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Thermodynamic Magic Square | Physical Chemistry PDF Download

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 | Physical Chemistry

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 | Physical Chemistry

 Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry

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

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 | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry                     Thermodynamic Magic Square | Physical Chemistry..........(2)
Thermodynamic Magic Square | Physical Chemistry                        Thermodynamic Magic Square | Physical Chemistry..............(3)
Thermodynamic Magic Square | Physical Chemistry                          Thermodynamic Magic Square | Physical Chemistry ................(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 | Physical Chemistry

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

Thermodynamic Magic Square | Physical Chemistry

Four Maxwell thermodynamic equation are given below:
Thermodynamic Magic Square | Physical Chemistry
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 | Physical Chemistry

dG = VdP – SdT
Thermodynamic Magic Square | Physical Chemistry

dA = –PdV – SdT
Thermodynamic Magic Square | Physical Chemistry
i.e.Thermodynamic Magic Square | Physical Chemistry

dV = TdS – PdV
Thermodynamic Magic Square | Physical Chemistry

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 | Physical Chemistry

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 | Physical Chemistry

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

 Thermodynamic Magic Square | Physical Chemistry

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 | Physical Chemistry

Thermodynamic Magic Square | Physical Chemistry         for ideal gas

PV = nRT

Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry

Thermodynamic Magic Square | Physical Chemistry …(1)

 Thermodynamic Magic Square | Physical Chemistry

 Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry  for ideal gas

PV = nRT

 Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry

 Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry

PV = nRT
Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry

Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry
Thermodynamic Magic Square | Physical Chemistry

The document Thermodynamic Magic Square | Physical Chemistry is a part of the Chemistry Course Physical Chemistry.
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FAQs on Thermodynamic Magic Square - Physical Chemistry

1. What are Maxwell's thermodynamic equations?
Ans. Maxwell's thermodynamic equations are a set of four equations that describe the relationships between the various thermodynamic properties of a system. They are derived from the laws of thermodynamics and provide a mathematical framework for understanding the behavior of thermodynamic systems.
2. What is the thermodynamic magic square?
Ans. The thermodynamic magic square is a graphical representation of Maxwell's thermodynamic equations. It consists of a square divided into four quadrants, with each quadrant representing one of the four equations. The magic square helps in visualizing and understanding the interconnections between different thermodynamic variables.
3. What is expansivity in thermodynamics?
Ans. Expansivity, also known as thermal expansion coefficient, is a measure of how much a material expands or contracts in response to changes in temperature. It quantifies the fractional change in volume per unit change in temperature. Expansivity is an important property in thermodynamics as it affects the behavior of materials under different temperature conditions.
4. What is compressibility in thermodynamics?
Ans. Compressibility is a measure of how much a substance can be compressed or reduced in volume by the application of external pressure. It is defined as the reciprocal of the bulk modulus of elasticity. Compressibility is an important parameter in thermodynamics as it determines how easily a substance can be compressed or how much its volume changes under pressure.
5. How are the thermodynamic magic square and Maxwell's equations useful in practical applications?
Ans. The thermodynamic magic square and Maxwell's equations are useful in various practical applications. They provide a mathematical framework for understanding and predicting the behavior of thermodynamic systems, such as heat engines, refrigeration systems, and chemical reactions. These equations help in designing and optimizing such systems by providing insights into the relationships between different thermodynamic variables.
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