# Thermal Physics and Differentials

A hypothetical substance has a compressibility k = a /V and a volume expansivity
B = 2bT /V , where a and b are constants and V is the molar volume. Show that the
equation of state is:
V = bT2 - aP + constant

To be honest I'm not entirely sure what I'm actually supposed to be doing with this question. Do I treat it as an ideal gas and therefore use that equation or is that completely wrong.

So far I have integrated each of the equations in respect to T, giving me,

(integral of) a/V dT = aT/V + c

(integral of) 2bT/V dT = bT^2/V + c

But now I'm stuck and can't seem to find a relevant relationship between P and what I've got there.

## Answers and Replies

vela
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Your first equation is wrong. Compressibility has to do with the change in volume due to a change in pressure, not temperature.

So do I integrate both in terms of dP?

vela
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Both what?

What are the (mathematical) definitions of compressibility and expansivity?

Well I think they're k = 1/P and B = 1/T but I've been given equations for both in the question.

vela
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Sorry, I meant the general definitions in terms of derivatives.

To be honest I really don't know. I've become very confused with all the derivatives and general definitions.

vela
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Well, you should start by looking those up in your textbook or notes. If you don't know the basic definitions of the quantities involved in the problem, it's no surprise that the problem is confusing.

http://en.wikipedia.org/wiki/Material_properties_(thermodynamics [Broken])

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