Physics · Thermodynamics · NEET
At constant volume (Cv) all the heat you give goes into raising internal energy U, so temperature rises fast with little heat. At constant pressure (Cp) the gas expands, so part of the heat is used to do work W = P dV against the surroundings and only the rest raises U. So for the same 1 kelvin rise you must supply MORE heat at constant pressure. Hence Cp is greater than Cv.
Start from the first law for 1 mole: dQ = dU + P dV. At constant volume, dV = 0, so Cv = dU/dT. At constant pressure, Cp = dU/dT + P(dV/dT). From the ideal gas law PV = RT, at constant P we get P dV = R dT, so P(dV/dT) = R. Therefore Cp = Cv + R, i.e. Cp - Cv = R. The extra term R is exactly the expansion work per mole per kelvin.
The clean form Cp - Cv = R is for MOLAR specific heats (heat per mole per kelvin). For n moles the total heat difference is nR, but the molar quantities still obey Cp - Cv = R. In NEET numericals always check whether Cp, Cv are molar (units J per mol per K) before applying it.
Cp is greater than Cv for almost all substances, but for solids and liquids the difference is very tiny because they barely expand when heated, so the expansion work is nearly zero. The clean result Cp - Cv = R is special to an ideal gas. For NEET, use Cp - Cv = R only for ideal gases.
No, not for a normal gas heated in the usual way. Because expansion at constant pressure always needs extra work, Cp is always greater than Cv. The difference R is positive, so Cp > Cv always holds for an ideal gas.
The volume (V) of a monatomic gas varies with its temperature (T) as a straight line through the origin from A to B (an isobaric process). The ratio of the work done by the gas to the heat absorbed by it, when it goes from A to B, is
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Mayer's relation states that for an ideal gas the molar specific heat at constant pressure minus the molar specific heat at constant volume equals the universal gas constant: Cp - Cv = R, where R = 8.314 J per mol per K.
Because at constant pressure the gas does extra expansion work, so more heat is needed for the same temperature rise than at constant volume where no work is done.
For an ideal gas Cp - Cv = R = 8.314 J per mol per K, which is about 2 calories per mol per K.
It is exact for an ideal gas. Real gases follow it very closely at low pressure and high temperature, where they behave nearly ideally.
Cv = (3/2)R and Cp = (5/2)R, so their difference is exactly R, and their ratio gamma = Cp/Cv = 5/3.
