An equation of state $($EOS$)$ in thermodynamics, defined narrowly,

refers to the two$-$variable function $(,),$ where $$ is the

pressure, $$ is the absolute temperature, and $$is the molar

volume. For example, the equation of state that describes an ideal

gas is $($ is the gas constant$)$:$($a$)$ Convert this integral form of the equation to the

corresponding differential form. In other words, what are the

functions $(,)$ and $(,),$ in the equation below?$($b$)$ If the change in temperature, $,$ and the change in molar

volume, $,$ are sufficiently small, then the above differential

equation can be used to estimate approximately the change in

pressure:Suppose the absolute temperature increases by $1\%,$ and the molar

volume decreases by $2\%,$ what is the percentage change of the

pressure?$($c$)$Determine the change in $$ when the gas goes from the

state of

$$to the state of

$,$ by integrating your differential equation in $($a$).$ Be specific

about the path you choose to perform the integration. Check that

your answer is the same as if you used the integral form

directly.$($d$)$The ideal gas EOS is simple and is a good approximation

at low pressures, but it is often inadequate for real applications.

For a real, non$-$ideal gas, a more accurate EOS is the van der Waals

EOS:where $$and $$are constants. Convert this integral

form to the corresponding differential form. Also check that the

differential $$P$$is exact by verifying that:$($e$)$ The constants $$ and $$are different for different

chemical species. To obtain the constants, we can measure the

critical point of the gas

$,$ which is a special state of a gas that satisfies the following

equations:By solving these two equations simultaneously, express $$ and $$

as functions of $$ and $.$ Hint: Relate $$ with the critical molar

volume $$ first, and use the van der Waals EOS to replace $$ with

$$ and $.$