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## Homework Statement

Show that at constant volume V and temperature T but decreasing number N=n*N[itex]_{A}[/itex] of particles the Van der Waals equation of state approaches the equation of state of an ideal gas.

Hint: Rearrange the equation of state into the explicit functional form P=P(v,T) and use x=1/v as a small parameter for a Taylor series P(x)=P(0)+dP/dx x + ...

## Homework Equations

Van der Waals equation of state for a real gas:

[itex]( P + \frac{a}{v^{2}} ) ( v - b ) = RT[/itex]

Taylor series expansion:

[itex]f(x)=f(a)+f'(a)(x-a)+\frac{f"(a)}{2!} (x-a)^{2} + ... [/itex]

## The Attempt at a Solution

Rearranging...

[itex]( P + \frac{a}{v^{2}} ) ( v - b ) = RT[/itex]

[itex]P + \frac{a}{v^{2}} = \frac{RT}{v-b}[/itex]

[itex]P = \frac{RT}{v-b} - \frac{a}{v^{2}}[/itex]

Now it's been a while since I've done a Taylor expansion so I don't seem to remember how to go about it. Could someone just point me in the right direction? Thanks!

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