Identifying the correct law: For a perfect (ideal) gas, the change in internal energy is directly proportional to the change in temperature. Which named law states this relationship?

Introduction / Context:
For ideal gases, internal energy is a function of temperature only. This foundational statement links microscopic molecular energy to macroscopic thermal behavior and is attributed to Joule's experiments and conclusions, commonly summarized as Joule's law for ideal gases.

Given Data / Assumptions:

• Working substance is a perfect (ideal) gas.
• Internal energy is denoted by u; temperature by T.
• Specific heats are finite and positive.

Concept / Approach:

The law states: for an ideal gas, u = u(T) only. Hence any change in internal energy depends solely on temperature change, not on pressure or volume separately. Mathematically, du = cv * dT for an ideal gas. If temperature does not change, internal energy does not change, regardless of volume or pressure changes alone.

Step-by-Step Solution:

Verification / Alternative check:

Free expansion of an ideal gas shows no temperature change and thus no change in internal energy despite a large volume change, confirming u depends on T only.

Why Other Options Are Wrong:

Boyle's law: P * V = constant at constant T.Charles' law: V ∝ T at constant P.Gay-Lussac law: P ∝ T at constant V.Avogadro's law: Equal volumes at same T and P contain equal numbers of molecules.

Common Pitfalls:

Assuming internal energy depends on pressure or volume for an ideal gas; that is true for real gases but not for ideal gases.

Final Answer:

Joule's law