Compressibility Factor| Formula and Conditions for JEE

The compressibility factor is a parameter that shows the deviation that occurred in the behaviour of the real gas from the ideal gas behaviour. To achieve the characteristics of an ideal gas, the real gas needs to sustain the condition of the ideal gas and verify the relationship of the ideal gas which is shown below:

PV_m=RT

However, the condition of pressure and volume disturbs the behaviour of real gases, and a deviation occurs in their properties. The compressibility factor is defined for the determination of that deviation in real gas behaviour.

Certain topics such as what is compressibility factor, compressibility factor formula, compressibility factor for real gases, compressibility factor at high pressure, compressibility factor for real gas formula, and so on, will be thoroughly discussed in this article.

What is the Compressibility Factor?

The compressibility factor is a parameter that determines the deviation of real gases from ideal gases. It is denoted by Z. It is a thermodynamic property that is used for analysing the behaviour of real gases. According to the thermodynamic point of view, compressibility is a correction factor to the ideal factor. 

For real gases, the compressibility factor deviates from the ideal value, as happened because of the intermolecular forces that occurred between the molecules of the gases. The gas is pronounced to be at a critical range, where it deviates a lot from the ideal behaviour of the gas. The variation in compressibility factor of gases occurred a lot at the same temperature and pressure conditions.  

Compressibility Factor Formula

According to the ideal gas law, a relationship is defined as;

PV_m=RT

For the non-ideal gas, this relationship would be given as;

PV_m=ZRT

Hence, 

$Z=\frac{P V_{m}}{R T}$

P = Pressure

Z = Compressibility factor

V_m= Molar volume of gas

R = Universal gas constant

T = Temperature

The compressibility factor can be determined as the ratio of the molar volume of the real gas at a certain pressure and temperature to the molar volume of an ideal gas at the same temperature and pressure. Mathematically, the compressibility factor formula is written as,

$Z=\dfrac{V_{actual}}{V_{ \text { ideal } }}$

V_actual = Molar volume of the real gas 

V_ideal = Molar volume of an ideal gas

For the ideal gas, the value of the compressibility factor depicts the ideal behaviour of the gas. Any deviation from the given value will be treated as non-ideal behaviour of the gas. 

Compressibility Factor at Higher Pressure Condition

At higher pressure conditions, the first impact on the gas molecule is the space issue. The space between the molecules of the gas started reducing. Now in such high-pressure conditions, the volume of the gas becomes appreciable relative to the total volume occupied by the gas. In that high pressurised condition, the amount of vacant space between the gases becomes less. Therefore, the gas becomes less compressible. If the pressure is low, then the gas molecules have practically no chance for attraction as they have so much vacant space between them. 

However, in the case of the high-pressure condition, the force of attraction is no longer insignificant. In such a case, the molecules of the gas become closer to each other if the temperature is low. When the temperature is low, then there must be lower kinetic energy in the molecules of the gases. Therefore, there will be less chance of colliding with each other. The image shown below depicts the force attraction between the gas molecules at low temperature and high pressure.

This image describes the intermolecular forces in the distribution of ideal gas and real gas under high-pressure conditions 

Compressibility Factor for Real Gas Formula

A generalised equation for the real gases proposed by Van der Waals equation in 1873 is shown below:

$\left(P+\dfrac{a}{V_{m}^{2}}\right)\left(V_{m}-b\right)=R T$

For the higher pressure for real gas, this given factor is neglected, therefore,

$\begin{aligned} &\dfrac{a}{V_{m}^{2}} \approx 0 \\ &P V-P b=R T \end{aligned}$

On rearranging and solving the above equation,

$\dfrac{P V}{R T}=1+\dfrac{P b}{R T}$

It is known that

$Z=\dfrac{P V}{R T}$

Therefore,

$Z=1+\dfrac{P b}{R T}$

Hence, the compressibility factor for real gas is $Z=1+\dfrac{P b}{R T}$

Conclusion

This article concludes with qualitative and interesting information on the compressibility factor. This article describes how the behaviour of the gases gets affected at higher pressure conditions and how the compressibility factor works in those conditions for real gases. This parameter has a value of 1 for the ideal gas, and real gases are found as an approximate value of this ideal value. Since the real gas value deviates from the ideal gas behaviour, the compressibility factor is used to analyse that deviation.