I have to show that for a van der Waals gas the critical temperature, volume and pressure are given by:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] T_c= \frac{8a}{27bR} [/tex]

[tex]V_c= 3nb [/tex]

[tex]p_c= \frac{a}{27b^2} [/tex]

I started off this way:

Van der Waals states that for a non ideal gas the pressure is:

[tex] P = \frac{nRT}{V-nb} - a \frac{n^2}{V^2} [/tex]

with a and b constants...

The point of inflection of the (V,p) graph is the critical volume + temperature. Therefore this point is given by the conditions:

[tex] \frac{\delta p}{\delta V} =0 [/tex]

[tex] \frac{\delta ^2 p}{\delta V^2} =0 [/tex]

I came to the conclusion that these conditions are:

[tex] \frac{\delta p}{\delta V} = - \frac{nRT}{(V-nb)^2} + \frac{2an^2}{V^3}=0 [/tex]

[tex] \frac{\delta ^2 p}{\delta V^2} = \frac{2 nRT}{(V-nb)^3} - \frac{6an^2}{V^4}=0 [/tex]

Now how do I obtain V /Tfrom these two equations?

I believe I need to set the second equation equal to 0 and solve for V & T. Then substitute the answer in the first one and show that it also equals 0.

Only: how do I get an expression for V & T?!?!?! I've done the following:

[tex] \frac{2 nRT}{(V-nb)^3} - \frac{6an^2}{V^4}=0 [/tex]

[tex] \frac{2 nRT}{(V-nb)^3} = \frac{6an^2}{V^4} [/tex]

[tex] \frac{nRT}{(V-nb)^3} = \frac{3an^2}{V^4} [/tex]

What steps should I take next? Please help me

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# Homework Help: Maths problem: deriving expressions for the critical temperature/volume/pressure

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