Principle of corresponding states

[decerto]

Homework Statement

Show that it is always possible to adjust measurement units such that a; b can be assigned any values
you want. This means that e.g. all van der Waals gases look exactly the same if the units are
accordingly adjusted. (This is what is called principle of corresponding states).

Homework Equations

$P=\frac{T}{v-b}-\frac{a}{v^2}$

The Attempt at a Solution

[/B]
So the question before this one was to work out the critical values for P, v and T for the van der waals equation of state + 3 other qualitatively similar gas models, the critical values were all of the form $P_c \propto \frac{a}{b^2}$, $T_c \propto \frac{a}{b}$ and $v_c \propto b$ but I am not sure if this is relevant.

The question is confusing me to be honest, shouldn't it be trivially true that you can change your definition of a metre or kilogram in different cases to get the same values for different gasses?

 

[DrClaude]

So the question before this one was to work out the critical values for P, v and T for the van der waals equation of state + 3 other qualitatively similar gas models, the critical values were all of the form $P_c \propto \frac{a}{b^2}$, $T_c \propto \frac{a}{b}$ and $v_c \propto b$ but I am not sure if this is relevant.

This is most certainly very relevant (hint!).

The question is confusing me to be honest, shouldn't it be trivially true that you can change your definition of a metre or kilogram in different cases to get the same values for different gasses?

For any pair of coefficients a and b? No, you can't do that by units alone. But you can do it using another rescaling, see my hint above.

 

[Chestermiller]

I agree that the problem statement isn't too clear. But, I would start out by writing:

$P_r=P/P_c$

$v_r=v/v_c$

$T_r=T/T_c$

These are the reduced pressure, the reduced volume, and the reduced temperature of the gas, respectively, and all three are dimensionless. I would then substitute for P, v, and T in the vdw equation and lump all the extra critical properties in with the a and b.

Chet

Ooops. I just saw Dr. Claude's post which basically suggests the same thing (a little more subtily).

 

[decerto]

I agree that the problem statement isn't too clear. But, I would start out by writing:

$P_r=P/P_c$

$v_r=v/v_c$

$T_r=T/T_c$

These are the reduced pressure, the reduced volume, and the reduced temperature of the gas, respectively, and all three are dimensionless. I would then substitute for P, v, and T in the vdw equation and lump all the extra critical properties in with the a and b.

Chet

Ooops. I just saw Dr. Claude's post which basically suggests the same thing (a little more subtily).

The question after this is about finding the reduced form of the equation which I have already done. This question seems to be about justifying it. I think Dr Claude is suggesting something different

 

[decerto]

This is most certainly very relevant (hint!).For any pair of coefficients a and b? No, you can't do that by units alone. But you can do it using another rescaling, see my hint above.

The following question asks me to put the equations in reduced form where $P_c$, $T_c$ and $v_c$ are equal to 1 if that is what your suggesting I do for this question as chestermiller says.

If not then does the fact you can write p, T and v in such a way as to be independent of a and b mean anything