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My book on thermal physics shows that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), the grand free energy, $\Phi$, tends to decrease. The definition for the grand free energy is $$ \Phi\equiv U-TS-\mu N. $$ Now, they wrote the entropy of the reservoir as: $$ S_R=1/T[U_T-U+P(V_T-V)-\mu(N_T-N)], $$ where subscript $R$ stands for reservoir, and subscript $T$ stands for total. I can see that that they’re using the thermodynamic identity, $$ dU=TdS-PdV+\mu dN, $$ but I don’t see why the pressure is assumed fixed here. Thermal equilibrium implies constant and equal $T$, and diffusive equilibrium implies constant and equal $\mu$. How can we conclude that $P$ must be constant too? As far as I know, we don’t have mechanical equilibrium.

I have twothree guesses:

  1. Either weWe are not in mechanical equilibrium, and it's just the reservoir's pressure that can be considered fixed (while the system's pressure may vary.

  2. Somehow thermal and diffusive equilibrium also imply mechanical equilibrium, so the pressure of the system and its surroundings (=reservoir) are always equal.

  3. Use the thermodynamic identity for the grand potential, which yields the following partial derivative: $$ -\left(\frac{\partial\Phi}{\partial V}\right)_{T,\mu}=P. $$ If would seem then that this partial derivative is the same for the reservoir, if our system transforms during diffusive and thermal equilibrium. Hm, I'm still confused.

Which one is correct?

EDIT

Oh, a third option would be to use the thermodynamic identity for the grand potential, which yields the following partial derivative: $$ -\left(\frac{\partial\Phi}{\partial V}\right)_{T,\mu}=P. $$ If would seem then that this partial derivative is the same for the reservoir, if our system transforms during diffusive and thermal equilibrium. Hm, I'm still confused.

My book on thermal physics shows that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), the grand free energy, $\Phi$, tends to decrease. The definition for the grand free energy is $$ \Phi\equiv U-TS-\mu N. $$ Now, they wrote the entropy of the reservoir as: $$ S_R=1/T[U_T-U+P(V_T-V)-\mu(N_T-N)], $$ where subscript $R$ stands for reservoir, and subscript $T$ stands for total. I can see that that they’re using the thermodynamic identity, $$ dU=TdS-PdV+\mu dN, $$ but I don’t see why the pressure is assumed fixed here. Thermal equilibrium implies constant and equal $T$, and diffusive equilibrium implies constant and equal $\mu$. How can we conclude that $P$ must be constant too? As far as I know, we don’t have mechanical equilibrium.

I have two guesses:

  1. Either we are not in mechanical equilibrium, and it's just the reservoir's pressure that can be considered fixed (while the system's pressure may vary.

  2. Somehow thermal and diffusive equilibrium also imply mechanical equilibrium, so the pressure of the system and its surroundings (=reservoir) are always equal.

Which one is correct?

EDIT

Oh, a third option would be to use the thermodynamic identity for the grand potential, which yields the following partial derivative: $$ -\left(\frac{\partial\Phi}{\partial V}\right)_{T,\mu}=P. $$ If would seem then that this partial derivative is the same for the reservoir, if our system transforms during diffusive and thermal equilibrium. Hm, I'm still confused.

My book on thermal physics shows that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), the grand free energy, $\Phi$, tends to decrease. The definition for the grand free energy is $$ \Phi\equiv U-TS-\mu N. $$ Now, they wrote the entropy of the reservoir as: $$ S_R=1/T[U_T-U+P(V_T-V)-\mu(N_T-N)], $$ where subscript $R$ stands for reservoir, and subscript $T$ stands for total. I can see that that they’re using the thermodynamic identity, $$ dU=TdS-PdV+\mu dN, $$ but I don’t see why the pressure is assumed fixed here. Thermal equilibrium implies constant and equal $T$, and diffusive equilibrium implies constant and equal $\mu$. How can we conclude that $P$ must be constant too? As far as I know, we don’t have mechanical equilibrium.

I have three guesses:

  1. We are not in mechanical equilibrium, and it's just the reservoir's pressure that can be considered fixed (while the system's pressure may vary.

  2. Somehow thermal and diffusive equilibrium also imply mechanical equilibrium, so the pressure of the system and its surroundings (=reservoir) are always equal.

