At constant volume with a fixed number of moles of gas, why does the pressure increase when temperature rises?

Introduction / Context:
For a closed, rigid container (constant volume) holding a fixed amount of an ideal or near-ideal gas, the relation P ∝ T emerges from the microscopic kinetic theory of gases. Understanding the molecular basis of this macroscopic behavior is crucial in thermodynamics and transport phenomena.

Given Data / Assumptions:

• Volume is constant; number of moles is fixed.
• Temperature increases by external heating.
• Gas behaves ideally or near-ideally over the range considered.

Concept / Approach:
Gas pressure arises from molecular impacts on container walls. The ideal-gas kinetic theory links temperature to the average translational kinetic energy. As temperature increases, molecules move faster on average, striking the walls more forcefully and somewhat more frequently, increasing momentum transfer per unit time and therefore the pressure at fixed volume and moles.

Step-by-Step Solution:

Verification / Alternative check:
From the ideal gas law, PV = nRT. At constant V and n, P ∝ T. Kinetic theory gives P = (2/3)(N/V)⟨E_trans⟩; since ⟨E_trans⟩ ∝ T, the proportionality is confirmed.

Why Other Options Are Wrong:

• Decrease in mean free path: Not the fundamental reason; mean free path often increases with temperature at fixed pressure; here pressure itself changes.
• Increased collision rate: A consequence of higher speeds, but the primary cause is increased average kinetic energy.
• Increase in molecular attraction: Intermolecular attractions do not cause the pressure rise; if anything, attractions reduce pressure from ideality.
• Generation of additional moles: Mole number is fixed by the stem.

Common Pitfalls:
Mixing cause and effect: collision rate rises because speed rises; speed rise is the temperature effect that fundamentally drives pressure upward.

Final Answer:
increase in average molecular speed