PHYSICAL CHEMISTRY IN BRIEF

CHAP. 3: FUNDAMENTALS OF THERMODYNAMICS [CONTENTS] 93

The volumeV 1 is often chosen as large as would allow for a system in its initial state to behave
as an ideal gas. In the limitV 1 → ∞at a constant amount of substance, equation (3.68)
rearranges to

U(T, V) =U◦(T 1 ) +

∫T

T 1

CV◦(T) dT+

∫V

∞

[

T

(

∂p

∂T

)

V

−p

]

dV. (3.69)

If the volume is constant, equation (3.68) simplifies to

U(T, V) =U(T 1 , V) +

∫T

T 1

CV(T, V) dT , [V]. (3.70)

For an isothermal process, equation (3.68) becomes

U(T, V) =U(T, V 1 ) +

∫V

V 1

[

T

(

∂p

∂T

)

V

−p

]

dV , [T]. (3.71)

3.5.2.2 Ideal gas.

For an ideal gas, the partial derivative of internal energy with respect to volume is zero, as
follows from (3.55) [see also the example in section3.4.5]. The internal energy of an ideal gas
is thus a function of temperature only (isothermal processes in an ideal gas are processes at
constant internal energy). Equation (3.70) for an ideal gas is

U◦(T) =U◦(T 1 ) +

∫T

T 1

CV◦(T)dT. (3.72)

3.5.2.3 Changes at phase transitions

The changes of internal energy during a crystalline transformation, melting and boiling (∆crystU,∆fusU
and ∆vapU) are calculated as follows

∆U = ∆H−p∆V , [T, p], (3.73)

where ∆Hand ∆V are changes of enthalpy and volume at the respective phase transition.