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# Behaviour of Real Gases: Deviation from Ideal Gas Behaviour

## Ideal and Real Gases

A gas which obeys the ideal gas equation, PV = nRT under all conditions of temperature and pressure is called an ‘ideal gas’.However, there is no gas which obeys the ideal gas equation under all conditions of temperature and pressure. Hence, the concept of ideal gas is only theoretical or hypothetical. The gases are found to obey the gas laws fairly well if the pressure is low or the temperature is high. Such gases are, therefore, known as ‘Real gases.’ All gases are real gases. However, it is found that gases which are soluble in water or are easily liquefiable, e. g. CO2, SO2, NH3 etc. show larger deviations than the gases like H2, O2, N2 etc.

## Differences between Ideal Gas and Real Gas

 Ideal Gases Real Gases Ideal Gases obey all gas laws under all conditions of temperature and pressure. Real Gases obey gas laws only at low pressures and high temperature. The volume occupied by the molecules is negligible as compared to the total volume occupied by the gas. The volume occupied by the molecules is not negligible as compared to the total volume of the gas. The force of attraction among the molecules are negligible. The force of attraction are not negligible at all temperatures and pressures. Obeys ideal gas equation PV = nRT Obey Van der Waals equation (P + an2/V2) (V - nb) = nRT

## Study of Deviations

To understand the deviations from ideal behaviour, let us first see how the real gases show deviations from Boyle’s law. According to Boyle’s law, PV = constant, at constant temperature. Hence, at constant temperature, plot of PV vs. P has to be a straight line which is parallel to x-axis. However, the real gases do not show such a behaviour as shown in figure no. 1 below. Fig No. 1 PV vs P for Real and Ideal Gas

From the plots, we observe that for gases like H2 and He, PV increases continuously with increase of pressure whereas for gases like CO, CH4 etc. PV first decreases with increase of pressure and reaches a minimum value and then increases continuously with increase of pressure. Similarly, if we plot experimental values of pressure versus volume at constant temperature (that is, for real gas) and theoretically calculated values from Boyle’s law (that is,for ideal gas) the two curves do not coincide as shown in figure no. 2. Fig. No. 2 Pressure vs Volume for Real and Ideal Gas

From above graphs, we observe that at higher pressure, volume which is observed is higher than that of calculated volume. At lower pressures, the observed and the calculated volumes approach each other.
Alternatively, upto what extent a real gas deviates from ideal behaviour can be studied using the terms of a quantity ‘Z’ which is known as the compressibility factor, and defined as: (i) For an ideal gas, as PV = nRT, Z = 1

(ii) For a real gas, as PV   nRT, Z ≠ 1.

Hence, two cases arise:
(a) When Z < 1, (For Example: for CH4, CO2 etc.) The gas is said to show negative deviation. Therefore gas will show more compression than expected from ideal behaviour.
This is caused by predominance of attractive forces among the molecules of these gases.
(b) When Z > 1, the gas is said to show positive deviation. This implies that the gas will show less compression than expected from ideal behaviour.
This is caused by the predominance of the strong repulsive forces among the molecules. Greater the departure in the value of Z from unity, greater are the deviations from ideal behaviour.

At the same temperature and pressure, the extent of deviation depends upon the nature of the gas, as shown in figure no. 3 Thus, at intermediate pressures, CO2 shows much larger negative deviation than H2 or N2. Fig. No. 3 Z vs P for Different Gases

For the same gas, at a particular pressure, the extent of deviation depends upon temperature, as shown in figure no. 4 for the case of N2 gas. Fig. No. 4 Z vs P for N2 gas at different temperatures

Plots in fig. no. 4 show that as the temperature increases, the minimum in the curve shifts upwards. Ultimately, a temperature is reached at which the value of Z remains close to 1 over an appreciable range of pressure. For Example, in case of N2, at 323 K, the value of Z remains close to 1 upto nearly 100 atmospheres.

The temperature at which a real gas behaves like an ideal gas over an appreciable pressure range is called Boyle temperature or Boyle point.
Further, from the plots shown in figure no. 3 and 4, it may be seen that at ordinary pressures (1-10 atm), Z is very near to 1, that is, the deviations from ideal behaviour are so small that the ideal gas laws can be applied.

