Thermodynamics - partial differential

[Beer-monster]

Homework Statement

Edit: Some more details.

The question concerns Joule-Thompson Expansion and I need to derive the relationship:

$$\left(\frac{\partial H}{\partial P}\right)_T =V+T\left(\frac{\partial S}{\partial P}\right)_T$$V is not constant as it is expansion but the enthalpy is specied as H(P,T) for the problem. Therefore I would assume that Volume is a function of P and T and not independent. However, I'm not 100% sure on this.

The Attempt at a Solution

$$dH=TdS+VdP$$

Therefore:

$$\left(\frac{\partial H}{\partial P}\right)_T = T\left(\frac{\partial S}{\partial P}\right)_T + V\left(\frac{\partial P}{\partial P}\right)_T = V+T\left(\frac{\partial S}{\partial P}\right)_T$$

Is this right? I know it would not be if I'm right and V is a function of P and T. Could someone please clear this up for me.Thanks

BM

 

[Mapes]

You got it, as long as you've implicitly used the chain rule and ignored the differentials as compared to non-infinitesimal values. That is,

$$\left(\frac{\partial}{\partial P}\right)_T\left(V\,dP\right)=\left(\frac{\partial V}{\partial P}\right)_T dP+V\left(\frac{\partial P}{\partial P}\right)_T=V.$$

The differentials can be ignored in any mix of differential and non-infinitesimal values. Does this make sense?

 

[Beer-monster]

I think I get it. As V is not constant and probably depends on P we can't ignore it in the differential and need to apply the product rule.

But doesn't the differential of V only equal 0 if V is constant or independent of P. In which case, why would we need to explicitly use the product rule?

 

[Mapes]

I'm not sure if this answers your question, but we must use the product rule every time we differentiate a product. It's safest to assume every variable depends on every other variable unless it's been specifically shown not to. In this case, $(\partial V/\partial P)_T\,dP$ goes away not because $V$ has any certain value, but because $dP$ is an infinitesimal value that vanishes when compared with the finite value $V$.

In summary, it's safer to use the product rule and cancel the appropriate terms than to work "fast and loose."

 

[Beer-monster]

So because dp is so small (approximately zero) compared to the differential $(\partial V/\partial P)_T$ we can ignore it provided the differential is of something non-infintesimal?

 

[Mapes]

Exactly. We can't automatically do the same thing for an equation like $dH=T\,dS+V\,dP$ because everything is infinitesimal.

 

[Beer-monster]

Yeah, that occurred to me and gave me a bit of a pause. But though differentials are made of infintesimal elements they are not themselves infintesimal so would be reduced to nothing if multiplied by dP.

However, if this was worked out in a problem the product rule would have to be expressed explicitly otherwise one would think that you were treating V as constant.