Define heat capacity. What are `C_(p) and C_(v)`? Show that `C_(p)-C_(v)=R.`

Explanation

Heat capacity of a substance is defined as the amount of heat required to raise its temperature through one degree.
The heat capacity at constant volume is represented by `C_(v).C_(v)` gives the measure of the change of internal energy ( E) of a system with temperature.
If heat is absorbed by the system at a constant pressure, heat capacity is represented by `C_(p)`. It is called heat capacity at constant pressure.
At constant pressure when heat is absorbed the volume of the gas increase. While the gas expands it does some work, for which extra heat amount is required. Hence `C_(p)` always greater than `C_(v)`. The `C_(p)` is equal to change in internal energy and the work done. The work done is equal to PV where V is the change in volume.
Relation between `C_(p) and C_(v)`:
For an ideal gas `H=E+PV`
Differentiating with temperature
`(dH)/(dT)=(dE)/(dT)+(d(PV))/(dT)`
For one mole of ideal gas PV = RT
`therefore (dH)/(dT)=(dE)/(dT)+(d(RT))/(dT)`
or `(dE)/(dT)+R(dT)/(dT)=(dE)/(dT)+R`
`therefore C_(p)=C_(v)+R or C_(p)-C_(v) or (C_(p))/(C_(v))=gamma`