  3. Use the thermodynamic identity for the grand potential, which yields the following partial derivative: $$ -\left(\frac{\partial\Phi}{\partial V}\right)_{T,\mu}=P. $$ If would seem then that this partial derivative is the same for the reservoir, if our system transforms during diffusive and thermal equilibrium. Hm, I'm still confused.

Which one is correct?

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Sha Vuklia
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My book on thermal physics shows that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), the grand free energy, $\Phi$, tends to decrease. The definition for the grand free energy is $$ \Phi\equiv U-TS-\mu N. $$ Now, they wrote the entropy of the reservoir as: $$ S_R=1/T[U_T-U+P(V_T-V)-\mu(N_T-N)], $$ where subscript $R$ stands for reservoir, and subscript $T$ stands for total. I can see that that they’re using the thermodynamic identity, $$ dU=TdS-PdV+\mu dN, $$ but I don’t see why the pressure is assumed fixed here. Thermal equilibrium implies constant and equal $T$, and diffusive equilibrium implies constant and equal $\mu$. How can we conclude that $P$ must be constant too? As far as I know, we don’t have mechanical equilibrium.

I have two guesses:

  1. Either we are not in mechanical equilibrium, and it's just the reservoir's pressure that can be considered fixed (while the system's pressure may vary.

  2. Somehow thermal and diffusive equilibrium also imply mechanical equilibrium, so the pressure of the system and its surroundings (=reservoir) are always equal.

Which one is correct?

EDIT

Oh, a third option would be to use the thermodynamic identity for the grand potential, which yields the following partial derivative: $$ -\left(\frac{\partial\Phi}{\partial V}\right)_{T,\mu}=P. $$ If would seem then that this partial derivative is the same for the reservoir, if our system transforms during diffusive and thermal equilibrium. Hm, I'm still confused.

My book on thermal physics shows that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), the grand free energy, $\Phi$, tends to decrease. The definition for the grand free energy is $$ \Phi\equiv U-TS-\mu N. $$ Now, they wrote the entropy of the reservoir as: $$ S_R=1/T[U_T-U+P(V_T-V)-\mu(N_T-N)], $$ where subscript $R$ stands for reservoir, and subscript $T$ stands for total. I can see that that they’re using the thermodynamic identity, $$ dU=TdS-PdV+\mu dN, $$ but I don’t see why the pressure is assumed fixed here. Thermal equilibrium implies constant and equal $T$, and diffusive equilibrium implies constant and equal $\mu$. How can we conclude that $P$ must be constant too? As far as I know, we don’t have mechanical equilibrium.

I have two guesses:

  1. Either we are not in mechanical equilibrium, and it's just the reservoir's pressure that can be considered fixed (while the system's pressure may vary.

  2. Somehow thermal and diffusive equilibrium also imply mechanical equilibrium, so the pressure of the system and its surroundings (=reservoir) are always equal.

Which one is correct?

My book on thermal physics shows that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), the grand free energy, $\Phi$, tends to decrease. The definition for the grand free energy is $$ \Phi\equiv U-TS-\mu N. $$ Now, they wrote the entropy of the reservoir as: $$ S_R=1/T[U_T-U+P(V_T-V)-\mu(N_T-N)], $$ where subscript $R$ stands for reservoir, and subscript $T$ stands for total. I can see that that they’re using the thermodynamic identity, $$ dU=TdS-PdV+\mu dN, $$ but I don’t see why the pressure is assumed fixed here. Thermal equilibrium implies constant and equal $T$, and diffusive equilibrium implies constant and equal $\mu$. How can we conclude that $P$ must be constant too? As far as I know, we don’t have mechanical equilibrium.

I have two guesses:

  1. Either we are not in mechanical equilibrium, and it's just the reservoir's pressure that can be considered fixed (while the system's pressure may vary.

  2. Somehow thermal and diffusive equilibrium also imply mechanical equilibrium, so the pressure of the system and its surroundings (=reservoir) are always equal.

Which one is correct?

EDIT

Oh, a third option would be to use the thermodynamic identity for the grand potential, which yields the following partial derivative: $$ -\left(\frac{\partial\Phi}{\partial V}\right)_{T,\mu}=P. $$ If would seem then that this partial derivative is the same for the reservoir, if our system transforms during diffusive and thermal equilibrium. Hm, I'm still confused.

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