The Maxwell Relations

Transcription

1 CHEM 331 Physical Chemistry Fall 2017 The Maxwell Relations We now turn to one last and very useful consequence of the thermodynamic state functions we have been considering. Each of these potentials (U, H, A, G) represent state functions. As such, integrals over their differentials (du, dh, da, dg) are path independent; it does not matter how the change is carried out, only the initial and final states of the system are important for the evaluation of U, H, A and G. This means that the differentials are exact. If a function F(x,y) is a state function and its differential is exact: df M(x,y) dx + N(x,y) dy then its mixed second partial derivatives are equal: Since and, we have: Established in 1871 by James Clerk Maxwell, the Maxwell Relations are derived from the fact mixed second partial derivatives of the thermodynamic potentials are equivalent in the manner demonstrated above for F(x,y). James Clerk Maxwell James_Clerk_Maxwell

2 For example, the differential of the state function H(S,P) can be written as: dh T ds + V dp ds + dp As we have already noted: T and V From the fact that the mixed second partial derivatives of H(S,P) must be equal: we can derive a Maxwell Relation by inserting the appropriate first derivatives into each of the above expressions: And now we have the Maxwell Relation: Applying this procedure to the four thermodynamic potentials U(S,V), H(S,P), A(T,V), and G(T,P), we obtain the four usual Maxwell Relations. Internal Energy du T ds - P dv ds + dv Enthalpy dh T ds + V dp ds + dp

3 Helmholtz Free Energy da - S dt - P dv dt + dv Gibbs Free Energy dg - S dt + V dp dt + dp The Maxwell Relations are themselves useful in transforming hard to measure thermodynamic derivatives into those that are much easier to measure. Certain kinds of data are easily measured, whereas others may involve a great investment of time and effort. Thus, for example; we may be engaged in high-pressure experiments in which we need to know how the enthalpy changes with pressure at constant temperature. We need, in other words, the derivative ( H/ P) T. We could determine the derivative by undertaking a series of calorimetric experiments over a range of pressure values, but calorimetric experiments are difficult and time-consuming. We want to determine the information in the easiest possible way. We have meters to measure temperature, pressure, and volume. We do not have entropy meters, enthalpy meters, free-energy meters, or Helmholtz-energy meters. Variables that are easily measurable include compressibilities, coefficients of thermal expansion, and heat capacities. J. Phillip Bromberg Physical Chemistry, 2 nd Ed. The situation is somewhat stronger than Bromberg suggests. It turns out that a very limited number of measured quantities is required to determine almost all the desired derivatives. In the practical applications of thermodynamics the experimental situation to be analyzed frequently dictates a partial derivative to be evaluated. For instance, we may be concerned with the analysis of the temperature change which is required to maintain the volume of a singlecomponent system constant if thepressure is increased slightly. This temperature change is evidently dt dp and consequently we are interested in an evaluation of the derivative ( T/ P) V,N. A general feature of the derivatives that arise in this way is that they are likely to involve constant mole numbers and that they generally involve both intensive and extensive parameters. Of all such derivatives, only three can be independent, and any given derivative can be expressed by an identity in terms of an arbitrarily chosen set of three basic derivatives. This set is conventionally chosen as C p, and. H.B. Callen Thermodynamics

4 Callen then proceeds with a proof of this last statement, a proof you are welcome to consider by turning to his text. The manipulation of hard to measure thermodynamic derivatives into those that are more easily measured is known as the reduction of derivatives. At this point we will demonstrate the reduction of derivatives by providing a couple well chosen of examples. It is at this point that we find ourselves returning to all of those cases during the progress of the class in which we indicated that a particular expression would be "proven later". We now provide those proofs. So, sit back, relax and enjoy the examples. Thermodynamic Equations of State We are familiar with several Equations of State; the Ideal Gas Equation, the van der Waals Equation, etc. Below we derive an Equation of State that is general and applicable for any system. We previously defined an Equation of State as a relationship between the variables P, T, and of the form: g(p,, T) 0 For a general system, we have: du T ds - P dv Dividing by dv and holding T constant, we have: T - P Applying the Maxwell Relation We obtain: T - P This can be rearranged into a more familiar form for an Equation of State, as specified above. P T - Another form of the Thermodynamic Equation of State, which can be derived in a similar manner (you may want to try this derivation), is given by the following:

5 V T + The Derivative of the Internal Energy wrspt Volume at Const. T We wish to reduce the derivative: We start with: U A + TS and then form the needed derivative: + T Now, we apply an appropriate Maxwell Relation: and get: + T Finally, we recognize: -P and This then gives us: -P + T - P + Notice we have converted the need to make U vs. V measurements into making measurements of and, which are much easier. For an Ideal Gas: P

6 So, And, - P + T - P + P 0; which is Joule's Law. For a van der Waals Gas: P So, And, - P + T This is a result we have used previously. Heat Capacity Difference; C p - C v Previously, we showed: C p - C v We can now use our above expression for the derivative of the internal energy wrspt V at constant T: C p - C v Now we use: and V to obtain the desired reduction: C p - C v Joule-Thomson Coefficient To obtain a reduction for the Joule-Thomson Coefficient: JT - Here we start with:

7 + T Applying the Maxwell Relation: we obtain: + T Recognizing: V and V we arrive at the final reduction: JT - - These and other important derivatives are given in the following appendicies: Basic Thermodynamic Equations First Derivatives of T, P, V, and S First Derivatives of U, H, A and G You should notice that the use of the Maxwell Relations is only one piece in the puzzle that is the path from our initial derivative to its final reduced form. Along this path we may also need to use an expression for du, ds, dh, da or dg as well as be able to manipulate derivatives generally. To this end, you should review the following three relationships involving partial derivatives:

8 Let's proceed with another example. I've drawn it from our "First Derivatives of U, H, A and G" handout. Another Example I've chosen for this example, almost at random, a derivative of the enthalpy: Now to the reduction of this derivative. Start with: dh T ds + V dp and form the needed derivative: Now apply a Maxwell Relation and invert the last derivative: T Finally, use our usual transformations to and : At this point you should peruse the "First Derivatives of U, H, A and G" and "First Derivatives of T, P, V, and S" apendicies and imagine how you might carry-out each of the indicated "reductions". Just for fun, you might want to pick out a few and prove the "reductions".

9 Appendix - Basic Thermodynamic Equations C v C P C P - C V JT

10 Appendix - First Derivatives of T, P, V, and S

11 Appendix - First Derivatives of U, H, A and G Internal Energy U Enthalpy H Helmholtz A Gibbs G