SciPhyChem: (A Level Chem) Face the real gas

Wednesday, 16 April 2014

(A Level Chem) Face the real gas

The ideal gas equation is PV = nRT ==> PV/nRT = 1. The value PV/nRT is expected to be constant at 1 as pressure or temperature is increased. But by experiments, PV/nRT deviates from 1 (gas becomes less ideal) at high pressures and low temperatures. Van der Waals realized that root of the problem comes from the assumptions made in the Kinetic Theory of gases.

Assumptions in an ideal gas	Elaboration
Molecules have 0 volume.	In the equation, V is the volume available for molecules to move around. The gas molecules don't occupy any space at all.
Intermolecular attraction is 0.	P is the pressure created by free molecules colliding with the walls of the container. There're no intermolecular forces to affect the speed of molecules and hence the impact of molecules with the walls of the container.

In an ideal gas, the volume of gas molecules is negligible and the intermolecular attraction is negligible...

Here's a question from a past Physics paper.

(Phy Nov 2009) An ideal gas occupies a container of volume 4.5 x 10³ cm³ at a pressure of 2.5 x 10⁵ Pa and a temperature of 290 K.
(a) Show that the number of atoms of gas in the container is 2.8 x 10²³.
(b) Atoms of a real gas each have a diameter of 1.2 x 10^-10 m.
(i) Estimate the volume occupied by 2.8 x 10²³ atoms of this gas.
(ii) Suggest whether the real gas does approximate to an ideal gas.

(a) PV = nRT
Number of moles of gas, n
= PV/RT
= (2.5 x 10⁵ Pa)(4.5 x 10³ cm³)/(8.31 J K^–1 mol^–1)(290 K)
= (2.5 x 10⁵ Pa)(4.5 x 10^-3 m³)/(8.31 J K^–1 mol^–1)(290 K)

Number of atoms
= number of moles, n x Avogadro's constant
= (2.5 x 10⁵ Pa)(4.5 x 10^-3 m³)/(8.31 J K^–1 mol^–1)(290 K) x 6.02 × 10²³ mol^–1
= 0.467 mol x 6.02 × 10²³ mol^–1
= 2.8 x 10²³

(b) (i) Volume of atom
= 4πr³/3
= 4π(1.2 x 10^-10 m)³/3

Volume of 2.8 x 10²³ atoms
= 2.8 x 10²³ x 4π(1.2 x 10^-10 m)³/3
= 2.0 x 10^-6 m³

(ii) Volume of atoms / volume of gas
= 2.0 x 10^-6 m³ / 4.5 x 10^-3 m³
= 0.00045

The volume of the atoms is a very small fraction of the volume of the gas. The real gas does approximates the ideal gas.

Gases deviate from ideal behavior at low pressures and low temperatures...

For a real gas, PV/nRT = 1 at very low pressures such as 1 atm. For many real gases, PV/nRT decreases rapidly to a minimum below 1 as pressure is increased. Thereafter, PV/nRT increases from below 1 to above 1.

Pressure	Value of PV/nRT	Explanation
At very low pressure of 1 atm	Close to 1	Molecules are far apart enough such that: the volume occupied by molecules compared to the whole gas is negligible the molecules exert negligible intermolecular forces to affect the speed at which they collide with the walls of the container
When pressure increases from 1 atm	Decreases rapidly below 1	Molecules are closer such that they are able to exert attractive intermolecular van der Waals forces. The molecules collide with the walls of the container with less speed. P is lower than expected. The attractive intermolecular forces cause the gas to shrink more rapidly.
As pressure increases further	Increases from below 1 to above 1	Molecules are so close that their electron clouds begin to intrude into one another's territory. Molecules begin to repel one another. The repulsion causes the molecules to collide with the walls of the container with greater speeds. P increases. The repulsive forces cause the gas to expand more rapidly. V increases.

