Thermodynamics : Thermodynamic Relations & real Gas Mixtures

1. Maxwell’s Equations

These are the set of equations that establish the relation between the partial derivatives of properties P, V, T, and S of a simple compressible system.

dU = TdS – PdV

dH = TdS + VdP

Helmholtz function

F = U – TS (availability of closed system)

Gibb’s function

G = H – TS (availability of open system)

For all real processes, the value of the Helmholtz function & Gibbs function decreases & attains a min value at equilibrium.

Helmholtz function

F = U – TS (availability of closed system)

Gibb’s function

G = H – TS (availability of open system)

For all real processes the value of Helmholtz function & Gibbs function decreases & attains a min value at equilibrium.

Hence Four Maxwell’s relations are,

COEFFICIENT OF VOLUME EXPANSION (β)

It is rate of change of volume with respect to temperature at constant pressure.

ISOTHERMAL COMPRESSIBILITY (K_T)

It shows the rate of change of volume with respect to pressure at a constant temperature or at isothermal conditions.

2. T-dS Equation

This is known as the first Tds equation

This is known as the second TdS equation.

T = Positive

C_p –C_v = +ve

C_p > C_v

_{3. Energy Equation}

Joule Thompson Coefficient (µ)

When a fluid passes through the porous plug, capillary tube, or valve, its pressure decreases. The throttling process is isenthalpic in nature. The temperature behavior of the fluid during throttling is described by the joule Thompson coefficient(µ),

                     Fig:1

                   Fig:2

Important points:

1. Joule Thomson coefficient is +ve in cooling region i.e slope of isenthalpic curve on T – P diagram is +ve in cooling region,
2. μ is –ve in heating region i.e the slope of isenthalpic curve on T – P diagram is –ve in the heating region,
3. There is nothing as a heating or cooling region for an ideal gas & the value of the joule Thomson coefficient is zero everywhere.

 4. Clausius Clapeyron Equations

Clausius Clapeyron equations is a relationship between saturation pressure, temperature, and enthalpy of vaporization and the specific volume of two phases involved. This equation helps in calculations of properties in two-phase regions.

                                Fig:3

The above equation is called the Clausius Clapeyron equation. It helps to determine enthalpy change associated with phase change by measuring pressure, temperature, and volume.

Gas Constant

• It has been experimentally observed that the ideal gas relation given closely approximately the p-V-T behaviour of real gases at low densities.
• At low pressure and high temperature, the density of a gas decreases and the gas behaves as an ideal gas under these conditions.

where, R_u = Universal gas constant, M = Molar mass, R = Gas constant.

Compressibility Factor

• Compressibility factor (correction factor) is measurement of deviation of gases from ideal gas behaviour.
• Compressibility factor (z):

• It can also be expressed as

• For ideal gases ⇒ z = 1
• For real gases ⇒ z is away from unity (> 1 or < 1)

Van der wall’s Equation of State:

• Two effects: Inter molecular attraction forces:

and b accounts for volume occupied by the gas molecules.

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