MAXWELL RELATIONS AND EXACT DIFFERENTIALS CLASS NOTES

MAXWELL RELATIONS AND EXACT

DIFFERENTIALS CLASS NOTES

Natural Variables

The variables kept constant in a process are the natural variables of that process. Thermodynamic potential are a function of its natural variables only.  Partial derivatives of that potential with respect to its natural variables determine the thermodynamic properties. Partial derivatives of that potential cannot determine the thermodynamic properties. It is true If the thermodynamic potential is not known in terms of its natural variables,

All the four thermodynamic potentials involve the use of Natural variables. Natural variables form from every combination of the T–S and P–V variables. Pairs of conjugate variables are excluded.  Conjugate pairs form from quantities µ_i and N_i  .

Where μ_i is the chemical potential for an i-type particle

N_i is the number of particles of type i in the system

Maxwell Relations

Second order partial differentials of a thermodynamic potential are based with respect to natural variables. But it is independent of the order of differentiation. T and S are thermal natural variables and p and v are mechanical natural variables. James Clerk Maxwell were the first to derive the Maxwell relations. Euler’s Reciprocity Law give Maxwell relations. The thermodynamic potentials are  given below:

∂²P/∂y ∂x    = ∂²P/∂y ∂x  ( Euler’s Reciprocity Law)

The four most common Maxwell equations are

dU = T dS –PdV = + (∂T/∂V)_S  = – (∂P/∂S)_v = ∂²U/ (∂S ∂V)

dH = T dS + V dP = + (∂T/∂P)_S  = + (∂V/∂S)_p = ∂²H/ (∂S ∂p)

dF = –S dT – P dV =  + (∂S/∂V)_T  = + (∂P/∂T)_v = — ∂²F/ (∂T ∂V)

dG = –S dT+ P dV = –(∂S/∂P)_T  = + (∂V/∂T)_P = ∂²G/ (∂T ∂P)

Second order partial differential equations give thermodynamic potentials. It requires the use of measurable thermodynamic parameters.

 Exact differentials in thermodynamics are dU, dW and  dQ

Non exact differentials in thermodynamics are dU, dQ, dW

Use of exact and non exact differentials

For example

dU =dQ-dW + Chemical potential—————–applicable for a non reversible  change

For a reversible change only

dQ = T dS——————————only for a reversible process

dW = p dV—————————– only for a reversible process

REFERENCES

1. Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971

2. Elements of Statistical Thermodynamics (2nd Edition), L.K. Nash, Addison-Wesley, 1974

3. Thermal Physics (2nd Edition), Kittel, Charles & Kroemer, Herbert (1980).

4. Encyclopedia of Physics (2nd Edition, W. H. Freeman Company. McGraw Hill,”), C.B. Parker, 1994

5. Thermodynamics – an Engineering Approach , Cengel, Yunus A., & Boles, Michael A, McGraw Hill, 2002

6. Statistical Physics (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008

7. Thermodynamics, From Concepts to Applications (2nd Edition), A. Shavit, C. Gutfinger, CRC Press (Taylor and Francis Group, USA), 2009