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Those are the definitions of each. They don't assume anything about the system and can always be applied. You are getting mixed up with the scenarios in which they are usually applied since nice things happen. For example, for a system at constant pressure (and number of particles) $\Delta H=Q$, where $Q$ is the heat that enters or leaves the system.

To add some more detail, this can be seen by substituting in the thermodynamic identity $$\text dU=T\text dS-P\text dV+\mu\text d N$$ into the differential of one of your potentials. For example, as mentioned above we have $$\text dH=\text dU+P\text dV+V\text dP$$ so then $$\text dH=T\text dS+V\text dP+\mu\text dN$$ i.e. at constant pressure and number of particles $\text dH=T\text dS=\text dQ$

You also say that the Gibbs free energy takes both mentioned assumptions "for granted", but see what happens if you do this process with the Gibbs free energy at constant temperature and pressure. It is a very important relation.

These processes are more generally called Legendre transformations

Those are the definitions of each. They don't assume anything about the system and can always be applied. You are getting mixed up with the scenarios in which they are usually applied since nice things happen. For example, for a system at constant pressure (and number of particles) $\Delta H=Q$, where $Q$ is the heat that enters or leaves the system.

To add some more detail, this can be seen by substituting in the thermodynamic identity $$\text dU=T\text dS-P\text dV+\mu\text d N$$ into the differential of one of your thermodynamic potentials. For example, as mentioned above we have $$\text dH=\text dU+P\text dV+V\text dP$$ so then $$\text dH=T\text dS+V\text dP+\mu\text dN$$ i.e. at constant pressure and number of particles $\text dH=T\text dS=\text dQ$

You also say that the Gibbs free energy takes both mentioned assumptions "for granted", but see what happens if you do this process with the Gibbs free energy at constant temperature and pressure. It is a very important relation.

These processes are more generally called Legendre transformations

Those are the definitions of each. They don't assume anything about the system and can always be applied. You are getting mixed up with the scenarios in which they are usually applied since nice things happen. For example, for a system at constant pressure (and number of particles) $\Delta H=Q$, where $Q$ is the heat that enters or leaves the system.

To add some more detail, this can be seen by substituting in the thermodynamic identity $$\text dU=T\text dS-P\text dV+\mu\text d N$$ into the differential of one of your potentials. For example, as mentioned above we have $$\text dH=\text dU+P\text dV+V\text dP$$ so then $$\text dH=T\text dS+V\text dP+\mu\text dN$$ i.e. at constant pressure and number of particles $\text dH=T\text dS=\text dQ$

You also say that the Gibbs free energy takes both mentioned assumptions "for granted", but see what happens if you do this process with the Gibbs free energy at constant temperature and pressure. It is a very important relation.

Those are the definitions of each. They don't assume anything about the system and can always be applied. You are getting mixed up with the scenarios in which they are usually applied since nice things happen. For example, for a system at constant pressure (and number of particles) $\Delta H=Q$, where $Q$ is the heat that enters or leaves the system.

To add some more detail, this can be seen by substituting in the thermodynamic identity $$\text dU=T\text dS-P\text dV+\mu\text d N$$ into the differential of one of your potentials. For example, as mentioned above we have $$\text dH=\text dU+P\text dV+V\text dP$$ so then $$\text dH=T\text dS+V\text dP+\mu\text dN$$ i.e. at constant pressure and number of particles $\text dH=T\text dS=\text dQ$

You also say that the Gibbs free energy takes both mentioned assumptions "for granted", but see what happens if you do this process with the Gibbs free energy at constant temperature and pressure. It is a very important relation.

These processes are more generally called Legendre transformations

Those are the definitions of each. They don't assume anything about the system and can always be applied. You are getting mixed up with the scenarios in which they are usually applied since nice things happen. For example, for a system at constant pressure (and number of particles) $\Delta H=Q$, where $Q$ is the heat that enters or leaves the system.

To add some more detail, this can be seen by substituting in the thermodynamic identity $$\text dU=T\text dS-P\text dV+\mu\text d N$$ into the differential of one of your potentials. For example, as mentioned above we have $$\text dH=\text dU+P\text dV+V\text dP$$ so then $$\text dH=V\text dP+\mu\text dN$$ i.e. at constant pressure and number of particles $\text dH=T\text dS=\text dQ$

You also say that the Gibbs free energy takes both mentioned assumptions "for granted", but see what happens if you do this process with the Gibbs free energy at constant temperature and pressure. It is a very important relation.

Those are the definitions of each. They don't assume anything about the system and can always be applied. You are getting mixed up with the scenarios in which they are usually applied since nice things happen. For example, for a system at constant pressure (and number of particles) $\Delta H=Q$, where $Q$ is the heat that enters or leaves the system.

To add some more detail, this can be seen by substituting in the thermodynamic identity $$\text dU=T\text dS-P\text dV+\mu\text d N$$ into the differential of one of your potentials. For example, as mentioned above we have $$\text dH=\text dU+P\text dV+V\text dP$$ so then $$\text dH=T\text dS+V\text dP+\mu\text dN$$ i.e. at constant pressure and number of particles $\text dH=T\text dS=\text dQ$

You also say that the Gibbs free energy takes both mentioned assumptions "for granted", but see what happens if you do this process with the Gibbs free energy at constant temperature and pressure. It is a very important relation.