The Kinetic Theory of Gases. Ideal Gases Revision Notes

In these revision notes for The Kinetic Theory of Gases. Ideal Gases, we cover the following key points:

• The definition of mole
• What is the Avogadro's number and what does it represent?
• How can we calculate the number of particles in a substance?
• What is an ideal gas?
• What is the formula of the Ideal Gas Law?
• What is the Boltzmann constant and how can we express the ideal gas law in terms of it?
• How to calculate work done by the gas in a thermodynamic process?
• How to find the work from a P - V graph?
• What are state variables and path-dependent variables?

The Kinetic Theory of Gases. Ideal Gases Revision Notes

The standard (SI) unit of amount of matter is not kilogram but mole (mol) instead. It represents a determined number of particles used as a standard of measurement for all materials. Mole is one of the 7 fundamental SI units.

By definition, one mole (in short mol) of a substance represents the mass of 6.02 × 10²³ particles of that substance.

The number 6.02 × 10²³ is known as Avogadro's number (N_A) and it represents the number of particles contained in 12 g of Carbon - 12.

In other words, 1 mol of carbon-12 has a mass of 12 grams and contains 6.022140857 x 1023 of carbon atoms (written in 10 significant figures). We say molar mass M of carbon-12 is 12 g/mol = 0.012 kg/mol.

We can calculate the number of moles n in a certain amount of a known substance of mass m and molar mass M through the equation

An ideal gas is a model (not a reality) used to describe easier the behaviour of gases. However, real gases in certain conditions can be assumed as ideal gases with a good approximation.

A gas must meet the following condition to be considered as ideal:

1. Particles do not collide with each other but only with the walls of the container. For this, they must have negligible dimensions (negligible volume).
2. Particles are equally sized and do not interact through intermolecular forces (attraction or repulsion) with other gas particles.
3. Particles move in random directions in accordance with the Newton's Laws of Motion.
4. Particles experience perfectly elastic collisions with the container walls, i.e. they don't lose energy during such collisions.

Thermodynamic systems at low pressure and high temperature have a similar behavior to ideal gases.

The Ideal Gas Law is the equation of state for an ideal gas, which establishes the relation between the four parameters of a gas sample. These four parameters are pressure, volume, temperature and number of moles of the gas sample.

The Ideal Gas Law is an empirical law of physics. It was derived from experiment and observation rather than from theory. The experiments show that the ratio of the products

is always constant. This constant is always the same for all gas samples. It is known as the ideal gas constant and is denoted by R. For ideal gases it has a value of R = 8.31 J/mol × K.

Therefore, we can write:

Rearranging in order to remove the fraction, we obtain the Ideal Gas Law:

In chemistry, there is another unit used for the ideal constant R. It stems from the fact that in chemistry pressure is usually measured in atmospheres (atm), volume in litres (L) and temperature in Kelvin degree (K).

There is a very important quantity known as the Boltzmann Constant, k, which is used to write the Ideal Gas Law in an alternative form. It is defined by the equation:

The value of Boltzmann Constant is k = 1.38 × 10^-23J/K. Therefore, in term of Boltzmann Constant the ideal gas law is written as:

Work in a thermodynamic process is calculated in terms of pressure and volume change. Its formula is

This expression gives the left part of the ideal gas law formula.

The formula of work in thermodynamics indicates that:

In a P - V diagram, work is mathematically represented through the area under the graph. This reasoning can be extended in the processes with non-constant pressure as well. Thus, even if the pressure is changing, we can divide the curve over P - V diagram into small, constant pressure steps. The total area under the curve is therefore the sum of small rectangular areas. As we increase the number of rectangles by taking thinner rectangles, the error in calculation becomes smaller.

When studying P - V diagrams, we reach two important conclusions:

1. Every point on a P - V diagram has a definite temperature for a fixed amount of gas.
2. The points on a P - V diagram are called "states". A gas can undergo an infinite number of intermediate states when it moves from the initial to the final state (from i to f).

As explained earlier, pressure, volume and temperature are known as "state variables".

Internal energy is also a state variable. Internal energy of a gas sample is a function of temperature only. For a definite amount of gas, every specific temperature value corresponds to a specific amount of internal energy. Therefore, for a given state, the internal energy of gas has a fixed value.

State variables are macroscopic parameters. For a given state, a gas sample has fixed values of P, V, T and U. Although the speed or energy of a certain molecule are in continuous change, the values of state variables are determined by an average of effects produced by billions of molecules.

Heat and work are not state variables. It is meaningless if we talk about heat or work contained in a system.

Heat and work depend not only on the initial and final states, but also on the intermediate states the gas undergoes during a thermodynamic process. Therefore, heat and work are not state variables, because they depend on the path followed between the initial and final states. Such variables are called "path-dependent variables".