b) For a thermodynamic system, isobaric coefficient of volume expansion (a) and
isothermal compressibility (B) are defined as
1 av
a=
)
и от
p
1 av
B=-
У др
T T
Show that for an isochoric change, ß dp = a dT.
Answers
Explanation:
Given,
\alpha = \frac{1}{V}(\frac{dV}{dT})_pα=
V
1
(
dT
dV
)
p
\beta = - \frac{1}{V}(\frac{dV}{dP})_Tβ=−
V
1
(
dP
dV
)
T
Now, from the ideal gas equation,
PV=nRTPV=nRT
V=\frac{nRT}{P}V=
P
nRT
Now, taking the differentiation with respect to the corresponding terms,
\frac{dV}{dT}=\frac{nR}{P}
dT
dV
=
P
nR
if P is constant then,
(\frac{dV}{dT})_P=\frac{nR}{P}...(i)(
dT
dV
)
P
=
P
nR
...(i)
now, taking the differentiation with respect to P,
(\frac{dV}{dP})_T=\frac{nRT}{P^2}...(ii)(
dP
dV
)
T
=
P
2
nRT
...(ii)
Now, taking the ratio of (i)(i) and (ii)(ii)
\frac{(\frac{dV}{dT})_P}{(\frac{dV}{dP})_T}=\frac{\frac{nR}{P}}{\frac{nRT}{P^2}}
(
dP
dV
)
T
(
dT
dV
)
P
=
P
2
nRT
P
nR
\frac{(dP)_T}{(dT)_P}=\frac{P}{T}
(dT)
P
(dP)
T
=
T
P
(dP)_T=(dT)_P\times \frac{P}{T}(dP)
T
=(dT)
P
×
T
P