Ideal Gas Law Problem

[Elzair]

I am having a little trouble on this problem:

For Water at 25^{\circ}C, (Volume Expansion Coefficient) \beta = 2.57 * 10^{-4}K^{-1} and (Isothermal Compressibility) \kappa = 4.52*10^{-10}Pa^{-1}. Suppose you increase the temperature of some water from 20^{\circ}C to 30^{\circ}C. How much Pressure must you apply to prevent it from expanding?

Can I treat this as an Isobaric expansion and an Isothermal compression?

I think I found the change in Volume like this: \frac{\Delta V}{V_{0}}=\beta\Delta T

How do I find the Pressure?

[tim_lou]

well, I suppose you wish to find ΔP/ΔT, holding V constant,

so, consider P as a function of T and V, we know the partial derivatives relating to V, but not T. So consider, V=const (we are holding V constant), or V(T,P)=const, then

dV=\left(\frac{\partial V}{\partial T}\right)_P dT + \left(\frac{\partial V}{\partial P}\right)_T dP=0

so that:
\left(\frac{\partial P}{\partial T}\right)_V=-\frac{\left(\partial V / \partial T\right)_P}{\left(\partial V/\partial P\right)_T}

now, look at the definition of \beta and \kappa, what are they (hint: they are related to partial derivatives of V)?

[Elzair]

Thanks, I think I got it. I will now post the answer here for others.

\frac{dV}{dT} = \frac{\beta}{\kappa}

therefore \Delta P = \frac{\beta}{\kappa}\Delta T

P.S. Does the Tex function on this forum feature a way to represent the therefore symbol (i.e. a triangle of three dots)?