Ideal gases (5.1.4)

The mole is the SI base unit for the amount of substance.

Amount of substance is a measure of the number of elementary particles (such as atoms or molecules) in a substance.

One mole of any substance contains the same number of elementary particles, equal to the Avogadro constant

It is important to note that the mole does not only apply to pure substances. In mixtures, you simply talk about moles of each component, not ‘a mole of the mixture as a whole’ unless defined carefully.

For example, one mole of carbon atoms, one mole of hydrogen atoms, and one mole of water molecules each contain particles.

Historically, one mole was defined as the amount of substance containing the same number of atoms as 12 g of carbon-12.

The number of atoms or molecules in a substance is:

where:

• is the number of moles of the substance in
• is the Avogadro constant.

The molar mass of a substance is the mass of one mole of that substance.

For an element, the molar mass (in grams per mole) is numerically equal to the mass number of the atom. The mass number is the top number in nuclide notation and represents the total number of protons and neutrons in the nucleus.

Nuclide notation:

where:

• is the mass number
• is the atomic number.

The molar mass of an element is therefore given by:

For example, hydrogen is written as The mass number is 1, so one mole of hydrogen atoms has a mass of:

For example, carbon-12 is written as The mass number is 12, so one mole of carbon-12 atoms has a mass of:

The molar mass of a molecule is the mass of one mole of that molecule. It is calculated by adding together the molar masses of all the atoms that comprise the molecule. An example of this is shown below.

A water molecule () contains two hydrogen atoms and one oxygen atom:

• Hydrogen has a molar mass of , and
• oxygen has a molar mass of .

So, the molar mass of () is:

The kinetic theory of gases describes the behaviour of the particles (atoms or molecules) in an ideal gas.

It relies on several simplifying assumptions compared to a real gas:

• An ideal gas contains a large number of molecules ≈ in random, rapid motion.
• Particles have negligible volume compared to the container; they are treated as point particles (volume ≈ 0).
• All collisions are perfectly elastic, so momentum and kinetic energy are conserved, including particle–particle and particle–wall collisions.
• The duration of collisions is negligible compared with the time between collisions.
• Inter-particle forces are negligible, ignoring small electrostatic interactions.

The kinetic theory of gases explains how an ideal gas exerts pressure. Particles move randomly and collide with each other and the container walls.

When a particle collides with a wall, the wall exerts a force on it, changing its momentum. By Newton’s second law, the average force on the particle is:

where:

• is the change in momentum of the particle due to the collision.
• is the time between collisions with that wall.

Kinetic theory assumes the particles collide elastically with the container walls. During a collision with a wall:

• the change in momentum parallel to the wall is zero, while
• the change in momentum perpendicular to the wall is non-zero.

For a particle of mass with perpendicular velocity component , the change in momentum is:

The average force on the particle by the wall during the time of collision :

By Newton’s second law, a force equals the rate of change of momentum. When a gas particle collides with a container wall and rebounds, its momentum changes, so the wall exerts a force on it, and by Newton’s third law, the particle exerts an equal and opposite force on the wall.

Countless particles strike the wall every second, producing a constant average force. This gives rise to a pressure on the container wall:

Where:

• is the total force on the wall due to all particle collisions, and
• is the cross-sectional area of the wall.

Before the kinetic theory of gases was developed, the behaviour of gases was described using three empirical gas laws:

• Boyle’s law
• Charles’ law
• Pressure law

It is useful to note that empirical means these laws were based on experimental observations rather than theory.

A gas is considered to be an ideal gas if it obeys all three laws exactly and contains a fixed amount of gas.

These laws were derived from experimental observations, allowing them to predict what happens when pressure, volume, or temperature changes; however, they do not explain why these changes occur.

The kinetic theory of gases explains these laws in terms of the motion and collisions of particles, linking pressure, volume, and temperature to the speed and kinetic energy of particles.

All three gas laws can be combined into a single equation, the ideal gas equation:

where:

• is the pressure of the gas
• is the volume
• is the temperature of an ideal gas
• is the number of moles of the gas
• is the molar mass constant.

In 1662, Robert Boyle discovered an empirical relationship between the pressure of a gas and its volume , assuming constant temperature and amount of gas.

Boyle’s law states that:

This means that if the volume of an ideal gas decreases, while the temperature remains constant, the pressure increases, and vice versa.

The relationship between volume and pressure in an ideal gas at a constant temperature is as follows:

• Constant temperature: The average kinetic energy of the particles remains unchanged.
• Volume and pressure:
  • Reducing the volume increases the frequency of particle collisions with the container walls, which, in turn, raises the pressure.
  • Conversely, increasing the volume reduces the collision frequency, thereby lowering the pressure.

In 1787, Gay-Lussac discovered a relationship between the pressure ( of a gas and its temperature , provided the volume and amount of the gas remain constant.

