PHYSICAL CHEMISTRY IN BRIEF

CHAP. 3: FUNDAMENTALS OF THERMODYNAMICS [CONTENTS] 94

Example

Derive the relation for the dependence of the internal energy of an ideal gas on temperature and

amount of substance if

Cp◦m=a+bT+cT^2.

Solution

In equation (3.72) we take into account thatCV◦(T) =nCV◦m, wherenis the amount of substance,

and that for an ideal gas the Mayer relation (3.67) applies. We obtain

U(T) = U(T 1 ) +n

∫T

T 1

(Cp◦m−R) dT

= U(T 1 ) +n

[

(a−R)(T−T 1 ) +

b

2

(T^2 −T 12 ) +

c

3

(T^3 −T 13 )

]

3.5.3 Enthalpy.

3.5.3.1 Temperature and pressure dependence for a homogeneous system

By integrating the total differential (3.52) with respect to the general prescription (3.30) we
obtain

H(T, p) =H(T 1 , p 1 ) +

∫T

T 1

Cp(T, p 1 ) dT+

∫p

p 1



V−T

(

∂V

∂T

)

p



dp. (3.74)

Ifp 1 = 0, the system in its initial state behaves as an ideal gas, and equation (3.74) rearranges
to

H(T, p) =H◦(T 1 ) +

∫T

T 1

C◦p(T) dT+

∫p

0



V−T

(

∂V

∂T

)

p



dp. (3.75)

If pressure stays unchanged during a thermodynamic process, equation (3.74) simplifies to

H(T, p) =H(T 1 , p) +

∫T

T 1

Cp(T, p) dT , [p]. (3.76)