Why are there are all these different kinds of "energy"? Here's a rough idea what the point is.
Well, if you have a system of particles, and you let it reach equilibrium, there are several tendencies:
- It tends to reach a state with lowest energy.
- It tends to reach a state with highest entropy.
- It tends to reach a state with largest volume.
However, since energy, entropy and volume are related--if you set any two of them, the third is determined--it's not possible to satisfy all three tendencies at once. So the system will reach some kind of compromise between those three tendencies. Exactly what compromise is found depends on the constraints acting on the system.
Let's look at a couple of examples. Consider the following setup: We have an insulated upright cylinder (no heat goes in or out). We have a moveable insulated plate that can move up and down in the cylinder, dividing it into two chambers, with variable volume. (So basically, we have a piston). We put a certain amount of gas into each chamber. We put pressure on the plate so that it moves slowly in one direction or the other. What is the equilibrium position of the plate?
Well, since no heat can flow in or out of either chamber, we know the entropy can't change (because for reversible processes,
change in entropy is given by \delta S = \delta Q/T where \delta Q is heat added and T is temperature). So the composite system (two chambers) will change so as to minimize its total energy. But the total volume can't change--it's constrained to be the volume of the cylinder.
So let U_1(V_1) be the energy of one of the chambers when its volume is V_1 and let U_2(V_2) be the energy of the other chamber when its volume is V_2.
Total energy : U_1(V_1) + U_2(V_2) = E.
Total volume : V_1 + V_2 = V
So we want to minimize E subject to the constraint that V is constant.
So to minimize one quantity subject to a constraint, we use the technique of Lagrange multipliers. Have you studied those in calculus? I'm hoping you did. If not, ask about them.
Let \lambda be the multiplier for the volume constraint. We minimize
U_1 + U_2 - \lambda(V_1 + V_2)
with respect to V_1 and V_2 by taking derivatives and setting them to zero:
\frac{\partial U_1}{\partial V_1} - \lambda = 0
\frac{\partial U_2}{\partial V_2} - \lambda = 0
So using thermodynamics, \frac{\partial U_1}{\partial V_1} = -P_1, the negative of the pressure of the first chamber, and similarly, \frac{\partial U_2}{\partial V_2} = -P_2, the negative of the pressure of the second chamber. So these two equations tell us that the pressures are equal at equilibrium, and the multiplier \lambda turns out to be -P, the negative of this common pressure.
So minimizing the total energy of the two chambers turns out to be equivalent to minimizing the quantity
U - \lambda V = U + PV for each chamber separately, with P held constant.
So in such a circumstance where the pressure is shared between the two subsystems, but heat is not, the thing that is minimized is enthalpy = H = U+PV for each subsystem separately. Equilibrium between the two chambers is achieved when H is minimum for each system, and P is the same for the two systems.
Now, suppose we change the problem so that the total energy is held constant (no energy comes into or out of the two-chamber system), but the plate is now no longer an insulator--heat can flow between the two chambers. In this case, you can't minimize total energy, since that's constant. But you can maximize the entropy. So we want to maximize the total entropy S subject to the constraints that the total energy and total volume are held constant.
Again using Lagrange multipliers, we want to maximize:
S_1 + S_2 -\lambda_1 (U_1 + U_2) -\lambda_2 (V_1 + V_2)
where \lambda_1 is the multiplier for the total energy constraint, and \lambda_2 is the multiplier for the total volume constraint. So maximizing gives us these equations:
\frac{\partial S_1}{\partial U_1} - \lambda_1 = 0
\frac{\partial S_2}{\partial U_2} - \lambda_1 = 0
\frac{\partial S_1}{\partial V_1} - \lambda_2 = 0
\frac{\partial S_2}{\partial V_2} - \lambda_2 = 0
Again, using thermodynamics, \frac{\partial S_1}{\partial U_1} = 1/T_1where T_1 is the temperature in the first chamber. \frac{\partial S_2}{\partial U_2} = 1/T_2where T_2 is the temperature in the second chamber. \frac{\partial S_1}{\partial V_1} = P_1/T_1where P_1 is the pressure in the first chamber. \frac{\partial S_2}{\partial V_2} = P_2/T_2where P_2 is the pressure in the second chamber. So our four equations tell us that at equilibrium:<br />
<br />
T_1 = T_2 = \lambda_1<br />
P_1/T_1 = P_2/T_2 = \lambda_2<br />
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So at equilibrium, the pressures and temperatures of the two chambers must be equal. So maximizing the total energy of the two chambers subject to the volume and total energy constraint is equivalent to maximizing the quantity:<br />
<br />
S - \lambda_1 U - \lambda_2 V = S - U/T - PV/T<br />
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for each chamber separately, with P and T held constant. So let's define a new quantity, G, the Gibbs Free Energy, which is defined by:<br />
<br />
G = U+PV-TS<br />
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Then maximizing the entropy subject to the volume and total energy constraints is equivalent to maximizing S - U/T - PV/T = -G/T for each chamber separately, with T and P held constant. Maximizing -G/T with T held constant is the same thing as minimizing G.<br />
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So in such a situation, the two systems are in equilibrium if G is minimum for each system separately and the temperature and pressures are equal.
