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 itype 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:
∂^{2}P/∂y ∂x = ∂^{2}P/∂y ∂x ( Euler’s Reciprocity Law)
The four most common Maxwell equations are
dU = T dS –PdV = + (∂T/∂V)_{S } = – (∂P/∂S)_{v} = ∂^{2}U/ (∂S ∂V)
dH = T dS + V dP = + (∂T/∂P)_{S } = + (∂V/∂S)_{p} = ∂^{2}H/ (∂S ∂p)
dF = –S dT – P dV = + (∂S/∂V)_{T } = + (∂P/∂T)_{v} = — ∂^{2}F/ (∂T ∂V)
dG = –S dT+ P dV = –(∂S/∂P)_{T } = + (∂V/∂T)_{P} = ∂^{2}G/ (∂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 =dQdW + 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

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

Elements of Statistical Thermodynamics (2nd Edition), L.K. Nash, AddisonWesley, 1974

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

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

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

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

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