WBCHSE Class 11 Chemistry For Deviation from Ideal Gas Behaviour Notes

Deviation from Ideal Gas Behaviour

Ideal gas:

A gas which obeys the equation pV = nRT under all conditions of temperature and pressure is called an ideal gas. In reality no gas does this. Deviations from the ideal gas behaviour occur as the temperature becomes low and pressure becomes high.

In other words, real gases obey the ideal gas equation at low pressure and high temperature. All gases condense at higher pressures and sufficiently low temperature.

That real gases deviate from the ideal gas behaviour is known to us. But, how much is the deviation and on what factors does it depend? Let us plot a graph of pressure against volume for ideal and real gases as shown in Figure.

The curve shows the deviation of the behaviour of a real gas from ideal gas behaviour but we cannot determine the quantity of deviation from it.

Let us introduce a quantity Z in tire ideal gas equation to account for the deviation of real gas behaviour from ideal gas behaviour. Then

⇒ \(p V=Z n R T \quad \text { or } \quad Z=\frac{p V}{n R T} \quad \text { or } \quad Z=\frac{p V_m}{R T} \quad\left(V_m\right. \text { is the molar volume of the gas), }\)

where Z = 1 for an ideal gas, i.e., \(p V_m / R T=1 \text { or } p V_m=R T\) for ideal gases. The quantity Z is called the compressibility factor. It is not equal to 1 in case of real gases, i.e., \(p V_m \neq R T\) for real gases. It can be negative or positive. The deviation of Z from 1 determines the deviation of a gas from ideal behaviour.

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When Z < 1, a gas is said to show negative deviation. This means the gas is more compressible than the ideal gas. When Z > 1, a gas is said to show positive deviation, which implies that it is less compressible than the ideal gas. Gases are more compressible due to the predominance of attractive forces and less compressible due to strong repulsive forces among their molecules.

Nature of gas One of the factors on which deviation from ideal behaviour depends is the nature of the gas. Different gases show different extents of deviation at the same temperature and pressure.

For example (at 300 K), CO2 and N₂ are more compressible at low pressure (negative deviation) and less compressible at high pressure. But hydrogen shows positive deviation for all pressures. At intermediate pressures the CO₂ curve shows a greater dip (larger negative deviation) indicating that it can be liquefied with greater ease.

Effect of pressure At very low pressure Z approaches unity for all gases and they exhibit nearly ideal behaviour. For pressures between 1-10 bar Z is close to 1 and the ideal gas equation can be applied to real gases. At high pressures all gases have Z> 1 as repulsive forces arc predominant.

Effect of temperature The deviation from ideal behaviour decreases as temperature increases. For every gas, there is a temperature at which the value of Z is close to unity for a fairly large range of pressure.

This temperature is known as Boyle temperature or Boyle point.

Remember that the larger the deviation from ideal gas behaviour, the more easily can a gas be liquefied.

For the same number of moles of a real gas and an ideal gas, we may write \(p V_{\text {real }}=Z n R T\) ….(1)

and \(p V_{\text {ideal }}=n R T\)…. (2)

Substituting for nRT from equation (2) in Equation (i), we get

⇒ \(p V_{\text {real }}=Z p V_{\text {ideal }}\)

or \(\mathrm{Z}=V_{\text {real }} / V_{\text {ideal }} \text {. }\)

From this equation it can be inferred that the compressibility factor of a gas is simply the ratio of its actual volume to the volume it would have occupied had it behaved ideally.

Causes of deviation from ideal behaviour: Real gases follow the ideal gas equation only at low pressure and high temperature. Why do they deviate from ideal behaviour otherwise? There must surely be something wrong with the assumptions made in the kinetic theory (when experimental observations do not tally with the predictions of a theory, there has to be something wrong with the assumptions made while formulating the theory). Two of the important assumptions made were that

1. The volume of the molecules is negligible compared to the volume of the gas and
2. The intermolecular force is negligible.

