Thermodynamics: Small Question On Gibb's Free Energy

In summary: So if I have two systems at the same temperature and pressure, the total internal energy (equilibrium internal energy) in the second system would be twice as much as the first system even though there are twice as many particles.In summary, the Gibb's free energy increases linearly with the number of particles. Pressure is an intensive variable and the entropy doubles when the size of the system is doubled.
  • #1
doubleB
19
0
Hi to everyone at the forum.
I've attached the question in word format so you can see it properly but
it is basically this:

-----
The Gibb's Free Energy is:

G(T, P, N) = U - TS + PV

Where N is the number of particles. (Only one type of particle is present).

Explain why the N-dependence of G is given by:

G(T, P, N) = Ng(T, P)

Where g(T, P) is independent of N.

-----

So far the only thing I can think is that the number of particles, N, is not dependent on anything else and can be factored out. But to be honest I don't understand the question very well and can see how my answer (if it is an answer) is rather flimsy.

Thank you for any help you can give,

Ben
 

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  • #2
Hi Ben,

Welcome to Physics Forums! To understand what's going on here, ask yourself the following question: what would happen to the Gibbs free energy if I doubled the size of the system while keeping all the intensive variables (like T and P) fixed?
 
  • #3
I thought that pressure was an extensive variable? In Gibb's free energy is it assumed that pressure is always constant?
I guess that if you doubled the size of the system, keeping pressure constant, then you would need twice as many particles. The Gibb's free energy would double as well?
Ben
 
  • #4
Nope, pressure is an intensive variable. And yes, the Gibbs free energy describes a system where you control the pressure. You are right that the Gibbs potential doubles, but I sense some uncertainty on your part. To understand what's going on a little better, look at your expression for the Gibbs potential in terms energy, entropy, and volume. Doubling the size of the system will clearly double the volume (if you keep pressure and temperature fixed). What does such a doubling do to the entropy and energy?

Once you're satisfied that the Gibbs potential doubles when you double N and keep T and P fixed, it's easy to show that the Gibbs potential can depend on N only linearly.
 
  • #5
God that was stupid of me! Of course pressure is extensive. I had (unwisely) consulted wikipedia where the entry is wrong for intensive/extensive.
Right well as far I can see I would say that by doubling the volume while keeping T and P fixed would result in a doubling of entropy and a halving of internal energy? Although now I'm feeling quite confused again; internal energy doesn't depend on anything but temperature, is that right?
In that case shouldn't it just remain constant if T is fixed?
Ben
 
  • #6
I think you just typed the wrong word, but just to be sure, pressure is intensive.

You're right about entropy, but why do you think the energy would be cut in half? That doesn't make any sense. The internal energy of an ideal gas depends on T and N, but the dependence on N is simple. Think of it this way, if you had a monoatomic ideal gas at temperature T, the internal energy per particle is 3/2 kT. So if I have another system with twice as many particles at the same temperature, what would happen to the total internal energy?
 
  • #7
Ok I see, so internal energy will double for twice the number of particles.
So overall will the - S and + V doubling sort of cancel out leaving the Gibb's energy doubled?
Also, why does the entropy double for double the size of the system? Is it because there are now twice as many particles in twice the volume and so they are more disordered?
Thanks for the continued help,
Ben
 
  • #8
Oops, just noticed I typed extensive again in that post before! Yep, pressure's intensive.
 

1. What is Gibb's Free Energy?

Gibb's Free Energy, also known as Gibbs Energy or G, is a thermodynamic property that measures the amount of energy in a system that is available to do work. It takes into account both the system's enthalpy (H) and entropy (S) to determine whether a reaction or process will be spontaneous or non-spontaneous.

2. How is Gibb's Free Energy calculated?

Gibb's Free Energy is calculated using the equation G = H - TS, where G is the Gibb's Free Energy, H is the enthalpy, T is the temperature in Kelvin, and S is the entropy. This equation can be used to determine the spontaneity of a reaction or process at a given temperature.

3. What is the significance of a negative Gibb's Free Energy?

A negative Gibb's Free Energy (ΔG < 0) indicates that a reaction or process is spontaneous and can occur without the input of external energy. This means that the system has a lower overall energy state after the reaction or process has taken place.

4. Can Gibb's Free Energy be used to predict the direction of a reaction?

Yes, Gibb's Free Energy can be used to predict the direction of a reaction. A negative ΔG indicates a spontaneous reaction proceeding in the forward direction, while a positive ΔG indicates a non-spontaneous reaction that will proceed in the reverse direction.

5. How is Gibb's Free Energy related to equilibrium?

At equilibrium, the Gibb's Free Energy is at its minimum value, indicating that the system has reached a stable state where no further changes will occur. This is because at equilibrium, the rates of the forward and reverse reactions are equal, resulting in a net change of zero in the free energy of the system.

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