Q.What is molar heat capacity at constant pressure (Cp) and constant volume (Cv)? Derive the relationship between the two (Cp and Cv). OR What is enthalpy of formation? State and explain Hess's law of constant heat summation.

Cv = heat capacity at constant volume (heat goes entirely into raising internal energy); Cp = heat capacity at constant pressure (heat also does expansion work); for one mole of an ideal gas, Cp - Cv = R.

Molar heat capacity at constant volume (Cv): the amount of heat required to raise the temperature of one mole of a gas by 1 degree (Kelvin or Celsius) while keeping the volume constant. At constant volume, the gas does no work of expansion (w = -P.delta V = 0, since delta V = 0), so by the first law of thermodynamics all the heat supplied goes entirely into increasing the internal energy:

qv = delta U = Cv . delta T, i.e. Cv = (dU/dT) at constant V.

Molar heat capacity at constant pressure (Cp): the amount of heat required to raise the temperature of one mole of a gas by 1 degree while keeping the pressure constant. At constant pressure, the gas is free to expand, so the heat supplied is used both to raise the internal energy AND to do work of expansion against the constant external pressure. This heat exchanged at constant pressure equals the change in enthalpy:

qp = delta H = Cp . delta T, i.e. Cp = (dH/dT) at constant P.

Derivation of the relation between Cp and Cv (for one mole of an ideal gas):

By definition, enthalpy H = U + PV.

For one mole of an ideal gas, PV = RT (ideal gas equation), so:

H = U + RT

Differentiating both sides with respect to temperature T:

dH/dT = dU/dT + R …