JEE Main & Advanced Physics Kinetic Theory of Gases Mayer's Formula          

(1) Out of two principle specific heats of a gas, \[{{C}_{P}}\] is more than \[{{C}_{V}}\] because in case of \[{{C}_{V}},\] volume of gas is kept constant and heat is required only for raising the temperature of one gram mole of the gas through \[{{1}^{o}}C\] or 1 K. Hence no heat, what so ever, is spent in expansion of the gas.

It means that heat supplied to the gas increases its internal energy only i.e. \[{{(\Delta Q)}_{V}}=\Delta U=\mu {{C}_{V}}\Delta T\]          ...(i)

(2) While in case of \[{{C}_{P}}\] the heat is used in two ways

(i) In increasing the temperature of the gas by \[\Delta T\]

(ii) In doing work, due to expansion at constant pressure \[(\Delta W)\]

So           \[{{(\Delta Q)}_{P}}=\Delta U+\Delta W=\mu \,{{C}_{P}}\Delta T\]                           ...(ii)

From equation (i) and (ii)    \[\mu \,{{C}_{P}}\Delta T-\mu \,{{C}_{V}}\Delta T=\Delta W\]

\[\Rightarrow \] \[\mu \,\Delta T({{C}_{P}}-{{C}_{V}})=P\Delta V\] \[\Rightarrow \] \[{{C}_{P}}-{{C}_{V}}=\frac{P\Delta V}{\mu \,\Delta T}=R\]

[For constant pressure, \[\Delta W=P\Delta V\] also from \[PV=\mu RT,\] \[P\Delta V=\mu R\Delta T\]]

This relation is called Mayer?s formula and shows that \[{{C}_{P}}>{{C}_{V}}\] i.e. molar specific heat at constant pressure is greater than that at constant volume.