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A relação fundamental:
(
∂
P
∂
T
)
V
=
(
∂
S
∂
V
)
T
{\displaystyle \left({\frac {\partial P}{\partial T}}\right)_{V}\;=\;\left({\frac {\partial S}{\partial V}}\right)_{T}}
anteriormente permite calcular a variação da entropia com a mudança de volume durante um processo isotermo. Para um gás obedecendo à equação de estado de Van der Waals, temos:
(
P
+
a
n
2
V
2
)
(
V
−
n
b
)
=
n
R
T
{\displaystyle \left(P\;+\;{\frac {an^{2}}{V^{2}}}\right)(V-nb)\;=\;nRT}
o que conduz a:
(
∂
P
∂
T
)
V
=
n
R
(
V
−
n
b
)
{\displaystyle \left({\frac {\partial P}{\partial T}}\right)_{V}\;=\;{\frac {nR}{(V-nb)}}}
A diferencial exata total da entropia escreve-se:
d
S
=
(
∂
S
∂
V
)
T
d
V
+
(
∂
S
∂
T
)
V
d
T
{\displaystyle dS\;=\;\left({\frac {\partial S}{\partial V}}\right)_{T}dV\;+\;\left({\frac {\partial S}{\partial T}}\right)_{V}dT}
o que, a temperatura constante, pode ser simplificado:
(
d
S
)
T
=
(
∂
S
∂
V
)
T
d
V
{\displaystyle (dS)_{T}\;=\;\left({\frac {\partial S}{\partial V}}\right)_{T}dV}
A integral escreve-se:
Δ
S
=
∫
V
1
V
2
(
d
S
)
T
=
∫
V
1
V
2
(
∂
S
∂
V
)
T
d
V
=
∫
V
1
V
2
n
R
V
−
n
b
d
V
=
n
R
ln
(
V
2
−
n
b
V
1
−
n
b
)
{\displaystyle \Delta S\;=\;\int _{V_{1}}^{V_{2}}(dS)_{T}\;=\;\int _{V_{1}}^{V_{2}}\left({\frac {\partial S}{\partial V}}\right)_{T}dV\;=\int _{V_{1}}^{V_{2}}{\frac {nR}{V-nb}}dV\;=\;nR\ln \left({\frac {V_{2}-nb}{V_{1}-nb}}\right)}
anteriormente :
(
∂
E
∂
V
)
T
=
T
(
∂
P
∂
T
)
V
−
P
{\displaystyle \left({\frac {\partial E}{\partial V}}\right)_{T}\;=\;T\left({\frac {\partial P}{\partial T}}\right)_{V}\;-\;P}
Para um gás de Van Der Waals, como demonstrado acima:
(
∂
P
∂
T
)
V
=
n
R
V
−
n
b
{\displaystyle \left({\frac {\partial P}{\partial T}}\right)_{V}\;=\;{\frac {nR}{V-nb}}}
portanto
(
∂
E
∂
V
)
T
=
n
R
T
V
−
n
b
−
P
{\displaystyle \left({\frac {\partial E}{\partial V}}\right)_{T}\;=\;{\frac {nRT}{V-nb}}-P}
(
∂
E
∂
V
)
T
=
a
n
2
V
2
{\displaystyle \left({\frac {\partial E}{\partial V}}\right)_{T}\;=\;{\frac {an^{2}}{V^{2}}}}
d
E
=
(
∂
E
∂
V
)
T
d
V
+
(
∂
E
∂
T
)
V
d
T
{\displaystyle dE\;=\;{\left({\frac {\partial E}{\partial V}}\right)}_{T}dV\;+\;{\left({\frac {\partial E}{\partial T}}\right)}_{V}dT}
(
d
E
)
T
=
(
∂
E
∂
V
)
T
d
V
{\displaystyle (dE)_{T}\;=\;{\left({\frac {\partial E}{\partial V}}\right)}_{T}dV}
(
d
E
)
T
=
a
n
2
V
2
d
V
{\displaystyle (dE)_{T}\;=\;{\frac {an^{2}}{V^{2}}}dV}
Δ
E
=
∫
V
1
V
2
(
d
E
)
T
=
∫
V
1
V
2
a
n
2
V
2
d
V
=
a
n
2
(
1
V
1
−
1
V
2
)
{\displaystyle \Delta E\;=\;\int _{V_{1}}^{V_{2}}(dE)_{T}\;=\;\int _{V_{1}}^{V_{2}}{\frac {an^{2}}{V^{2}}}dV\;=\;an^{2}\left({\frac {1}{V_{1}}}-{\frac {1}{V_{2}}}\right)}
