Definition – specific heat at constant volume (cv) Which statement correctly defines the specific heat at constant volume, cv, for a gas?

Introduction / Context:
Specific heats quantify how much heat is needed to change temperature under specified constraints. Distinguishing between cv and cp is fundamental for analyzing compression/expansion processes and for computing changes in internal energy and enthalpy of ideal gases.

Given Data / Assumptions:

• Unit mass basis is used (per kilogram); molar versions exist but are not implied unless stated.
• Temperature change is one degree (e.g., 1 K).
• Constraint is either constant volume (cv) or constant pressure (cp).

Concept / Approach:

By definition, cv is the heat required per unit mass per degree temperature rise at constant volume, while cp is the analogous quantity at constant pressure. For an ideal gas, increments obey du = cv * dT and dh = cp * dT, and the difference cp − cv = R (per unit mass). The constraint matters because at constant pressure the system does boundary work as it expands, requiring more heat than at constant volume to achieve the same temperature rise (cp > cv for gases).

Step-by-Step Solution:

Verification / Alternative check:

Thermodynamic tables list cv and cp separately; for air at room temperature, cv ≈ 0.718 kJ/kg·K and cp ≈ 1.005 kJ/kg·K, illustrating cp > cv.

Why Other Options Are Wrong:

Constant pressure and water-specific statements do not define cv on a per-mass-of-gas basis.“Any one of the above” is incorrect because only the constant-volume statement matches.

Common Pitfalls:

Mixing molar and mass-specific bases (cv,m vs cv); always check the basis and the path constraint.

Final Answer:

the amount of heat required to raise the temperature of unit mass of gas through one degree, at constant volume