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88                                                   Essentials of Physical Chemistry

            note that living systems have to expend energy to constantly combat the increase of the randomness
            in ‘‘biological errors.’’ While very clever repair enzymes do amazing things to reduce randomness,
            the process of aging is a manifestation of increasing entropy.


            SUMMARY OF THE SECOND LAW OF THERMODYNAMICS
            We have discovered there is a state variable ‘‘S’’ that represents an amount of randomness as in a
            phase change and is related to the reversible heat change and the temperature at which it occurs as

                                                   T
                                     þ             ð 2
                                       dq rev  DH    C P          T 2
                                                       dT ¼ C P ln    :
                                        T     T      T            T 1
                                DS ¼       ¼     ¼
                                                   T 1
            We also know that S tends to increase spontaneously; it is a measure of spontaneity in a closed
            system. However, an important motivation for the Carnot cycle proof is to show the very important
            relationship: dS ¼ dq rev =T. This leads to a cascade of important relationships that we present
            rapidly here because we are eager to get to some ‘‘essential’’ equations in our condensed course.
            Perhaps you have not realized that our process of discovery of S will open a door to more helpful
            thermodynamic relationships.


            EIGHT BASIC EQUATIONS OF THERMODYNAMICS
            Now that we know that dq rev ¼ TdS we, can cut a huge time-saving swath through all of
            thermodynamics and concentrate on the ‘‘essentials.’’

              1. The following assumes all work is reversible PV (gas) work so that dw ¼ PdV.
              2. The following assumes all heat changes are computed reversibly so that dq ¼ TdS.

            With these two conditions, we can leap forward over 100 years of developments in thermodynamics
            and that is why it was ‘‘essential’’ to do the Carnot cycle proof.
              From the first law: dU ¼ dq þ dw ¼ dq   Pdv so we have dU ¼ TdS   PdV. We have already
            noted that the energy tends to go down while entropy tends to go up, so in nature there is really a
            constant trade-off occurring between these two tendencies and what really matters is the difference
            between the two tendencies. The first treatment of this trade-off was given by H. L. F. von
            Helmholtz (1821–1894), who defined a new function ‘‘A’’ so that A   U   TS where we have to
            multiply S by a temperature to get an energy unit. Now, consider the first law again with addition
            and subtraction of SdT to find what is now known as the ‘‘Helmholtz free energy, A.’’

                dU ¼ TdS   PdV þ SdT   SdT ¼ TdS þ SdT   PdV   SdT ¼ d(TS)   PdV   SdT

            and we find

             dU   d(TS) ¼ SdT   PdV ¼ d(U   TS) ¼ SdT   PdV ¼ dA     or dA ¼ SdT   PdV:

            We see from this that if both T and V are held constant dT ¼ 0 and dV ¼ 0, dA ¼ 0 so that under
            those conditions (U   TS) ¼ 0 and so the Helmholtz energy A indicates an equilibrium; a balanced
            trade-off between increasing entropy and decreasing energy. That is very interesting and a math-
            ematical truth under reversible heat and work conditions. However, it turns out that it is not very
            useful in the laboratory, since it implies that pressures must be the only variable if T and V are held
            constant. Certain experiments can be designed to meet these conditions, but more likely the pressure
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