition stays fixed; reactions change chemical internal energy even at the same temperature.

Heat capacity C V,m is defined through temperature response at constant volume, but its use in ΔU = nC V,mΔT for an ideal gas is not limited to paths at constant volume. Since U is a state function of temperature alone in this model, the same endpoint ΔU applies along any path. The actual heat q along another path can differ because work differs.

For a monatomic ideal gas in a simple classical temperature range, C V,m = 3R/2. For diatomic gases at ordinary temperatures, a value near 5R/2 is often used when rotational modes are active and vibration is not significantly excited. These are model approximations, not universal constants across all temperatures and molecules. Use the value specified by the problem when precision matters.

Step-by-step reasoning

1. Confirm ideal gas, fixed amount and composition. 2. Read T i and T f and compute ΔT. 3. Select a supplied or justified C V,m. 4. Calculate ΔU = nC V,mΔT or integrate if C V,m varies. 5. Use ΔU = q + w to find a missing transfer for the actual path.

Visual explanation

Draw a U-versus-T line rising for an ideal gas over a limited range. Plot two points at the same T but different volumes at the same U height. Draw an arrow along the line for heating, labelled nC V,mΔT, and a horizontal arrow for an isothermal volume change with ΔU = 0.

Real-world analogy

A scoreboard may depend only on points scored, not on the route players took around the field. Ideal-gas U depends on temperature state, while heat and work reveal the route by which that temperature was reached. The analogy is limited because molecular energy has specific degrees of freedom.

Real-world example

Compressing an ideal gas in an insulated piston-cylinder does positive work on it. With q = 0, ΔU > 0, so its temperature rises. The first law gives the energy change, and heat capacity converts that change into a predicted temperature rise.

Why?

Why can C V,m be used for ΔU on a non-constant-volume ideal-gas path? U is a state function of T alone for the ideal gas, so the same temperature endpoints give the same ΔU regardless of how volume changed.

Common misconception

“An ideal gas has zero internal energy because particles have no attractions.” Ideal-gas particles still have kinetic and internal molecular energy. The model states that U depends on temperature, not that U or ΔU always vanishes.

Worked example

Two moles of an ideal gas with C V,m = 20.8 J mol⁻¹ K⁻¹ warm from 300 K to 330 K. ΔT = +30 K, so ΔU = (2)(20.8)(30) = +1248 J. If 1800 J of heat enters along the actual path, the first law gives w = ΔU − q = 1248 − 1800 = −552 J. The gas did work on the surroundings while its internal energy rose.

Quick check

1. What is ΔU for a fixed-composition ideal gas at constant temperature? Answer: Zero, regardless of the path between the stated ideal-gas endpoints.

Exam focus

Use heat capacity units consistently and keep q separate from ΔU unless the process and work conditions justify equality. Note that C V,m can depend on temperature and that real gases are not exactly ideal.

Advanced insight

In kinetic theory, a monatomic ideal gas's translational motion gives U = 3nRT/2 in the classical model, hence C V,m = 3R/2. Additional molecular rotations and vibrations alter the heat capacity. Quantum energy spacing determines when those modes contribute appreciably.

Summary

For a fixed-composition ideal gas, U depends only on T. Over a range where C V,m is nearly constant, ΔU ≈ nC V,mΔT; for isothermal change, ΔU = 0. Heat and work along the actual path still require first-law bookkeeping.

Practice questions

1. Find ΔU for 1.0 mol of an ideal gas with C V,m = 25 J mol⁻¹ K⁻¹ cooling by 20 K. Answer: ΔU = (1.0)(25)(−20) = −500 J. 2. An ideal gas isothermally expands while doing 700 J of work on surroundings. What is q? Answer: w = −700 J and ΔU = 0, so q = +700 J. 3. Why might a real gas change temperature in insulated free expansion? Answer: Intermolecular interactions can make its internal energy depend on volume as well as temperature.