The pressure law states that:

This means that if the temperature of an ideal gas in a container of constant volume increases, the pressure increases in the same proportion, and vice versa.

The observed relationship is due to the following particle-level explanation:

• Increased temperature means increased kinetic energy: Raising the temperature of the gas increases the average kinetic energy of its constituent particles.

• Faster movement and collisions: These particles, moving faster, collide with the walls of the container both more frequently and with a greater change in momentum during each collision.

• Pressure rises in a fixed volume: Since the volume of the container is held constant, the increased collision frequency and force directly translate into an increase in pressure as the temperature rises.

Charles’ law describes how the volume of a gas varies with temperature when the pressure and amount of gas are held constant. It states that:

This means that if the temperature of a gas is increased while keeping pressure constant, the volume of the gas must increase.

This phenomenon is attributed to the following mechanism:

• As the temperature increases, the gas particles accelerate, resulting in more forceful collisions with the container’s walls.
• To maintain a constant pressure, the volume of the gas must increase (expand).
• This expansion reduces the collision frequency with the walls, thereby counteracting the effect of the more forceful collisions.

The three gas laws provide the following relationships:

• Boyle’s law:
• Pressure law:
• Charles’ law:

These can be combined to give:

The constant depends on the amount of gas. For moles, it is written as , where is the molar gas constant.

The value of is and was determined experimentally using one mole of gas at room temperature and atmospheric pressure, but it applies to any ideal gas at any temperature or pressure.

The final version for the equation of state for an ideal gas is represented as:

where:

• is the pressure in Pa
• is the volume in m³
• is the amount of gas in moles
• is the molar gas constant
• is the temperature in K.

This equation links the microscopic motion of particles (kinetic theory) to the macroscopic gas laws (measurable properties such as pressure, volume, and temperature).

The pressure and volume of an ideal gas can be explained by the microscopic motion of its particles.

The more particles there are, or the faster or heavier they are, the greater the pressure they exert on the container walls.

These properties are linked by the equation:

where:

• is pressure in Pa
• is volume in m³
• is the number of particles
• is the mass of one particle in kg
• is the molecular mean square speed of the particles in It is the average of the squares of the particle speeds and depends directly on the temperature of the gas.

Assuming that the volume of the gas is a constant, this equation shows that:

• If increases, there are more collisions, so the pressure increases.
• If increases, each collision transfers more momentum, so the pressure increases.
• If increases, particles move faster, transferring more momentum per collision, so the pressure increases.

The particles in an ideal gas move in random directions, so the average velocity is zero. For this reason, particle motion is typically described in terms of speed, rather than velocity. The mean square speed is the average of the squares of the particle speeds. As it is a speed squared, the units will be .

For example, if four particles have speeds of:

• ,
• ,
• , and

their mean square speed can be calculated as follows:

The root mean square speed is defined as the square root of the mean square speed :

The root mean square speed has units of .

The Boltzmann constant links temperature to the energy of individual particles:

Where:

• is the molar gas constant, and
• is the Avogadro constant.

While applies to one mole of gas, applies to a single particle, making it useful in microscopic descriptions of gases.

The ideal gas equation is usually written in terms of moles :

However, we can rewrite it in terms of the number of particles . Since:

Where:

• is pressure,
• is volume,
• is the boltzamann constant, and
• is temperature

The kinetic energy of a particle of mass moving at a speed is:

For a particle in an ideal gas, the mean kinetic energy is:

where:

• is the mass of one particle in (kg)
• is the mean square speed of the particles in

The pressure and volume of an ideal gas can be explained by the microscopic motion of its particles. By combining the macroscopic gas law with kinetic theory, we can relate the average kinetic energy of particles to the gas’s temperature. The ideal gas law in terms of particles is:

The pressure and volume of an ideal gas are also related through kinetic theory:

Where:

• is the mass of one particle, and
• is the mean square speed of the particles.

Equating the two right-hand sides of the equations gives:

The left-hand side is equal to the mean kinetic energy of the particles so:

Where:

• is the molar gas constant, and
• is the temperature in .

The internal energy of a substance is the sum of the random kinetic and potential energies of its particles:

• In an ideal gas, electrostatic forces between particles are negligible, so the potential energy is zero.
• The internal energy is therefore entirely due to the kinetic energy of the particles.

The mean kinetic energy of the particles in an ideal gas is equal to:

The internal energy of a gas is directly proportional to its temperature, as the mean kinetic energy of the gas molecules is proportional to the temperature.

The total internal energy of an ideal gas is the sum of the kinetic energies of all particles:

Where:

• is the total internal energy of the ideal gas in (J)
• is the number of particles in the gas
• is the Boltzmann constant
• is the absolute temperature of the gas in (K)

Doubling the temperature doubles both the mean kinetic energy and the total internal energy. Therefore, the internal energy of an ideal gas is proportional to its absolute temperature.