• It is true that at low pressure and high temperature, or even under normal conditions of temperature and pressure, the volume of the molecules of a gas is a very small fraction of the volume of the gas.
• If we consider the molecules of a gas to be of radius 2.0 x 10^-10 m the volume of the molecules is only 0.1% of the volume of the gas at room temperature and 1 bar pressure. But what happens when pressure increases or temperature falls? The total volume of the gas decreases, while the volume of the molecules remains unchanged. Under such conditions, the volume of the molecules is no longer a negligible fraction of the total volume of the gas.
• The second assumption that intermolecular forces are negligible cannot really hold under all conditions of temperature and pressure. Consider the fact that gases can be condensed into liquids. Surely this would not have been possible had the molecules of gases really been totally independent of each other.
• At high pressure or low temperature, the volume of a gas decreases and the same number of molecules are crowded in less space, so the attraction between them becomes appreciable and can no longer be neglected.
• The number of collisions with the walls of the container decreases due to intermolecular interaction. This results in lower values of pressure than would have been the case in the absence of intermolecular interaction. It has been determined that if oxygen gas behaved ideally, it would have exerted a pressure of 1.003 bar at 0°C.

van der Waals equation: The ideal gas equation was modified to describe the behaviour of real gases by J D van der Waals, a Dutch physicist. He introduced two correction terms to account for the volume of the molecules and the attraction between them.

Volume correction If the molecules of a gas occupy an appreciable volume then the actual volume available for their movement inside a container is less than the observed volume of the container, say V.

Let us suppose that the volume of 1 mol of molecules is v and that there are n moles of molecules in a container of volume V. Then the space available for the molecules to move about should be V- nν.

Actually, the effective volume of molecules in motion is calculated to be four times their real volume. If the molecules are assumed to be spheres having radius r, then volume of one molecule of the gas is \(4 / 3 \pi r^3\).

Therefore, effective volume would be 4 x \(4 / 3 \pi r^3\)(for one molecule). Thus, the effective volume of 1 mol of molecules is 4u and that of n moles of molecules is 4nv. Suppose we let b = 4z>.

Then corrected volume = V – b, when V contains 1 mol of molecules.

And corrected volume = V -nb, when V contains n moles of molecules.

This quantity b is known as the excluded volume or co-volume. It depends on the nature of the real gas and its units are L mol^-1.

Pressure correction The intermolecular attraction does not allow the molecules to move as freely as they would had they been completely independent of each other. The pressure exerted by a gas is due to the collisions of its molecules with the walls of the vessel.

If the molecules hitting the walls are being pulled back by other molecules, the collisions will naturally be less vigorous than if the molecules had been completely independent. In other words, a real gas exerts less pressure than the ideal gas because of the force of attraction between its molecules, or the observed pressure is less than the ideal pressure. Let us call the ideal pressure p, and the observed pressure p. The relation between them turns out to be

∴ \(p_i=p+\frac{a n^2}{V^2}\)

where n is the number of moles, V is the volume of the gas, and A is a constant which depends on the nature of the gas or the force of attraction between the molecules. The units of a are atm L² mol^-2.

If we substitute the corrected values of volume and pressure in the ideal gas equation, we get the equation of state for real gases.

⇒ \(\left(p+\frac{a n^2}{V^2}\right)(V-n b)=n R T\) for n moles

or \(\left(p+\frac{a}{V_m^2}\right)\left(V_m-b\right)=R T\) for 1 mol,

where \(V_{\mathrm{m}}=\) molar volume.

The term \(\left(\frac{a}{V_m^2}\right)\) is also called the internal pressure of the gas.

The equation of state for real gases is also known as the van der Waals equation and a and b are called van der Waals constants, which are different for different gases. p, V and T are the observed values of pressure, volume and temperature of a gas.

Significance of van der Waals constants The constant a is the measure of the intermolecular forces between the molecules of a gas. The greater the value of a, the greater is the intermolecular force of attraction.

The constant b is a measure of the effective size of the molecules of a gas. When the molecular size is the predominant factor influencing the behaviour of a gas, Z > 1 the gas shows positive deviation from ideal behaviour. On the other hand, when the effect of intermolecular attraction predominates, Z < 1 and the gas shows negative deviation from ideal behaviour.

The molecules of hydrogen and helium are very small, so the intermolecular force in these gases is negligible. This is why the molecular size factor predominates and Z is always greater than unity.

