278 4 Thermodynamics and Statistical Physics
4.39 By Maxwell’s first equation
(
∂S
∂V
)
T=
(
∂P
∂T
)
V(1)
dS=dU+PdV
T(2)
using (2) in (1)
(
∂U
∂V)
T=T
(
∂P
∂T
)
−P
For perfect gases,P=RT
( V
∂U
∂V
)
T=
RT
V
−P= 0
Thus, temperature remaining constant, the internal energy of an ideal gas
is independent of the volume.4.40
dP
dT=
L
T(ν 2 −ν 1 )dT=T
L
(ν 2 −ν 1 )dP=
373(1677−1)(2× 106 )
546 × 4. 2 × 107
= 55. 1 ◦C
4.41 ν 1 =1cm^3 ;v 2 =
1
0. 091
= 10 .981 cm^3dP=LdT
T(ν 2 −ν 1 )=
80 × 4. 2 × 107 × 1
(− 1 +273)(10. 981 − 1 .0)
= 1. 238 ×
106 dynes
cm^2= 1 .24 atmP 2 =P 1 +dP= 1. 0 + 1. 24 = 2 .24 atm4.42 ν 1 =
1
ρ 1=
1
1. 145
= 0 .873 cm^3 /gν 2 =1
ρ 2=
1
0. 981
= 1 .019 cm^3 /gdT=T(ν 2 −ν 1 )dP
L=