For ideal gases it is best to prove this using the equation of state pV = nRT.
This expression gives a linear relation between the concentration and pressure : p = M RT which can then be used to show that neither equilibrium nor partial pressures change. A further advantage is the reduction of Kc to Kp using the same equation , then one uses partial pressures to quantify equilibria. Finally the entire process must be isothermal ( ideal gases undergo free expansion at const tempreature ) otherwise there would be shift in equilibrium.
An alternative method is the use of dalton's law alone. It can be shown using this method that the partial pressures do not change when the equilibrium does not shift after the addition of an inert gas.
Consider a system in which a certain gas-phase chemical reaction rests at a dynamic equilibrium state. If the gases obey dalton's law of partial pressures, then it can be shown that the addition of an inert gas does not change the partial pressures of the components as long as the total volume is constant.
For the ith component, let * denote its initial state. Dalton's law at thermal equilibrium :
Suppose the system is perturbed by an addition of an inert gas, then since no chemical reaction occurs the mole number of each gas remains a constant. Let ( small ) delta p represent the pressure of the inert gas in the closed system, and ( small ) delta n the amount of the gas added. Again according to dalton's law :
The absence of * signifies the final equilibrium state reached by the gases. Pi is the difference in total pressure between the initial and final states.
The difference between the partial pressures for the ith component is eq (1) :
Applying dalton's law for the inert gas :
The type of an ideal gas is irrelevant. Therefore for an amount delta n of a gas present initially in the mixture ( with constant composition ) must have a partial pressure delta p :
Combining the last two equations leads to :
Therefore the change in total pressure of the system is due to the addition of the inert gas. The partial pressure of each gas has not changed as can be verified by substituting the value of small delta p into eq (1) which would vanish.
For real gases :
The study of real gas-phase systems must include a quantitative analysis of fugacities. In general fugacities are thermodynamic pressures related to real pressures by the relation :
Phi is called the fugacity coefficient and it is a function of T , p , the type of gas, and the nature of its surrounding medium. If the gas is dominated by attractive forces in the domain ( p , T ) of application , then phi < 1 . Similarily if the gas is dominated by repulsive forces then phi > 1. Ofcourse for ideal behaviour the total net forces are zero and phi is unity.
Let "id" represent the ideal state of a gas. Using the equation dG = Vdp - SdT , at constant temperature :
mu is the chemical potential. From the definition of fugacity :
Replacing this formula in the previous equation :
Substituting the compressibility/compression factor and knowing that:
To solve this integral Z must be known. In general Z can be expressed as a power series in terms of pressure. As long as the pressure is not significantly high the series can be truncated at the 2nd term. It follows then :
Since the virial coefficient is independent of pressure then the precise expression for the fugacity coefficient becomes :
This equation shows that the fugacity coefficient is a function of pressure and the 2nd virial coefficient which is a function of temperature and the potential energy of the medium. In general the thermodynamic equilibrium constant for real gases can be written in the form :
Kp is the equilibrium constant in terms of real pressures. v is the stoichiometric number. Based on a specific equation of state beta is determined in terms of certain parameters.
Adding a small amount of an inert gas changes the temperature and pressure of the system as well as its potential energy. However if one assumes the gas interacts weakly with the species present, then the 2nd virial coefficient becomes a function of temperature alone. Furthermore, assuming that the thermodynamic equilibrium constant changes slightly with temperature ( i.e. the change in enthalpy for adding an inert gas is approximately zero ) will make the calculations easier.
Since the total volume is constant , the partial pressures can be calculated if the change in temperature is known from an equation of state f(P,V,T) = 0. However such an equation must include volume corrections for all the species involved.
The 2nd virial coefficient can then be easily calculated.
For example for the Van der waal eq :
Therefore the total volume, initial partial pressures, and initial temperature must be known as well as the final temperature of the system. The new fugacities can then be calculated. How equilibrium shifts is easily determined.
The great failure of this method is in neglecting the interactions between the molecules of one gas and another. This is because the fugacity coefficients were determined for each gas as if it were not interacting with another gas. Correcting this requires a completely different approach and a modified equation of state to account for the attractive forces among the gases. I haven't given this much thought for now, but I suppose some results from statistical thermodynamics can be used to take into account the potential energy difference.