Introduction

- In today’s article, we are going to study about ‘
**Maxwell equations**‘ How beneficial it is and what is the basics of concept of these equations you should know before getting a detailed information on it.

# Maxwell equations/ relation

The state of homogeneous system is completely determine if we know its mass and any two of the thermodynamic variables (p,v,t,u,s) maxwell’s relation relate the (p,v,t,u,s).

## Maxwell’s first relation – from first thermodynamic law

We have,

dU = dQ – dW

dU = dQ – PdV

And dU = TdS- PdV …(1)

Internal energy is function of s and dV then by partial differentiation,

( ∂U /∂S)at constant V = T …(2)

Similarly, ( ∂U /∂V)at constant S = -P …(3)

By the property of exact Differentiation,

[ ∂/∂V (∂U/ ∂V) at constant V] at constant S = [ ∂/∂S (∂U/∂V)at constant S] at constant V

( ∂T /∂V)at constant S =- ( ∂P /∂S)at constant V …(3)

## Maxwell’s second equation

By the definition of helmholtz free energy

F= U- TS …(1)

For small changes equation first becomes,

dF= dU – SdT – TdS

We know dU – TdS = -PdV

dF= – SdT – PdV

Thus f is a function of V and T, so that

( dF/dT)at constant V = -S …(2)

And ( dF/dV)at constant T = -P …(3)

Definition of exact differentiation

( [ ∂/∂V (∂F/ ∂T) at constant V] at constant T = [ ∂/∂T (∂F/∂V)at constant T] at constant V

-( ∂S /∂V)at constant T = -( ∂P /∂T)at constant V

( ∂S /∂V)at constant T = ( ∂P /∂T)at constant V

## Maxwell’s third relation

By the definition of enthalpy

H= U+ PV

For small changes,

dH= dU+ PdV+ VdP …(2)

Now we know that TdS= dU+ PdV

dH= TdS+ VdP

H is a function of S And P

( ∂H /∂S)at constant P = T …(3)

And ( ∂H /∂P)at constant S = V …(4)

By the definition of exact differentiation equation

( [ ∂/∂P (∂H/ ∂S) at constant P] at constant S = [ ∂/∂S (∂H/∂P)at constant S] at constant P

( ∂T /∂P)at constant S = ( ∂V /∂S)at constant P

## Maxwell’s fourth relation

By the definition of Gibbs free energy,

G= H- TS …(1)

G= U+ PV – TS

(∴ H= U+ PV)

dG= dU + PdV + VdP -TdS -SdT

dG= TdS – TdS + VdP -SdT

And hence we get,

dG= VdP- SdT

G is a function of T and P,

( ∂G /∂P)at constant T = V …(2)

(∂G /∂T)at constant V = -S …(3)

By the definition of exact differential equation

( [ ∂/∂P (∂G/ ∂T) at constant P] at constant T = [ ∂/∂T (∂G/∂P)at constant T] at constant P

(∂S /∂P)at constant T = – (∂V /∂T)at constant P

### Conclusion

So our article is finished and after completely reading this article, one can easily tell what is **Maxwell equations**. And we have also discussed some more topics like** Maxwell equations**, Maxwell’s first equation, Maxwell’s second relation, Maxwell’s third relation, Maxwell’s fourth relation etc.

So one can say that they got a detailed information about **Maxwell equations**** **and basic information about other topics.

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