Eight Basic Equations of Thermodynamics

Now that we know that dqiev = TdS we, can cut a huge time-saving swath through all of thermodynamics and concentrate on the essentials.  

The following assumes aU work is reversible PV (gas) work so that dw = —PdV. 

The following assumes all heat changes are computed reversibly so that dq = T dS. 

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 = dq + dw = dq Pdv so we have dU =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 5 by a temperature to get an energy unit. Now, consider the first law again with addition and subtraction of S dT to find what is now known as the Helmholtz free energy. A.  

---