Thermodynamics problem, Enthelpy zero for an ideal gas

[gibbsboson]

Homework Statement

Show that

[itex]\left(\frac{\partial H}{\partial T}\right)_{T} = 0 [/itex]

for an ideal gas

Homework Equations

The question required me to first solve

[itex]\left(\frac{\partial U}{\partial T}\right)_{P}[/itex] [itex] = C_{P}[/itex] - [itex]P\left(\frac{\partial V}{\partial T}\right)_{P}[/itex]

but I am unsure if I would use this for the rest of the question

The Attempt at a Solution

I have already shown that [itex]\left(\frac{\partial C_{V}}{\partial V}\right)_{T} = 0 [/itex] for an ideal gas but I am struggling to manage this one. I can show it is zero when I have this equation to begin with
[itex]dH = \left(\frac{\partial H}{\partial T}\right)_{V}dT[/itex] + [itex]\left(\frac{\partial H}{\partial T}\right)_{T}dV[/itex]
But I am unsure how to get to this point in the first place, so any help here would be excellent.

 

[Chestermiller]

Hi Gibbsboson. Welcome to Physics Forums.

I think that the equation you are trying to show is written incorrectly. It should read the partial with respect to P.

Have you learned the general equation for dH in terms of dT and dP for a pure species? If so, what is it?

Chet

 

[gibbsboson]

Hi Chestermiller

Sorry it should be [itex]\left(\frac{\partial H}{\partial V}\right)_{T} = 0 [/itex]

I don't think I have learned that equation. The only other equation involving enthalpy is H = U + PV. Or I also have dH = TdS + VdP, but I don't think I can use entropy here.

A point in the right direction would be brilliant because I have been struggling with this for a while now

Thanks in advance,

GB

 

[Chestermiller]

Hi Chestermiller

Sorry it should be [itex]\left(\frac{\partial H}{\partial V}\right)_{T} = 0 [/itex]

I don't think I have learned that equation. The only other equation involving enthalpy is H = U + PV. Or I also have dH = TdS + VdP, but I don't think I can use entropy here.

A point in the right direction would be brilliant because I have been struggling with this for a while now

Thanks in advance,

GB

Start out with dH = TdS + VdP, and take the partial of this equation with respect to V at constant T. This will give you a term involving the partial of S with respect to V at constant T. The Maxwell relationship you need to evaluate this derives from the equation for dA.

Give it a shot.

Chet

 

[gibbsboson]

Sorry for being a little slow. I end up with

[itex]\left(\frac{\partial H}{\partial V}\right)_{T} = T\left(\frac{\partial P}{\partial T}\right)_{V} + V\left(\frac{\partial P}{\partial V}\right)_{T}[/itex]

and I see no way of getting rid of this. When I substitute the ideal gas formula this doesn't cancel. Where should I go from here?

Thanks

GB

 

[Chestermiller]

Sorry for being a little slow. I end up with

[itex]\left(\frac{\partial H}{\partial V}\right)_{T} = T\left(\frac{\partial P}{\partial T}\right)_{V} + V\left(\frac{\partial P}{\partial V}\right)_{T}[/itex]

and I see no way of getting rid of this. When I substitute the ideal gas formula this doesn't cancel. Where should I go from here?

Thanks

GB

That's funny. When I substitute the ideal gas formula into this equation, it cancels for me. Please check your "arithmetic."

Chet

 

[gibbsboson]

Apologies, I was being stupid. Got it now.

Thanks for your help

GB