Lecture-Notes (Thermodynamics) - niser

36 CHAPTER 4. ENTROPY AND SECOND LAW OF THERMODYNAMICS
Using (4.6), we get
∂P
∂T V
where
= −
α = 1
∂V
V ∂T P
kT = − 1
∂V
V ∂P T
kS = − 1
∂V
V ∂P
∂z
= −
∂x y
∂f/∂x
∂f/∂z .
1
(∂T/∂V ) P (∂V/∂P) T
S
= − (∂V/∂T) P
(∂V/∂P) T
= α
kT
(coefficient of thermal expansion),
(isothermal compressibility),
(adiabatic compressibility).
We substitute in
∂U
δQ = TdS = CV dT + + P dV
∂V T
expressions (4.7) and (4.5) and get the following useful relations.
, (4.7)
1) The absorbed heat is expressed in terms of directly measurable coefficients as
where T and V are independent variables.
2) If T and P are used as independent variables, then
δQ = TdS = CV dT + α
TdV , (4.8)
kT
TdS = CPdT − αTV dP (4.9)
3) If V and P are used as independent variables, dT can be rewritten in terms of dV
and dP as
∂T ∂T
dT = dV + dP =
∂V P ∂P V
1 kT
+ dP .
αV α
The corresponding TdS equation is then
TdS = CP
dV +
αV
CPkT
α
− αTV dP . (4.10)
There is also an important connection between CP and CV (which follows from eq.
(4.8) and (4.9)):
CP − CV = α2
TV = −T
kT
∂V 2
∂T P
∂V
∂P T
> 0 . (4.11)