Physics · Kinetic Theory · NEET
When you heat a gas at constant volume, all the heat goes only into raising the internal energy (temperature). When you heat the same gas at constant pressure, the gas expands, so part of the heat also does work pushing the surroundings. Therefore you must supply more heat per degree at constant pressure, making Cp larger than Cv. The exact extra amount for one mole is R.
R is the universal gas constant, R = 8.314 J per mole per kelvin (about 2 cal per mole per kelvin). It appears here because for one mole of an ideal gas the work done during a 1 kelvin rise at constant pressure equals R. That work is the whole difference between Cp and Cv.
Yes. NCERT states clearly that Cp - Cv = R is true for any ideal gas, whether mono, di or polyatomic. Only the individual values change: monatomic Cv = 3/2 R and Cp = 5/2 R; rigid diatomic Cv = 5/2 R and Cp = 7/2 R. In every case the difference is exactly R.
In the form Cp - Cv = R, both Cp and Cv are MOLAR heat capacities (per mole), so it is written for one mole. If you use specific heats per gram (small cp and cv), the relation becomes cp - cv = R/M, where M is the molar mass. Mixing the two forms is the most common mistake.
The first law gives dQ = dU + dW. At constant volume dW = 0, so dQ = dU and that heat only raises temperature. At constant pressure the gas expands, so dW = P dV = R dT for one mole. This work term R is exactly what makes Cp bigger than Cv.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
For one mole of an ideal gas, Cp - Cv = R, where Cp and Cv are the molar heat capacities at constant pressure and constant volume, and R is the universal gas constant (8.314 J per mole per kelvin).
It equals R = 8.314 J per mole per kelvin, which is about 2 calories per mole per kelvin. This value is the same for every ideal gas.
The extra heat at constant pressure only pays for the work of expansion, P dV = R dT per mole, which depends on the gas law PV = RT and not on the internal structure of the molecule. So the difference is always R.
Start from the first law dQ = dU + P dV. At constant volume Cv = dU/dT. At constant pressure Cp = dU/dT + P(dV/dT). For one mole PV = RT gives P(dV/dT) = R, so Cp = Cv + R, i.e. Cp - Cv = R.
Gamma = Cp/Cv is the ratio of specific heats. Combined with Cp - Cv = R, you get Cv = R/(gamma - 1) and Cp = gamma R/(gamma - 1). For a monatomic gas gamma = 5/3, for a rigid diatomic gas gamma = 7/5.
