6. Thermodynamic relations and processes

6.1 Maxwell relations

For a system with one constituent with fixed number of particles, from the first and second laws, and from (5.10), we have the basic relations

(6.1)

The quantities on the left are all perfect differentials. For a general differential of the form

(6.2)

to be a perfect differential, the necessary and sufficient condition is

(6.3)

Applying this to the four differentials in (6.1), we get

(6.4)

These four relations are called the Maxwell relations.

A Mathematical Result

Let be three variables, of which only two are independent. Taking to be a

function of and , we can write

(6.5)

If now we take and as the independent variables, we can write

(6.6)

Upon substituting this result into (6.5), we get

(6.7)

Since we are considering and as independent variables now, this equation immediately yields the relations

(6.8)

These relations can be rewritten as

(6.9)

6.2 Other relations

The TdS equations

The entropy is a function of the state of the system. We can take it to be a function of any two of the three variables . Taking to be a function of P and , we write

(6.10)

For the first term on the right hand side, we can use

(6.11)

where is the specific heat at constant pressure. Further, using the last of the Maxwell

relations, we can now write (6.10) as

(6.12)

The coefficient of volumetric expansion (due to heating) is defined by

(6.13)

Equation (6.12) can thus be rewritten as

(6.14)

If we take S to be a function of V and T,

(6.15)

Again the first term on the right hand side can be expressed in terms of , the specific heat at constant volume, using

(6.16)

Further using the Maxwell relations, we get

(6.17)

Equations (6.12) (or (6.14)) and (6.17) are known as the equations.

Equations for specific heats

Equating the two expressions for , we get

(6.18)

By the equation of state, we can write p as a function of V and T, so that

(6.19)

Using this in (6.18), we find

(6.20)

However, using (6.9), taking and , we have

(6.21)

Thus the coefficient of in (6.20) vanishes and we can simplify it as

(6.22)

where we have used (6.21) again. We have already defined the coefficient of volumetric expansion α. The isothermal compressibility is defined by

(6.23)

In terms of these we can express as

(6.24)

This equation is very useful in calculating from measurements of and α and . Further, for all substances, . Thus we see from this equation that (The result can be proved in statistical mechanics.)

In the equations, if , and dp are related adiabatically, and we get

(6.25)

This gives

(6.26)

We have the following relations among the terms involved in this expression,

(6.27)

Using these we find

(6.28)

Going back to the Maxwell relations and using the expressions for , we find

(6.29)

these immediately yield the relations

(6.30)

Gibbs-Helmholtz relation

Since the Helmholtz free energy is defined as ,

(6.31)

This gives immediately

(6.32)

Using this equation for entropy, we find

(6.33)

This is known as the Gibbs-Helmholtz relation. If is known as a function of and , we can use these to obtain , 𝒫 and . Thus all thermodynamic variables can be obtained from as a function of and .

6.3 Joule-Kelvin expansion

The expansion of a gas through a small opening or a porous plug with the pressure on either side being maintained is called Joule-Kelvin expansion. (It is sometimes referred to as the Joule-Thomson expansion, since Thomson was Lord Kelvin’s original name.) The pressures are maintained by the flow of gases, but for the theoretical discussion we may think of them as being maintained by pistons which move in or out to keep the pressure the same. The values of the pressures on the two sides of the plug are not the same. The gas undergoes a decrease in volume on one side as the molecules move through the opening to the other side. The volume on the other side increases as molecules move in. The whole system is adiabatically sealed, so that the net flow of heat in or out is zero.

Since , we can write, from the first law,

(6.34)

Consider the gas on one side starting with volume going down to zero while on the other side the volume increases from zero to . Integrating (6.34), we find

(6.35)

This yields the relation

(6.36)

Thus the enthalpy on either side of the opening is the same. It is isenthalpic expansion. The change in the temperature of the gas is given by

(6.37)

The quantity

(6.38)

is called the Joule-Kelvin coefficient. From the variation of we have

(6.39)

so that, considering an isenthalpic process we get

(6.40)

This gives a convenient formula for . Depending on whether this coefficient is positive or negative, there will be heating or cooling of the gas upon expansion by this process. By choosing a range of pressures for which is negative, this process can be used for cooling and eventual liquefaction of gases.