# Thermodynamics problem

1. Jun 11, 2007

### neelakash

1. The problem statement, all variables and given/known data

(∂s/∂T)_x>0 for all processes where x is an arbitrary intensive or extensive variable of of the system.

2. Relevant equations
3. The attempt at a solution

I do understand that we have to exploit some property of the properties of the system.Say all of them are exact differential or so...But cannot understand how to do this Identity...

2. Jun 11, 2007

### neelakash

Hope this makes sense...Please check it.

First, we recall the definition of "Specific Heat At Constant X", where
"X" is an arbitrary intensive variable (like pressure) or extensive
variable (like volume):

{Specific Heat At Constant X} = c_X = (1/m)*(dQ/dT)X

----> dQ = m*(c_X)*dT ..... for constant X

where "m" is system mass. Thus, from the definition of entropy &
specific entropy (s = S/m), and the Second Law Of Thermodynamics
(for cyclic process), we have:

dS = dQ/T > 0

----> dS = {m*(c_X)*dT}/T > 0 ..... for constant X

----> ds = {(c_X)*dT}/T > 0 (<--- dividing by m, & s = S/m)

----> (∂s/∂T)_X = (c_X)/T > 0 (<--- dividing by dT & at const X)

----> (∂s/∂T)_X > 0