Example 1. Calculate the pressure exerted by 2 mol of ammonia gas at 27°C in a flask whose volume is 0.9 L. Given that a = 4.17 atm L² mol^-2, b = 0.0371 L mol^-1, R = 0.082 L atm K^-1 mol^-1.
Solution:

Given that a = 4.17 atm L² mol^-2, b = 0.0371 L mol^-1, R = 0.082 L atm K^-1 mol^-1.

According to the van der Waals equation

⇒ \(\left(p+\frac{a n^2}{V^2}\right)(V-n b)=n R T\)

∴ \(\quad p=\frac{n R T}{V-n b}-\frac{a n^2}{V^2}\)

= \(\frac{2 \times 0.082 \times 300}{0.9-2 \times 0.0371}-\frac{4.17 \times(2)^2}{(0.9)^2}\) = 39.0 atm.

Example 2. 1 mol of CO₂ occupies 0.35 L at 300 K and 60 atm pressure. Determine the compressibility factor of the gas.
Solution:

Given

1 mol of CO₂ occupies 0.35 L at 300 K and 60 atm pressure.

∴ \(Z=\frac{p V}{n R T}=\frac{60 \times 0.35}{1 \times 0.082 \times 300}=0.8536\)

Liquefaction of gases and critical point: Gases can be liquefied by the increase of pressure and decrease of temperature. If we consider the microscopic properties of a gas, then high pressure brings the molecules of a gas close together to allow attractive forces to operate and cause condensation.

In case of decrease of temperature, the kinetic energy of molecules decreases. As a result the slow moving molecules come closer to one another and ultimately the gas changes into the liquid state.

• Thomas Andrew in 1863 discovered the essential condition for liquefaction. He studied the pressure-volume relation of CO₂ at different temperatures. Figure shows the isotherms of carbon dioxide gas obtained by plotting data collected by Andrew. (An isotherm is a curve obtained at constant temperature conditions.) The isotherm obtained at a relatively high temperture, say 50°C, is close to that for an ideal gas.
• This shows that at this temperature CO₂ follows Boyle’s law rather closely. Now consider an isotherm at low temperature, say 13°C. The deviation from the expected theoretical curve is much more pronounced here.

• The isotherm PQRS, consists of three parts PQ, QR and RS. Let us see what happens when a gas is compressed at this temperature. As the pressure is increased, the behaviour of the gas follows the curve PQ which is roughly in accordance with Boyle’s law, i.e., the volume of the gas decreases. When the point Q is reached, the gas liquefies partially exhibiting considerable change in volume.
• Attempts to increase the pressure result in a sudden decrease in volume with no change in pressure, along QR. Finally, at R all the gas liquefies. Once the complete liquefaction has occurred even the application of a large pressure results only in a very little change in volume of the liquid as liquids are relatively incompressible.

Andrew observed that CO₂ could be liquefied below 31.1°C but not at and above this temperature. This is called the critical temperature, T_c. At temperatures below T_c, p-V isotherms contain a flat portion in the middle corresponding to liquid-vapour equilibrium (Andrew called the gas a vapour below Tc when it was in equilibrium with the liquid).

• What if we try to compress the gas along the isotherm at T_c? Starting at A an increase in pressure results in a decrease in volume. At point C, the surface separating the two phases, viz vapour and liquid does not appear.
• This is called the critical point. At this point, the densities of both the liquid and the vapour phase are the same so that they cannot be distinguished. Therefore, further increase in pressure does not result in liquefaction.
• The region below the dashed line in shows the liquid-vapour region where the two phases are in equilibrium. Any gas can be liquefied by applying pressure only if its isotherm passes through this region and this happens only below the critical temperature.
• The pressure at the critical point is called critical pressure (P_c) and the corresponding volume occupied by one mole of the gas is called critical volume (V_c).

The observations of Andrew regarding CO₂ are generally valid for all other gases. The value of critical temperature, pressure and volume may vary but the nature of isotherms at different temperatures is almost the same.

Those gases whose critical temperatures lie below room temperature, such as nitrogen and oxygen cannot be liquefied by pressure alone without cooling. Generally gases below their critical temperatures are called vapours. Critical constants for some gases are given in Table.