Define two principal specific heats of a gas . Which is greater and why ?

Explanation

(i) Specific heat of a gas at constant volume `(C_(v))` : It is defined as the amount of heat energy required to raise the temperature of one gram of gas through `1^(@)C` or 1 K , when volume of the gas is kept constant .
It is measured in `"cal.g"^(-1).K^(-1)" or " j.g^(-1).K^(-1)`.
(ii) Specific heat of a gas at constant presure `(C_(p))` : It is defined as the amount of heat energy required to raise the temperature of one gram of gas through `1^(@)C` or 1 K , when pressure of the gas is kept constant .
It is also measured in `"cal.g"^(-1).K^(-1)" or " j.g^(-1).K^(-1)`.
Out of the two principal specific heats of a gas `C_(p)gtC_(v)` . This can be justified as follows :
(a) When heat is given to gas at constant volume , it is only used increasing the internal energy of the gas . i.e in raising the temperature of the gas and no heat is soent in the expansion of the gas .
(b) When heat is given to a gas at constant pressure , it is spent in two ways :
(i) Part of the heat is increasing the internal energy of the gas and hence the temperature of the gas .
(ii) Remaining amount of heat is used in doing work i.e., in the expansion of the gas against the exteranl pressure.
Therefore to raise the temperature of 1 mole of gas through `1^(@)C` or 1 K , more heat is required at constant pressure `(C_(p))` than at constant volume `(C_(p))`.
Hence , `C_(p)gtC_(v)`.