Mathematical Fundamentals Used in Thermodynamics

In this article we will discuss about the mathematical fundamentals and equations used in thermodynamics.

Mathematical Fundamentals:

We will now discuss briefly certain mathematical manipulations which are repeatedly useful and important from the point of view of the subject.

The differential dz in Eq. (2) must be an exact differential (i.e., a point function), and an equation relating point functions must meet the test for exactness. This test is that the magnitude of second mixed partial derivatives of a function are independent of order of differentiation and hence equal.

Maxwell’s Equations:

The Maxwell relations are derived by applying the exactness test to the functions for the various properties:

(c) Helmholtz function (A).

The Eqs. (1), (2), (3) and (4) are called Maxwell’s equations. They apply to pure substances in process involving heat, work and internal energy.

Co-Efficient of Volume Expansion or Volume Expansivity (β):

The co-efficient of volume expansion is defined as the change in volume per unit volume for a unit change of temperature at constant pressure.

It is mathematically expressed as-

When β is positive, an isothermal increase in pressure results in rejected heat. When β is negative, an isothermal increase in pressure results in heat added.

Isothermal Compressibility:

The isothermal compressibility or the isothermal co-efficient of compressibility is defined as the change in volume per unit volume for unit change in pressure, at constant temperature. It is denoted by K and given by,

Note:

The negative sign indicates that volume decreases as pressure increases.

Adiabatic Compressibility:

Adiabatic compressibility is defined as the change in volume per unit volume for unit change in pressure at constant entropy. It is denoted by K_S and given by the expression.

Note:

Negative sign indicates that the volume decreases as pressure increases.

The ratio of isothermal and adiabatic compressibilities is equal to the ratio of specific heats.

As γ is always greater than unity, K is always greater than K_S.

Ratio of Coefficient of Volume Expansion and Isothermal Compressibility:

Coefficient of volume expansion is given by,

Thus, the ratio of coefficient of volume expansion and isothermal compressibility is equal to the change in pressure with temperature at constant volume.

General Equation for Change in Internal Energy:

We will now derive certain equations for change in internal energy, in terms of pressure, volume and temperature, by considering the following conditions:

TdS Equation:

From the law of thermodynamics, we have

dQ = dU + PdV

Specific Heat:

Specific heat of a substance is defined as amount of heat required to raise a unit mass of a substance through unit rise in temperature.

There are two processes whose specific heats are important viz.

1. Constant volume C_V and

2. Constant pressure C_P.

1. Specific Heat at Constant Volume (C_V):

At constant volume, the specific heat is given by,

2. Specific Heat at Constant Pressure (C_p):

Specific Heat Relations:

We know that,

The first TdS equation is given by

Joule Thomson Coefficient:

Consider a throttling process as shown in Fig. 8.1.

The velocity of fluid, whose pressure is reduced by a throttling process, increases at the restriction with a consequent decrease in enthalpy. But this increase in kinetic energy is dissipated by friction, as the eddies die down after restriction.

The steady flow energy equation implies that if the flow is adiabatic and if the fluid velocity before restriction is equal to that downstream of it, the enthalpy of fluid is restored. Throttling is therefore an Isenthalpic process.

The change in temperature with drop in pressure, at constant enthalpy, is defined as the Joule-Thomson Coefficient (μ).

μ varies with both temperature and pressure of the gas. The magnitude of μ is a measure of deviation of a gas from perfect gas behaviour.

For real gases, μ may be either positive or negative, depending on the thermodynamic state of the gas. If μ is greater than zero, it implies that the gas temperature decreases with decrease in pressure. If μ is less than zero, the gas temperature increases with decreases in pressure.

When μ = 0, the temperature of gas remains constant with throttling. The temperature at which μ = 0 is called as Inversion temperature for a given pressure.

Equation of State:

The ideal gas equation of state PV_S = RT can be established from the postulates of kinetic theory of gases developed by Clerk Maxwell.

The two main assumptions made in deriving the perfect gas laws are:

(1) The gas molecules are mere mass points occupying no space or negligibly small space compared to the volume of the gas.

(2) There is very little or no attraction or repulsion between the molecules.

In practice, all real gases deviate from the ideal gas laws. The deviation is small at ordinary temperatures and pressures but high at high pressure and low temperature. This is because all real gas molecules do occupy some space and to attract or repulse each other. At high pressure and low temperature, the intermolecular attraction and repulsion forces increase and also the volume of molecules becomes appreciable compared to total gas volume. Hence the ideal gas equation is not suitable for real gases at high pressure and low temperature. Some equations used to correct the ideal gas equation are given below.

(a) Van der Waal’s Equation:

J. D. Van der Waal, a Dutch physicist, was the first to correct the ideal gas equation PV_S = RT. He applied the laws of mechanics to individual molecules and introduced two correction terms in the ideal gas equation.

Van der Waal’s equation is given by,

Coefficient a was introduced to account for the existence of mutual attraction between the molecules.

Coefficient b was introduced to account for the volume of molecules, and is known as co-volume.

The term (a/V²_S) is called the force of cohesion.

Real gases conform more closely with Van der Waal’s equation of state than the ideal gas equation of state, particularly at high pressures. But is not obeyed by real gases in all ranges of pressures and temperatures.

(b) Clausius Equation:

The Clausius equation is a modification of Van der Waal’s equation and it neglects the mutual attraction between molecules. Thus, the Clausius equation is of the form –

P(V_s – b) = RT

Where b = Volume occupied by all molecules.

(c) Beattie Bridgeman Equation:

In 1928, Beattie-Bridgeman developed an equation of state which gives accu­rate results. The equation is of the form –

The constants Ao, Bo, a, b and c have to be determined experimentally for each gas.

The Beattie-Bridgeman equation does not give satisfactory results in the critical point region, but is fairly accurate (within 2 percent) in regions where density is less than 0.8 times the critical density.

(d) Bertholet Equation:

Bertholet equation is similar to the Van der Waal’s equation and uses the same constants a as in Van der Waal’s equation. The equation is of the form –

The Bertholet equation is of limited accuracy and gives properties accurate within 1 percent.

(e) Dieterici Equation:

Dieterici equation of state gives very accurate properties in the neighbourhood of the critical point and on the critical isotherm. The equation is of limited accuracy and may produce large errors when used away from critical region. The equation is of the form –

Where a and b are constants.

(f) Redlich-Kwong Equation:

The equation developed by Redlich-Kwong has the following form –

Where a and b are constants.

The Redlich-Kwong equation gives good results at high pressures and is fairly accurate for temperature above the critical value.

(g) Benedict-Webb-Rubin Equation:

The Benedict-Webb-Rubin equation is used for regions of high density i.e., where the density is approximately twice the critical density.

The equation has the form –

The eight constants A_o, B_o, C_o, a, b, c, x and γ are to be determined experimentally.

(h) Virial Equation:

The relation between PV_s and P may be expressed in the form of power series, as shown below.

Both the expressions in Eqs. (i) and (ii) are known as virial expansions or virial equations of state. These equations were first introduced by Dutch physicist Kammerlingh Onnes.

In the above equations, B, C, D, B, C, etc. are called virial coefficients. B and B are known as second viral coefficients, C and C are known as third virial coefficients and so on.

There exists a relation between the virial coefficients in Eqs (i) and (ii) as follows –

Compressibility Factor:

We know that the virial equation of state is of the form –

Here, the ratio (PV_S/RT) is known as the compressibility factor, denoted by Z.

For ideal gas, Z = 1.

For real gas, Z is determined experimentally. The magnitude of Z for a particular gas at a particular pressure and temperature is a measure of deviation of the gas from ideal gas behaviours.