Derive the relationship between `C_(p) and C_(v)`.

Explanation

Let `C_(v)` is the heat capacity at constant volume and `C_(p)` is the heat capacity at the constant pressure.
At constant volume, `q_(v) = C_(v) Delta T = Delta U`
At constant pressure `q_(p) = C_(p) = Delta T = Delta H`
The difference between `C_(p) and C_(v)` can be derived for an ideal gas as follows :
For a mole of an ideal gas `Delta H = Delta U = Delta(pV)` = `Delta U + Delta(Rt)`
= `Delta U + R Delta T`
`therefore Delta H = Delta U + R Delta T`
By substituting the vlaue of `Delta H and Delta U`, we get
`C_(P) Delta T = C_(v) Delta T + R Delta T`
`C_(p) = C_(v) + R`
`C_(p) - C_(v) = R`
This relationship is know as Mayer.s relationship. This tell us that `C_(p)` is greater than `C_(v)` by gas constant R.