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Count Degrees of Freedom to Get Cv, Cp and Gamma

Every question on gas heat capacities reduces to one number: the degrees of freedom f of the molecule. From f you get the molar heat capacities, their ratio γ, and the internal energy, and the relation between Cp and Cv is fixed by R.

NSEJS · Thermal Physics · 8 linked questions · 100% coverage

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From f to Cv, Cp and γ

For an ideal gas,

Cv = (f/2) R, Cp = Cv + R = (f/2 + 1) R and γ = Cp/Cv = 1 + 2/f

A monatomic gas has f = 3 (Cv = 3R/2, γ = 5/3), a diatomic gas has f = 5 (Cv = 5R/2, γ = 7/5) and a non-linear triatomic gas has f = 6 (Cv = 3R, γ = 4/3), all with vibration neglected.

If a gas has R/

Cv = 0.4 then Cv = 2.5 R, so f = 5 and the molecules are diatomic

For an ideal gas the specific heat does not depend on temperature in this model.

Heat used for work at constant pressure

When a gas is heated at constant pressure, the heat splits into the change in internal energy and the work done: Q = n Cp ΔT, W = n R ΔT, so W/Q = R/Cp = (γ - 1)/γ.

For γ = 4/3 this gives (1/3)/(4/3) = 25% of the heat used for external work

Cp - Cv = R holds per mole;

per gram, cp -

cv = R/M, so a light gas has a larger gram-specific-heat difference: hydrogen (M = 2) against oxygen (M = 32) gives a = 16 b

Internal energy and heat at constant volume

The internal energy of an ideal gas is U = (f/2) n R T for each component, so a mixture adds up component by component: 2 mol of oxygen (f = 5) and 4 mol of argon (f = 3) at temperature T have U = (5/2)(2)RT + (3/2)(4)RT = 11 RT.

The heat to warm a gas at constant volume

Q = n Cv ΔT, with n = m/M: 1 g of helium is 1/4 mol, so Q = (1/4)(3R/2)(T2 - T1) = (3/8) R (T2 - T1)

OlympIQ exam lens

✓f = 3 (monatomic), 5 (diatomic), 6 (non-linear triatomic), vibration neglected
✓Cv = (f/2)R, Cp = Cv + R, γ = 1 + 2/f
✓Fraction of constant-pressure heat that does work = R/Cp = (γ - 1)/γ
✓Per gram, cp - cv = R/M, so lighter gases have larger values
✓U = (f/2) nRT for each gas in a mixture; Q at constant volume = n Cv ΔT

OlympIQ trap

Do not mix per-mole and per-gram quantities. Cp - Cv = R holds per mole, but per gram it is R/M, which is why hydrogen's difference is sixteen times oxygen's, not equal to it.

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