Ideal gas in a rigid tyre:\nA car tyre (fixed volume 0.057 m³) contains air at 300 kPa and 300 K. After a long drive, the pressure rises to 330 kPa. Assuming air is ideal and Cv = 21 J/mol·K, find the change in internal energy of the air per mole (J/mol).

Introduction / Context:
This numerical tests thermodynamics of an ideal gas in a rigid container. When volume is fixed, any pressure change is directly linked to temperature change. For an ideal gas, the change in internal energy depends only on temperature via ΔU = n * Cv * ΔT. Since the tyre volume is constant, we can relate the final temperature to the pressure rise and compute the per-mole energy change relevant in automotive and process thermal analyses.

Given Data / Assumptions:

• Rigid vessel (tyre): V = constant = 0.057 m³ (volume value not needed for per-mole change).
• Initial state: P1 = 300 kPa, T1 = 300 K.
• Final state: P2 = 330 kPa, T2 = ?, V constant.
• Air behaves ideally; Cv = 21 J/mol·K.
• Composition and amount (n) remain fixed; no mass leaks.

Concept / Approach:
For an ideal gas at constant volume, P ∝ T (since P = n R T / V). Therefore, T2 = T1 * (P2/P1). The internal energy change per mole is Δu = Cv * ΔT. No work is done (W = 0) because the volume is fixed; any heat transfer shows up entirely as a change in internal energy.

Step-by-Step Solution:
Relate T and P at constant V: T2/T1 = P2/P1 = 330/300 = 1.10.Compute T2: T2 = 300 * 1.10 = 330 K.Find ΔT: ΔT = 330 − 300 = 30 K.Per-mole internal energy change: Δu = Cv * ΔT = 21 * 30 = 630 J/mol.Match with options: 630 J/mol.

Verification / Alternative check:
Use the ideal-gas energy relation U = n * Cv * T. The ratio method is consistent with P ∝ T at fixed V and n. The numeric increase is modest, fitting a 10% pressure rise.

Why Other Options Are Wrong:
380, 760, and 880 J/mol correspond to incorrect ΔT assumptions or misuse of Cp instead of Cv, or failure to apply the P–T proportionality at constant volume.

Common Pitfalls:
Using Cp rather than Cv; forgetting absolute temperature; incorrectly incorporating tyre volume when a per-mole change is requested; assuming work at constant volume.

Final Answer:
630