Relation between CP and CV in gaseous systems
No external work is being done when a gas is heated at constant volume i.e. gas uses all the haet which is given to it for increasing its internal energy. Hence if temperature of one mole of a gas is raised through 1oC, the molar heat capacity is given itself at constant volume by increase in internal energy.
But when a gas is heated at constant pressure there will be expansion of gas i.e. increase in volume take place and some external work will b done. For this some extra heat is required which should be given to the gas to perform the external work.
Hence the Molar Heat capacity of a gas at constant pressure must be greater than Molar Heat capacity of a gas at constant volume.
CP > CV
When gas is heated through 1oC at constant pressure, the difference between these will give the work done by one mole of the gas in expansion.
As we know that at constant pressure work done by gas in expansion is given mathematically as:
w= P∆V
For one mole of an ideal gas:
PV = RT ……………………….. (1)
When temperature is raised by 1oC from T to T + 1 so that volume becomes V + ∆V, then
P (V + ∆V) = R (T +1) ………………………………… (2)
Subtracting equation (1) from equation (2), we get:
P∆V = R
Thus, At constant pressure work done by one mole of the an ideal gas in expansion when heated through 1oC is equal to R. hence,
CP – CV = R
Thus, the difference between molar heat capacity of a gas at constant pressure, CP and at constant volume, CV is equal to the gas constant R. i.e. 1.987 cal or 8.314 J
