Heating at constant volume (closed system gas): For a gas heated at constant volume from T1 to T2, which expression gives the heat supplied? (m = mass, cv = specific heat at constant volume, cp = specific heat at constant pressure, R = gas constant)

Introduction / Context:
For closed systems undergoing heating with no boundary work, the first law reduces to a direct relation between heat transfer and change in internal energy. At constant volume, boundary work is zero, and the temperature rise translates to internal energy rise determined by cv.

Given Data / Assumptions:

• Ideal or calorically perfect gas approximation.
• Volume held constant during the process (W = 0).
• Temperature increases from T1 to T2.

Concept / Approach:
First law for a closed system: Q − W = ΔU. At constant volume, W = ∫p dV = 0, so Q = ΔU. For an ideal gas, ΔU = m * cv * (T2 − T1). Therefore, the heat supplied equals mcv(T2 − T1). The cp formula applies to constant-pressure processes because enthalpy change ΔH = m * cp * (T2 − T1) equals Q at constant pressure (neglecting kinetic/potential changes).

Step-by-Step Solution:

Verification / Alternative check:
Since enthalpy H = U + pV and pV changes with temperature at constant volume for ideal gases, cp is not the correct proportionality for Q at constant volume; cv is.

Why Other Options Are Wrong:

• mR(T2 − T1): R relates cp and cv (cp − cv = R); not equal to Q at constant volume.
• mcp(T2 − T1) and mcp(T2 + T1): Apply to constant-pressure heating or are dimensionally or physically incorrect.

Common Pitfalls:
Confusing cp and cv; forgetting that boundary work vanishes at constant volume so heat equals internal energy change.

Final Answer:
mcv(T2 - T1)