## Significance of compressibility factor

The significance of compressibility factor can be further understood from the following derivation: If the gas shows ideal behaviour, Substituting this value of nRT/P in eqn. (1), we get Thus, compressibility factor is defined as the ratio of the actual molar volume of the gas (For Example: experimentally observed value) to the calculated molar volume (considering it as an ideal gas) at the same temperature and pressure.

## Causes of Deviation from Ideal Behaviour

As stated above, the real gases obey ideal gas equation (PV = nRT) only if the pressure is low the temperature is high. However, if the pressure is high or the temperature is low, the real gases show marked deviations from ideal behaviour. The reasons for such a behaviour shown by the real gases have been found to be as follows:

The derivation of the gas laws (and hence of the ideal gas equation) is based upon the Kinetic Theory of Gases which in turn is based upon certain assumptions. Thus, there must be something wrong with certain assumptions. A careful study shows that at high pressure or low temperature, two assumptions of Kinetic Theory of Gases are fails:

• When compared to the total volume of the gas, the volume occupied by the gas molecules is negligible.
• The forces of attraction or repulsion between the gas molecules are negligible.

The above two assumptions are true only if the pressure is low or the temperature is high so that the distance between the molecules is large. However, if the pressure is high or the temperature is low, the gas molecules come close together. Hence, under these conditions:

• The forces of attraction or repulsion between the molecules.
• The volume occupied by the gas may be so small that the volume occupied by the molecules may not be negligible.

## Equation of State for the Real Gases (van der Waals Equation)

To explain the behaviour of real gases, J .D. van der Waals, in 1873, modified the ideal gas equation applying appropriate corrections so as to take into account

• The volume of the gas molecules
• The forces of attraction between the gas molecules

He put forward the modified equation, known after him as van der Waals equation. The equation is

For 1 mole of the gas, For n moles of the gas, Where ‘a’ and ‘b’ van der Waals constant.There values depend upon nature of gas.

## Significance of Van der Waals Constants

• Van der Waals constant ‘a’: Its value is a measure of the magnitude of the attractive forces among the molecules of the gas. There would be large intermolecular forces of attraction if the value ‘a’, is large.
• Van der Waals constant ‘b’: Its value is a measure of the effective size of the gas molecules. Its value is equal to four times the actual volume of the gas molecules. It is called Excluded Volume or Co-volume.

## Units of van der Waals Constants

• Units of ‘a’: As p = am2/V2, therefore a = (p × V2) / n2 = atm L2 mol-2 or bar dm6 mol-2
• Units of ‘b’: As volume correction v = n b, therefore b = v/n = Lmol-1 or dm3mol-1

## Explanation of the Behaviour of Real Gases by van der Waals Equation

• At Very Low Pressures, V is very large. Hence, the correction term a/V2 is so small that it can be neglected, Similarly, the correction term ‘b’ can also be neglected in comparison to V. Thus, van der Waals equation reduces to the form PV = RT. This explains why at very low pressures, the real gases behave like ideal gases.
• At Moderate Pressures, V decreases. Hence, a/V2 increases and cannot be neglected. However, is still large enough in comparision to ‘b’ so that ‘b’ can be neglected. Thus, van der Waals equation becomes Thus, compressibility factor is less than 1. So at when at constant temperature, pressure is increased, V decreases so that the factor a/RTV increases. This explains why initially a dip in the plot of Z versus P is observed.

• At High Pressures, V is so small that ‘b’ cannot be neglected in comparison to V. The factor a/V2 is no doubt large but as P is very high, a/V2 can be neglected in comparison to P. Thus, van der Waals equation reduces to the form: Thus, compressibility factor is greater than 1. As P is increased (at constant T), the factor Pb/RT increases. This explains why after minima in the curves, the compressibility factor increases continuously with pressure.

• At High Temperatures: V is very large (at a given pressure) so that both the correction factors (a/V2 and b) become negligible as in case (i). Hence, at high temperature, real gases behave like ideal gas.

## Explanation of the exceptional behaviour of hydrogen and helium

From figure no. 3, it may be seen that for H2 and He, the compressibility factor Z is always greater than 1 and increases with increase of pressure. This is because H2 and He being very small molecules, the intermolecular forces of attraction in them are negligible, that is, ‘a’ is very small so that a/v2 is negligible. The van der Waals equation, therefore, becomes Thus, PV/RT that is, Z > 1 and increases with increase in the value of P at constant T.

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