## [SOLVED] thermal physics

1. The problem statement, all variables and given/known data
the differential of enthalpy is given by dH=TdS-VdP

2. Relevant equations
Beginning with the above relationship, find the equation for the differential of the entropy and then show that (d(1/T)/dp)|H=(d(V/T)/dH)|p

3. The attempt at a solution

this is what i got so far...
i rearranged dH=TdS-VdP to make S the subject which gives dS=dH/T+Vdp/T
then since S=S(H,p) and using the definition of the differential of S;
dS=(dS/dH)|p*dH+(dS/dp)|H*dp
i got then 1/T=(dS/dH)|p & V/T=(dS/dp)|H
i dont know what to do next and to how to exactly reach the final term of (d(1/T)/dp)|H=(d(V/T)/dH)|p

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 Blog Entries: 1 Recognitions: Gold Member Science Advisor Staff Emeritus Welcome to PF billybob, You're on the right lines and almost there. What do you know about the mixed second partial derivatives of an exact differential?

 Quote by Hootenanny Welcome to PF billybob, You're on the right lines and almost there. What do you know about the mixed second partial derivatives of an exact differential?
exactness condition
i know that they are equal, i thought i would end up somewhere along those lines so..

dS=(dS/dH)|p*dH+(dS/dp)|H*dp

=M(H,p)dH+N(H,p)dp

dM/dp=dN/dH

i see where its getting next bit algebra is getting tricky..

do i cheat and just sub (1/T) for M and and V/T for N?

i dont understand how to show this mathematically nicely

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## [SOLVED] thermal physics

 Quote by billybob5588 exactness condition i know that they are equal, i thought i would end up somewhere along those lines so.. dS=(dS/dH)|p*dH+(dS/dp)|H*dp
Correct.
 Quote by billybob5588 =M(H,p)dH+N(H,p)dp dM/dp=dN/dH
I'm not quite sure what you doing here, but you know that,

$$\left(\frac{\partial S}{\partial H}\right)_p =\frac{1}{T}$$

$$\left(\frac{\partial S}{\partial P}\right)_H =\frac{V}{T}$$

And you know that,

$$\left[\frac{\partial}{\partial P}\left(\frac{\partial S}{\partial H}\right)_p\right]_H = \left[\frac{\partial}{\partial H}\left(\frac{\partial S}{\partial P}\right)_H\right]_P$$

Can you see what you need to do next (it's a really simple step)?

 Quote by Hootenanny Correct. I'm not quite sure what you doing here, but you know that, $$\left(\frac{\partial S}{\partial H}\right)_p =\frac{1}{T}$$ $$\left(\frac{\partial S}{\partial P}\right)_H =\frac{V}{T}$$ And you know that, $$\left[\frac{\partial}{\partial P}\left(\frac{\partial S}{\partial H}\right)_p\right] = \left[\frac{\partial}{\partial H}\left(\frac{\partial S}{\partial P}\right)_H\right]$$ Can you see what you need to do next (it's a really simple step)?
subbing in becames..
(d/dp)(1/T)=(d/dH)(V/T)
i cant see how the |H and |p get switched around

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 Quote by billybob5588 subbing in becames.. (d/dp)(1/T)=(d/dH)(V/T)
Correct.
 Quote by billybob5588 i cant see how the |H and |p get switched around
Sorry my bad, I left out the subscripts of what I'm holding constant in my previous post. Does it make more sense now?

 Quote by Hootenanny Correct. Sorry my bad, I left out the subscripts of what I'm holding constant in my previous post. Does it make more sense now?
Yes! thanks you so much appreciate it heaps

going to attempt part (b) of the question ill post here if i having difficulties

thanks again!

