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Obtaining the equation of state for a gas

I have this question (q3) and I have absolutely no idea how to approach it. I have never seen anything vaguely similar to it. Please help!

photo (4).JPG

Thanks!
Original post by DonnieBrasco
I have this question (q3) and I have absolutely no idea how to approach it. I have never seen anything vaguely similar to it. Please help!

photo (4).JPG

Thanks!


Peculiar question. Especially since what they have given you is already an equation of state! It gives the Gibbs free energy as a function of pressure and temperature. Since both G, p and T are state functions, this is an equation of state, by definition!
(edited 10 years ago)
Reply 2
Avatar for Bradshaw
Bradshaw
6
Original post by Plato's Trousers
Peculiar question. Especially since what they have given you is already an equation of state! It gives the Gibbs free energy as a function of pressure and temperature. Since both G, p and T are state functions, this is an equation of state, by definition!


What they are looking for I think (although I may well be wrong) is something like pV=RT which is the equation of state for an ideal gas.

Use dG/dp at constant T = V and work out what pV equals for this non ideal gas.
Reply 3
Avatar for Bradshaw
Bradshaw
6
At this point it's also common to bring the RT to the other side to give a virial expansion in terms of densities I think!

I.e pV/NT = 1 + ...

Edit: actually thinking about it you'll have a quadratic in p so probably discard this part!
(edited 10 years ago)
Original post by Bradshaw
At this point it's also common to bring the RT to the other side to give a virial expansion in terms of densities I think!

I.e pV/NT = 1 + ...

Edit: actually thinking about it you'll have a quadratic in p so probably discard this part!



Original post by Bradshaw
What they are looking for I think (although I may well be wrong) is something like pV=RT which is the equation of state for an ideal gas.

Use dG/dp at constant T = V and work out what pV equals for this non ideal gas.



Original post by Plato's Trousers
Peculiar question. Especially since what they have given you is already an equation of state! It gives the Gibbs free energy as a function of pressure and temperature. Since both G, p and T are state functions, this is an equation of state, by definition!



Original post by DonnieBrasco
I have this question (q3) and I have absolutely no idea how to approach it. I have never seen anything vaguely similar to it. Please help!

photo (4).JPG

Thanks!


It's the virial expansion equation of state....?

Probably you just have to rearrange it, to what form I don't know though! Very odd question though what with it already being an equation of state.
Original post by Bradshaw
What they are looking for I think (although I may well be wrong) is something like pV=RT which is the equation of state for an ideal gas.

Use dG/dp at constant T = V and work out what pV equals for this non ideal gas.


Yes, I wondered if they just wanted the derivative w.r.t. pressure. That's usually what they're asking for when they give you a state function as a function of another state function. (For example giving you U as a function of T and texpecting the derivative dU/dT = Cv).

But I can't remember what dG/dp gives you.
(edited 10 years ago)
Reply 6
Avatar for Borek
Borek
4
Original post by Plato's Trousers
But I can't remember what dG/dp gives you.


I don't remember neither, but what are books for? :wink:

(∂G∂p)T=V\left(\frac {\partial G}{\partial p}\right)_T = V
Original post by Borek
I don't remember neither, but what are books for? :wink:

(∂G∂p)T=V\left(\frac {\partial G}{\partial p}\right)_T = V


ok, so the question gives us

Gm=RTln⁡p+A+Bp+Cp22+Dp33G_m=RT \ln p +A+Bp+\dfrac{Cp^2}{2}+\dfrac{Dp^3}{3}

so

(∂G∂p)T=RTp+B+Cp+Dp2\left(\dfrac{\partial G}{\partial p}\right)_T=\dfrac{RT}{p}+B+Cp+Dp^2

and since

(∂G∂p)T=V\left(\dfrac{\partial G}{\partial p}\right)_T=V

we have

V=RTp+B+Cp+Dp2V=\dfrac{RT}{p}+B+Cp+Dp^2

or

pV=RT+pB+Cp2+Dp3pV=RT+pB+Cp^2+Dp^3

which I guess is the equation of state they are after for this non-ideal gas.

(note there is no nn as this is the molar Gibbs energy, so I should really have written Vm, but you get the idea)
(edited 10 years ago)

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