Two Heat Capacities
For a gas, 'heat per degree' depends on how you heat it.
Lesson at a glanceShow the key points
- s = ΔQ/(mΔT) (27.1). ΔQ = ΔU + ΔW and ΔW depends on the path, so s depends on the process.
- s_V at constant volume and s_p at constant pressure (27.2–27.3).
- Molar: C_V = (ΔQ/nΔT)_V, C_p = (ΔQ/nΔT)_p (27.4–27.5), J K⁻¹ mol⁻¹.
- Solids and liquids barely expand, so for them the process hardly matters.
Ask “how much heat warms this gas by one degree?” and the honest answer is “it depends how you do it”. Gases need a process attached to their heat capacity.
Heat capacity depends on the process
The specific heat capacity is the heat per unit mass per degree of temperature rise:
Take a gas from state a to state b. The temperature change is fixed by the end states, but by the first law ΔQ = ΔU + ΔW, and the work ΔW depends on the path. A path with more area under it needs more heat for the same ΔT. So for a gas, s depends on the process.
Two processes are singled out:
Per mole instead of per kilogram, the molar heat capacities:
0.56 g of nitrogen (28 g/mol) in a rigid flask is warmed from 20°C to 40°C. Cᵥ = 20.8 J K⁻¹ mol⁻¹. How much heat is needed?
A 0.50 m³ tank holds helium at 1.0 × 10⁵ Pa and 300 K. Cᵥ = 12.5 J K⁻¹ mol⁻¹. How much heat raises it to 350 K (ignore the tank’s expansion)?
- 1For a gas, the heat for a given ΔT depends on the process (through the work done).
- 2Specific heats s_V and s_p; molar heat capacities C_V and C_p.
- 3For solids and liquids the difference is small.
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