# Note 5 - Note 5. Equilibrium I: Free Energy 5.1 Helmholtz...

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1 Note 5. Equilibrium I: Free Energy 5.1 Helmholtz free energy 5.1.1 Derivation In the previous section, I alluded to a new state function which will help us sort out the total entropy change. This new state function is called the free energy. When we created the enthalpy state function, we said that we wanted something to reflect the change in heat at constant pressure. In this case, we have a very different goal. In the previous section, we saw that using just entropy alone required us to consider the entropy changes of both the system and its surroundings. This is often difficult to do and best to avoid if possible. But how can we avoid doing this? One way around this is to saw that we want to build a state function which reflects the nature of equilibrium. To find such a state function, we return to the idea of maximum work. In the last unit, we defined equilibrium as the case when a system can do maximum work. For example, a reversible reaction can do the maximum amount of work. We will now introduce thermodynamic functions which can indicate whether one can do the maximum amount of work and thus tell us whether we are at equilibrium. Also, we will in this section always consider systems at constant temperature. For a reversible reaction, we have rev rev dq dw dU + = Since we can relate the change in reversible heat to entropy by T dq dS rev = We can write TdS dw dU rev + = rearranging, we get TdS dU dw rev = Thus, it is natural to define a new state function, called the Helmholtz free energy A TS U A = since (at constant temperature) rev dw TdS dU dA = =

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2 One way to think of the free energy is that it is a measure of the maximum work the system can do on its surroundings . Just as mechanical systems work to decrease U , molecular systems (at constant T and V ) try to reduce A to reach equilibrium. What does this mean? In mechanical systems, we expect spontaneous processes to occur only if dU < 0 . Apples spontaneously fall from trees, but they never spontaneously leap from the ground into the sky. Similarly, spontaneous processes in molecular systems occur only if dA < 0 . If there is a state with lower free energy, then the system will eventually get there. What happens at equilibrium? At equilibrium, we expect that the system will do no work and thus dA = 0. This naturally parallels dU = 0 in mechanical systems. 5.2 Another way to look at free energy and equilibria In the Note 3, we talked about T dq dS and said that dS > dq/T for spontaneous processes and dS = dq/T for reversible processes. Now, we’ll see what the interpretations of this are for other state functions. Let’s take dU for example. From the first law, we have dU = dq + dw and thus dq=dU - dw . For a gas, we have dw = -PdV and thus dq = dU + PdV . Putting this into the in equality above, we get TdS dU + PdV . Rearranging terms, we get PdV TdS dU Thus, we see at constant S and V we have dS = 0 and dV = 0 , thus leading to the inequality 0 dU
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## This note was uploaded on 09/30/2008 for the course CH 353M taught by Professor Lim during the Spring '08 term at University of Texas at Austin.

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Note 5 - Note 5. Equilibrium I: Free Energy 5.1 Helmholtz...

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