Physics · Thermodynamics · NEET
A gas can be heated in two very different ways. If you keep the volume fixed, all the heat goes into raising temperature (internal energy), so you use Cv. If you keep the pressure fixed, the gas expands and pushes the piston, so some heat is spent doing work and less is left to raise temperature; you need more heat, so you use Cp. Solids and liquids barely expand, so their two values are almost equal and we usually quote one number. Gases expand a lot, so Cp and Cv are clearly different.
Specific heat capacity s is heat per unit mass per kelvin, unit J per kg per K. Molar specific heat capacity C is heat per mole per kelvin, unit J per mol per K. For gases NEET almost always uses the molar form, written Cp and Cv, because gas amounts are counted in moles (n). The relation Cp - Cv = R only works for the molar versions, since R is a per-mole constant.
At constant volume the gas does no work (W = P times change in V = 0). By the first law Q = change in U + W, so Q = change in U. Dividing by the temperature change for 1 mole gives Cv = change in U divided by change in T. This is why Cv is the natural link between heat and internal energy, and why for any process the change in internal energy of an ideal gas is n Cv times change in T, even when the volume is not constant.
Start from Cv, which comes from degrees of freedom f: Cv = (f/2) R. Monatomic gas (like He, Ar) has f = 3, so Cv = 3/2 R and Cp = 5/2 R. Diatomic gas (like O2, N2) has f = 5 at NEET level, so Cv = 5/2 R and Cp = 7/2 R. Always add R to Cv to get Cp because Cp - Cv = R. Do not mix these up: the bigger number is always Cp.
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 Cp - Cv = R for one mole of an ideal gas, where R is the universal gas constant (about 8.31 J per mol per K). It shows the extra heat needed at constant pressure equals the work done during expansion for a 1 K rise.
No. For an ideal gas Cp is always greater than Cv by exactly R. Cp equals Cv only in the imaginary case where the gas cannot do work, which does not happen for real gases that expand.
The ratio Cp/Cv is called gamma, the ratio of specific heats or adiabatic index. It is 5/3 (about 1.67) for monatomic gases and 7/5 (1.4) for diatomic gases, and appears in the adiabatic equation P V^gamma = constant.
Always use Cv. For an ideal gas the change in internal energy is n Cv times the change in temperature for every process, including isobaric and adiabatic, because internal energy depends only on temperature.
Gas amounts are counted in moles, and equal moles of any ideal gas behave similarly. The per-mole form lets Mayer's relation Cp - Cv = R stay simple, since R is defined per mole.
