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If I consider an ideal gas with constant number of particles, I can easily think an experiment where I can control $(V, T)$ or $(P, T)$ independently (state variables). But I cannot imagine an experiment where I can control $(V, P)$ independently.

$V = V(P, T)$. Consider a cylinder with a gas and a piston. Adiabatic walls. Water circuit that passes through the cylinder to control temperature. Pressure control if the cylinder is a room with a controlled pressure (mobile piston).

$P = P(V, T)$. Same as before but you move the piston with a machine.

$T = T(P, V)$. No idea.

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  • $\begingroup$ Hint: are V and P independent variables for an ideal gas? Think about the ideal gas law and try to understand why this is not the case. $\endgroup$
    – Giuseppe Basile
    Commented Sep 27 at 10:36
  • $\begingroup$ All variables are interdependent. But when you write the equation of state, for example V = V(T, P), you are mathematically considering T and P as independent variables, and I have shown you an experiment where you can control both independently. Of course, in the adiabatic cylinder, if you remove the water circuit and you move the piston, pressure and temperature will change accordingly. $\endgroup$
    – Abel Gutiérrez
    Commented Sep 27 at 10:39
  • $\begingroup$ @GiuseppeBasile The OP is asking if there is a way to 1) Hold the pressure constant while using a change in volume to change temperature, and 2) Hold the volume constant while using a changing pressure to change the temperature. $\endgroup$
    – BioPhysicist
    Commented Sep 27 at 11:14
  • $\begingroup$ yeah, I was not thinking, now I get the point, I need to think of an example without changing the number of moles. $\endgroup$
    – Giuseppe Basile
    Commented Sep 27 at 11:29
  • $\begingroup$ Are you asking if we can change pressure and volume independently for an ideal gas of constant mass? $\endgroup$
    – Bob D
    Commented Sep 27 at 13:15

2 Answers 2

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This is really Roger V.’s answer, restated: More generally, you can alter $P$ and $V$ in a system independently by, say, changing the amount of matter or allowing a phase change or a chemical reaction to occur (altering $N$), or allowing heat transfer or an irreversible process to occur (altering $S$), or doing another type of work than pressure–volume work (altering extensive variable $X$—polarization or magnetization, for instance).

But you’ve precluded all of these by specifying a pure ideal gas at equilibrium, subject to only pressure–volume work, with no possibility for heat generation and no way to change the temperature except to set the temperature of the container walls. By this choice, the system has been simplified and idealized to the point where pressure and volume can’t be changed independently except by altering the surrounding temperature.

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An equation of state can have many different forms, and mathematically inclined people would simply write it as $f(p,V,T)=0$.

E.g., for an ideal gas $$ PV=nRT\Rightarrow\\ f(p,v,T)=PV- nRT\\ P(V,T)=\frac{nRT}{V},\\ V(P,T)=\frac{nRT}{P}, \\ T(P,V)=\frac{PV}{nR} $$

The actual question however seems to have more to do with the fact that $P,V$ are conjugate variables. In this sense, it is more meaningful to look at the expression (or differential) of the internal energy: $$ dU=TdS - pdV+\mu dN. $$ One can perform various Legendre transformations to obtain from this different thermodynamic potentials. However, since $P,V$ are conjugate variables, one can never make both of them to act as independent variables.

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    $\begingroup$ I think the OP is more interested in how one can physically vary only one of $P$ or $V$ in order to change $T$ rather than mathematical manipulations $\endgroup$
    – BioPhysicist
    Commented Sep 27 at 11:24
  • $\begingroup$ @BioPhysicist Indeed, but in this case the focus on the equation of state is misplaced. I extended the answer to address both sides of the question. $\endgroup$
    – Roger V.
    Commented Sep 27 at 11:29
  • $\begingroup$ Yeah, I think the point you made is correct, pressure and volume are intrinsically related, while T is not dependent, in principle, on either of those for an ideal gas. $\endgroup$
    – Giuseppe Basile
    Commented Sep 27 at 11:32
  • $\begingroup$ I agree, the question doesn't specifically rely on understanding of "equation of state" $\endgroup$
    – BioPhysicist
    Commented Sep 27 at 11:59
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    $\begingroup$ @GiorgioP-DoomsdayClockIsAt-90 I don't quite understand the purpose of your comment: if you think the answer is different, you should write an expanded answer - I (and probably others) will be interested to learn more about it. References could be helpful as well. $\endgroup$
    – Roger V.
    Commented Sep 28 at 5:46

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