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 Quote by billybob5588 Yes! thanks you so much appreciate it heaps going to attempt part (b) of the question ill post here if i having difficulties thanks again!
 now for part (b) of this question Using the equation of state for an ideal gas and the definition of enthalpy (H=U+pV), express 1/T and V/T as function of p and H and show that the relationship in part (a) is valid for an ideal gas. trying to understand what this question means.. i know pV=nRT, do i sub that in enthalpy equation which becomes H=U+nRT? want 1/T(p,H) and V/T(p,H) i really don't know where to start for this question and how to approach it. some hints will be much appreciated; thanks in advance.
 Blog Entries: 1 Recognitions: Gold Member Science Advisor Staff Emeritus So what you want is $$\frac{1}{T}=f(p,H)\hspace{2cm}\frac{V}{T}=F(p,h)$$ With respect to the definition of the enthalpy, bearing in mind that we are dealing with an ideal gas (no potential energy), can you write U as a function of T?

 Quote by Hootenanny So what you want is $$\frac{1}{T}=f(p,H)\hspace{2cm}\frac{V}{T}=F(p,h)$$ With respect to the definition of the enthalpy, bearing in mind that we are dealing with an ideal gas (no potential energy), can you write U as a function of T?
i thought U=0 like you said above, since its an ideal gas which makes H=pV;
i am sorry i don't understand writing U as a function of T since U is 0;

so V=H/p

pV=nRT => p(H/p)=nRT => T=H/nR

that right?

sorry i am completely puzzled by this

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 Quote by billybob5588 i thought U=0 like you said above, since its an ideal gas which makes H=pV;
Not quite. U is the internal energy, which roughly speaking is the sum of the potential energies and kinetic energies of the gas. Now as I said previously, since the gas is ideal, there is no potential energy; but there is still kinetic energy. Now, how is the kinetic energy of an ideal gas related to it's temperature? (HINT: Equipartition theorem).

 Quote by Hootenanny Not quite. U is the internal energy, which roughly speaking is the sum of the potential energies and kinetic energies of the gas. Now as I said previously, since the gas is ideal, there is no potential energy; but there is still kinetic energy. Now, how is the kinetic energy of an ideal gas related to it's temperature? (HINT: Equipartition theorem).
the relationship is;
change is kinetic energy means a change in temperature (proportional to each other).

equipartition theorem = 3/2RT

write U as a function of T is is U=3/2RT or U=C*T (where C is a constant)

so H=U+pV

H=CT+pV
that correct? and then make V/T and 1/T the subject?

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 Quote by billybob5588 the relationship is; change is kinetic energy means a change in temperature (proportional to each other). equipartition theorem = 3/2RT write U as a function of T is is U=3/2RT or U=C*T (where C is a constant) so H=U+pV H=CT+pV that correct? and then make V/T and 1/T the subject?
Are you sure it's R? Apart from that it looks good.

 Quote by Hootenanny Are you sure it's R? Apart from that it looks good.
its 3/2kT sorry;

ok so rearranging to make 1/T and V/T the subject i get
1/T=1/(H-pV)
V/T=H/TP - 1/P => for this one im not sure cause V/T is not the clear subject there is a T term in the equation

and now i should plug it into equation in part (a) then solve?

i thank you for your continuing help

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 Quote by billybob5588 its 3/2kT sorry
Better
 Quote by billybob5588 1/T=1/(H-pV)
I think your missing a factor of C here.
 Quote by billybob5588 V/T=H/TP - 1/P => for this one im not sure cause
For this one, I think it would be must easier to use the ideal gas eqaution and simply find V/T as a function of P.

 Quote by Hootenanny Better I think your missing a factor of C here. For this one, I think it would be must easier to use the ideal gas eqaution and simply find V/T as a function of P.
sorry my algebra is a bit weak

H=kT+pV where pV=nRT
H=kT+nRT
i dont see how this can help
and here probably alot more worse i know this is wrong but im trying everything.
H=kT+n^2*R^2*T^2/Vp (i got this from p=nRT/V and V=nRT/p and taking the product of this yeilds n^2*R^2*T^2/Vp)

this is embarassing

maths...