Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Physical Chemistry

IIT JAM : Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

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Maxwell Thermodynamic Equations

In thermodynamics, the Maxwell equations are a set of equations derived by application of Euler's reciprocity relation to the thermodynamic characteristic functions. 

The Maxwell relations, first derived by James Clerk Maxwell, are the following expressions:

  • dH = TdS + VdP
  • dG = –SdT + VdP
  • dA = –PdV – SdT
  • dV = TdS – PdV

Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Thermodynamic Magic Square

The magic square plays a very important role in thermodynamic. By using magic square we find many important relationships between thermodynamic function or quantity.

These relations are known as Maxwell thermodynamic equations. S, P, V, T are at the corner of the square & known as thermodynamic coordinates.
We can find the Maxwell thermodynamic equation by using magic square.

Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

 Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

dA = (–P)dV + (–S)dT
dA = –PdV – SdT …(3)
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev 

dG = VdP + (–S)dT
dG = VdP – SdT …(4) i.e.,

we can find the Maxwell thermodynamic equation by using magic square.

Maxwell Relationship between Thermodynamic Coordinates
i.e., S, P, V, T. 

 Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev                     Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev..........(2)
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev                        Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev..............(3)
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev                          Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev ................(4)

We can find the relation between thermodynamic coordinates by using Euler’s theorem.
We know that if z s the function of x & y then z = f(x, y)

Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

if z is a state function then it follows Euler’s theorem
i.e.    Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Four Maxwell thermodynamic equation are given below:
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
dH = TdS + VdP
dG = VdP – SdT
dA = –PdV – SdT
dV = TdS – PdV

We know that H, G, A & V are state function i.e., it follow the Euler’s theorem.

dH = TdS + VdP


i.e.    Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

dG = VdP – SdT
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

dA = –PdV – SdT
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
i.e.Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

dV = TdS – PdV
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Expansivity and Compressibility

The gas expands on heating through the expansion of gases is much more than that of liquids.  Similar gas compressed on increasing the pressure.

“The variation of volume V with temperature T, keeping pressure P constant is called the coefficient of thermal expansion or expansivity. It is denoted by α.  Thus

 Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Similarly “the variation of V with P, keeping T constant, is called the coefficient of isothermal compressibility or compressibility. It is denoted by β

 Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Another coefficient, isochoric thermal expansion coefficient, when the variation of P with T, keeping V constant is represented by γ

 Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Q.1.  Find the value of α, β, γ for an ideal gas?
Sol. Ideal gas equation is
PV = nRT         (for n mole)

PV = RT           (for 1 mole)

 Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev         for ideal gas

PV = nRT

Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev …(1)

 Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

 Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev  for ideal gas

PV = nRT

 Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

 Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

PV = nRT
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev
Thermodynamic Magic Square - Thermodynamic Chemistry Notes | EduRev

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