At high temperatures, molecules collide with the walls of container at high speeds with little effect by intermolecular van der Waals forces. At low temperatures close to the temperature at which the gas changes to liquid/solid, intermolecular van der Waals forces are significant. For each temperature, the change in PV/nRT with pressure follows the above description.

At high pressures and low temperatures, the volume of molecules compared to the gas becomes significant and the intermolecular van der Waals forces become significant enough to affect the pressure...

To account for the deviation from ideal gas behavior, van der Waals made the following corrections to P and V in PV = nRT.

The ideal gas volume available for molecules to move through is changed to V – nb where n = number of moles and b = constant that's related to the volume of molecules.
Molecules may collide and stick to one another (inelastic collision) due to intermolecular forces. The probability of two molecules being at the same place is proportional to n/V x n/V = n²/V². The reduction in pressure due to inelastic collisions is proportional to n²/V². The ideal gas pressure is then changed to (P + an²/V²) where a = constant.

Let's again write down the ideal gas equation PV = nRT ==> P = nRT/V.

If we put in the correction due to the volume of molecules, we get:

P = nRT(V – nb)

Next, we account for the reduction in pressure due to the inelastic collisions and we get:

P = nRT/(V – nb) – an²/V²

There's no need to memorize this intimidating equation. You just need to understand why gases deviate from the ideal behavior which is already tough enough.

(Chem Jun 2004) (i) When an evacuated glass bulb of volume 63.8 cm³ is filled with a gas at 24 ºC and 99.5 kPa, the mass increases by 0.103 g. Deduce whether the gas is ammonia, nitrogen or argon. (ii) Explain why ammonia is most likely of these three gases to deviate from ideal gas behavior.

(i) PV = nRT
Number of moles of the gas, n
= PV/RT
= (99.5 x kPa)(63.8 cm³)/(8.31 J K^–1 mol^–1)(24 + 273.15)
= (99.5 x 10³ Pa)(63.8 x 10^-6 m³)/(8.31 J K^–1 mol^–1)(24 + 273.15 K)
= 0.002571 mol

Ammonia: mass increase = 0.002571 mol x (14.01 + 3.01) g/mol = 0.0438 g
Nitrogen: mass increase = 0.002571 mol x (14.01 + 14.01) g/mol = 0.0720 g
Argon: mass increase = 0.002571 mol x 39.95 g/mol = 1.03 g
The gas is argon.

(ii) The ammonia molecule consists of an electronegative nitrogen atom with a lone pair of electrons bonded to three hydrogen atoms. Ammonia therefore consists of polar molecules. Nitrogen consists of non-polar molecules and argon consists of atoms. The van der Waals forces between the polar ammonia molecules is the greatest. This van der Waals forces between ammonia molecules would greatly affect the speeds at which the molecules impact the walls of the container. Therefore, ammonia is mostly likely to deviate from ideal gas behavior.

Try these questions.

(Chem 2011) At room temperature and pressure, chlorine does not behave as an ideal gas. At which temperature and pressure would the behavior of chlorine become more ideal?

	Pressure / kPa	Temperature / K
A	50	200
B	50	400
C	200	200
D	200	400

(Chem Nov 2006) For an ideal gas, the plot of pV against p is a straight line. For a real gas, such a plot shows a deviation from ideal behavior. The plots of pV against three real gases ammonia, hydrogen and nitrogen are shown below.

What are the identities of X-Z?

(Chem 2004/2013) (a) State two assumptions of ideal gas behavior. (b) (i) State the conditions of temperature and pressure under which real gases behave least like an ideal gas. (ii) Explain why real gases do not behave ideally under these conditions. (c) Explain the meanings of the terms p, V and T in the ideal gas equation. Give units for each to correspond with the value of R in the Data Booklet.
(Chem Jun 2011) Place ammonia, neon and nitrogen in decreasing order of ideal behavior. Explain your answer.
(Chem Nov 2009) Suggest one reason why CO₂ does not behave as an ideal gas.