Physics · Kinetic Theory · NEET
At constant volume, all the heat you give only raises the temperature (internal energy). At constant pressure, the gas expands and pushes the piston, so it spends some heat doing work. To get the same 1 K rise you must supply that extra work-energy too. So Cp = Cv + (work done) = Cv + R. Cp is always bigger by exactly R.
A gas expands a lot when heated, so how much it expands depends on the process (constant volume vs constant pressure), giving two different heat values. A solid barely changes volume when heated, so the work done is tiny and both values are almost equal. That is why we quote one specific heat for solids and two for gases.
Mayer's relation Cp - Cv = R uses MOLAR specific heats (heat per mole per kelvin), and R is the universal gas constant = 8.314 J per mol per K. If you use specific heats per gram (small c), then cp - cv = r where r = R/M is the specific gas constant for that gas. Always match: molar with R, per-gram with R/M.
Specific heat capacity is heat per unit MASS (per gram or per kg) per kelvin. Molar specific heat is heat per unit MOLE per kelvin. NEET kinetic theory uses molar specific heats Cp and Cv, because formulas like Cv = (f/2)R and Cp - Cv = R are written per mole. Multiply molar value by number of moles to get total heat.
By equipartition, internal energy per mole is U = (f/2)RT where f is degrees of freedom. So Cv = dU/dT = (f/2)R, and Cp = Cv + R = (f/2 + 1)R. For monatomic f=3, diatomic (rigid) f=5, polyatomic f=6. From these you get gamma = Cp/Cv = 1 + 2/f.
The values of Cp/Cv for hydrogen, helium and another ideal diatomic gas X (whose molecules are not rigid but have an additional vibrational mode) are respectively equal to:
An ideal gas is made of polyatomic molecules. Each molecule has three translational, three rotational and f vibrational modes. If the ratio of heat capacities Cp/Cv of the gas is 8/7, then the value of f is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Cv is the heat needed to warm 1 mole of a gas by 1 K when the volume stays fixed. Cp is the heat needed to warm 1 mole by 1 K when the pressure stays fixed. Cp is larger because the gas also expands and does work.
For an ideal gas, Cp - Cv = R, where R = 8.314 J per mole per kelvin. It says the difference between the two molar specific heats always equals the universal gas constant.
Gamma (the adiabatic ratio) is gamma = Cp/Cv. It equals 5/3 for monatomic gases, 7/5 for rigid diatomic gases, and about 4/3 for polyatomic gases. Formula: gamma = 1 + 2/f.
For an ideal gas they are treated as constants that depend only on the degrees of freedom (the type of molecule). Real gases show small changes with temperature, but NEET uses the ideal fixed values.
Molar specific heats are measured in joules per mole per kelvin (J/mol/K). For example, Cv of a monatomic gas is (3/2)R = 12.47 J/mol/K.
