Question

Using Maxwell relation derive the following Tds equation

Tds = c_p dT – T \(\left(\cfrac{∂v}{∂T}\right)_p\) dp

Answer 1

s = f (T, p)

Tds = T\(\left(\cfrac{∂s}{∂T}\right)_p\) dT + T \(\left(\cfrac{∂s}{∂p}\right)_T\) dp....(i) 

c_p = T\(\left(\cfrac{∂s}{∂T}\right)_p\)

Also, \(\left(\cfrac{∂s}{∂p}\right)_T\) = - \(\left(\cfrac{∂v}{∂T}\right)_p\) 

......Maxwell relation Substituting these in eqn. (i), we get

T_ds = c_p dT – T \(\left(\cfrac{∂v}{∂T}\right)_p\) dp