## DS v dV s

Similar equalities can be obtained using the other Gibbs equations that relate other properties, i.e., enthalpy, H = H (S, p) and {dH = T dS + V dp}, the Gibbs function G = G(T,p) and {dG = - SdT + V dp}, and the Helmholtz potential F = F(T,V) and {dF = -S dT - p dV}. These are also useful in the elimination of some of the non-measurable terms and thermodynamic properties.

In Somerton and Arnas (1985), the theory as well as the use of the method is given. For the general equation of {dZ = M dY + N dX}, the Jacobian formulation can be written as {[Z,^] = M[Y,^] + N[X,^]}, where is any other thermodynamic property. Also, the equalities [Z,Z] = 0, [Z,^]=-[^,Z], and d(X, Y) | =1 are very useful. From fundamentals of mathematics we have d( X, Y)) "

dY" = j. Using the first law of thermodynamics for a cycle, we get

(<SQ = <SW) since (<<dU)= 0 . In Gibbs form, {dU=TdS - pdV} giving for a cycle [T ds = p dV] which in Jacobian form becomes [T,S] = [p,V]. The systematic use of these will permit one to convert nonmeasurable terms into measurable ones including p, V, T, cp, cv, a, and kt.

The general methodology, therefore, is

1. Write down the given in terms of Jacobians.

2. Reduce by Maxwell equations using the various Gibbs equations.

3. Use the definition of cp, cv, a, and ktto further reduce the given equation.

4. If everything is done correctly, the result should only containp, v, T, cp, cv, a, and kt. If not, an error has been made; you need to go back and redo everything.

If this methodology is not used, then one goes around and around until a solution is obtained and one does not know if the route taken is the correct one. The advantage of the method of Jacobians is that the result shows if a mistake has been made.

The examples given in Somerton and Arnas (1985) are for learning the methodology. We will also look at other relations and get the physical significance of some other important thermodynamic phenomena, for example, throttling,

dp I

dp d1

U i cp and the speed of sound in any medium, dp i dv )t s

As an example, start with cv = T| — | . Therefore, following the me, d T, thodology given,

STEP 1:

.dT Jv 3(T, v) ^ 3(p, T) STEP 2: However, the denominator is by definition {-v kt}. Therefore.