Gay–Lussac (pressure) law – proportionality at constant volume According to the pressure law, the absolute pressure of a fixed mass of a perfect gas varies __________ with its absolute temperature when the volume is held constant.

Introduction / Context:
Gay–Lussac’s pressure law (also called Amontons’ law) describes how pressure changes with temperature for a gas in a rigid container. It is one of the fundamental ideal-gas relationships used in thermodynamics and gas dynamics.

Given Data / Assumptions:

• Fixed mass of a perfect gas.
• Volume constant (isochoric process).
• Absolute scales for pressure and temperature are used.

Concept / Approach:
From the ideal-gas equation pV = mRT, at constant V and m, pressure is proportional to temperature: p ∝ T. Thus, when temperature increases, pressure increases in direct proportion provided the gas behaves ideally and the container volume does not change.

Step-by-Step Solution:
1) Write pV = mRT.2) For fixed V and m, rearrange to p = (mR/V) * T.3) Coefficient (mR/V) is constant → p ∝ T.4) Therefore select “directly.”

Verification / Alternative check:
Plotting p against T for an isochoric process yields a straight line through the origin when using absolute units, consistent with the law and common laboratory observations with pressure gauges and gas bulbs.

Why Other Options Are Wrong:

• Indirectly/logarithmically/square root/independent: none match the linear proportionality dictated by the ideal-gas equation at constant volume.

Common Pitfalls:
Using Celsius instead of kelvin; a linear relation holds only with absolute temperature.

Final Answer:
directly