Ideal Gas Law Chang ein VolumeProblem

AI Thread Summary
The discussion revolves around calculating the pressure required to prevent the expansion of water when its temperature is increased from 20°C to 30°C. The user applies the volume expansion coefficient and isothermal compressibility to determine the change in volume and subsequently the pressure change. They derive the relationship between pressure change and temperature change using the definitions of the coefficients involved. The final formula presented is ΔP = (β/κ)ΔT, which connects the change in pressure to the temperature change while holding volume constant. Additionally, the user inquires about representing the "therefore" symbol in the forum's Tex function.
Elzair
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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?
 
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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)?
 
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Thanks!

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)?
 
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