- #1
matematikuvol
- 192
- 0
If in some thermodynamics system preasure [tex]P[/tex] and temperature [tex]T[/tex] are constant then Gibbs potential has minimum.
[tex]G=U-TS+PV[/tex]
Variation of [tex]G[/tex] is
[tex]\delta G=\delta U-T \delta S+P \delta V[/tex]
[tex]U=U(S,V)[/tex]
[tex]\delta U=(\frac{\partial U}{\partial S})_V\delta S+(\frac{\partial U}{\partial V})_S\delta V+\frac{1}{2}(\frac{\partial^2 U}{\partial S^2})_V(\delta S)^2+\frac{1}{2}(\frac{\partial^2 U}{\partial V^2})_S(\delta V)^2+\frac{\partial^2 U}{\partial S\partial V}\delta S \delta V[/tex]
If we use Maxwell relation we get
[tex]\delta G=\frac{1}{2}(\frac{\partial^2 U}{\partial S^2})_V(\delta S)^2+\frac{1}{2}(\frac{\partial^2 U}{\partial V^2})_S(\delta V)^2+\frac{\partial^2 U}{\partial S\partial V}\delta S \delta V[/tex]
and from here
[tex]\delta G>0[/tex]
Is it true from [tex]G[/tex] is minimum. Or [tex]\delta G=0, \delta^2 G>0[/tex]. Tnx for your answer.
[tex]G=U-TS+PV[/tex]
Variation of [tex]G[/tex] is
[tex]\delta G=\delta U-T \delta S+P \delta V[/tex]
[tex]U=U(S,V)[/tex]
[tex]\delta U=(\frac{\partial U}{\partial S})_V\delta S+(\frac{\partial U}{\partial V})_S\delta V+\frac{1}{2}(\frac{\partial^2 U}{\partial S^2})_V(\delta S)^2+\frac{1}{2}(\frac{\partial^2 U}{\partial V^2})_S(\delta V)^2+\frac{\partial^2 U}{\partial S\partial V}\delta S \delta V[/tex]
If we use Maxwell relation we get
[tex]\delta G=\frac{1}{2}(\frac{\partial^2 U}{\partial S^2})_V(\delta S)^2+\frac{1}{2}(\frac{\partial^2 U}{\partial V^2})_S(\delta V)^2+\frac{\partial^2 U}{\partial S\partial V}\delta S \delta V[/tex]
and from here
[tex]\delta G>0[/tex]
Is it true from [tex]G[/tex] is minimum. Or [tex]\delta G=0, \delta^2 G>0[/tex]. Tnx for your answer.
Last edited: