Heat capacity (denoted by C) is the increment in heat dq required to increase the temperature by an
amount dT.
There are several sources which can contribute to heat capacity. Specific Heat Capacity is simply the
heat capacity under a specific condition, for example no change in volume. This is expressed by
∂Q
∂S
C x=
=T
where x denotes the constant variable
∂T x
∂T x
Mayer relation is the connection between specific heat capacity under constant volume C V and
specific heat capacity under constant pressure C p for ideal gas and equals
C P – C V =R
   
CV
U T , V => dU =
   
∂U
∂T
dT 
V
∂U
∂V
dV
T
dU =dQ – dW =dQ – PdV .
 
 
dQ=
CP :
∂U
∂T
dQ=dU =
under V=const.
∂U
∂T
V
 
 
 
     
   
∂U
∂T
P
P
C P=
∂V
∂T
∂Q
∂T
P
dT 
P
∂V
∂T
V T , P ==> dV =
∂U
∂T
dT 
∂U
∂T
dQ=dU dW =
P
P
P
dQ
dT
∂U
∂T
∂U
∂P
P
P
∂U
∂P
V
∂U
∂T
V
dP
dPPdV
T
∂V
∂P
dP
T
∂V
∂P
P
T
∂V
∂T
=
T
∂U
∂P
dT 
dT 
=
   
 
 
 
  
 
 CV =
dT
U T , P => dU =
dQ =
dT
V
dP
T
P
side note :
H =U PV
==>
     
∂H
∂T
=
P
∂U
∂T
P
P
∂V
∂T
=C P
For an ideal gas, U only depends on T. Therefore
and : C P=C V P
 
∂V
∂T
 
∂U
∂T
=
P
dU
=CV
dT
P
PV = RT  PdV VdP=RdT  P
and we get Mayer relation:
 dH =dU  PdV VdP 
P
 
dV
dP
∂V
V
=R  P
dT
dT
∂T
C P=C V R or C P – C V =R
=R
P