Derivatives of thermodynamic potentials edit

The thermodynamic square is mostly used to compute the derivative of any thermodynamic potential of interest. Suppose for example one desires to compute the derivative of the internal energy ${\displaystyle U}$ . The following procedure should be considered:

1. Place oneself in the thermodynamic potential of interest, namely (${\displaystyle G}$ , ${\displaystyle H}$ , ${\displaystyle U}$ , ${\displaystyle F}$ ). In our example, that would be ${\displaystyle U}$ .
2. The two opposite corners of the potential of interest represent the coefficients of the overall result. If the coefficient lies on the left hand side of the square, a negative sign should be added. In our example, an intermediate result would be ${\displaystyle dU=-p\,[{\text{Differential}}]+T\,[{\text{Differential}}]}$ .
3. In the opposite corner of each coefficient, you will find the associated differential. In our example, the opposite corner to ${\displaystyle p}$  would be ${\displaystyle V}$  (volume) and the opposite corner for ${\displaystyle T}$  would be ${\displaystyle S}$  (entropy). In our example, an interim result would be: ${\displaystyle dU=-p\,dV+T\,dS}$ . Notice that the sign convention will affect only the coefficients, not the differentials.
4. Finally, always add ${\displaystyle \mu \,dN}$ , where ${\displaystyle \mu }$  denotes the chemical potential. Therefore, we would have: ${\displaystyle dU=-p\,dV+T\,dS+\mu \,dN}$ .

The Gibbs–Duhem equation can be derived by using this technique. Notice though that the final addition of the differential of the chemical potential has to be generalized.

Maxwell relations edit

The thermodynamic square can also be used to find the first-order derivatives in the common Maxwell relations. The following procedure should be considered:

1. Looking at the four corners of the square and make a ${\displaystyle \sqcup }$  shape with the quantities of interest.
2. Read the ${\displaystyle \sqcup }$  shape in two different ways by seeing it as L and ⅃. The L will give one side of the relation and the ⅃ will give the other. Note that the partial derivative is taken along the vertical stem of L (and ⅃) while the last corner is held constant.
3. Use L to find ${\displaystyle LHS=\left({\partial S \over \partial p}\right)_{T}}$ .
4. Similarly, use ⅃ to find ${\displaystyle RHS=-\left({\partial V \over \partial T}\right)_{p}}$ . Again, notice that the sign convention affects only the variable held constant in the partial derivative, not the differentials.
5. Finally, use above equations to get the Maxwell relation: ${\displaystyle \left({\partial S \over \partial p}\right)_{T}=-\left({\partial V \over \partial T}\right)_{p}}$ .

By rotating the ${\displaystyle \sqcup }$  shape (randomly, for example by 90 degrees counterclockwise into a ${\displaystyle \sqsupset }$  shape) other relations such as: ${\displaystyle \left({\partial p \over \partial T}\right)_{V}=\left({\partial S \over \partial V}\right)_{T}}$  can be found.

Natural variables of thermodynamic potentials edit

Finally, the potential at the center of each side is a natural function of the variables at the corner of that side. So, ${\displaystyle G}$  is a natural function of ${\displaystyle p}$  and ${\displaystyle T}$ , and ${\displaystyle U}$  is a natural function of ${\displaystyle S}$  and ${\displaystyle V}$ .

• Bejan, Adrian. Advanced Engineering Thermodynamics, John Wiley & Sons, 3rd ed., 2006, p. 231 ("star diagram"). ISBN 978-0-471-67763-5
• Ganguly, Jibamitra (2009). "3.5 Thermodynamic Square: A Mnemonic Tool". Thermodynamics in Earth and Planetary Sciences. Springer. pp. 59–60. ISBN 978-3-540-77306-1.
• Klauder, L. T. Jr (1968). "Generalization of Thermodynamic Square". American Journal of Physics. 36 (6): 556–557. Bibcode:1968AmJPh..36..556K. doi:10.1119/1.1974977.