Physics (9702) | Ideal gases

Welcome to the World of Ideal Gases! (9702 Physics)

Hello future Physicists! This chapter, "Ideal Gases," is where we connect the tiny, invisible world of atoms and molecules to the large, measurable properties of gases—like pressure and temperature.
This is a really important bridge between classical mechanics and thermal physics. Don't worry if some concepts seem abstract; we will use simple analogies to make sure you grasp the concepts firmly!

In this section, we will learn how to describe the behavior of gases using powerful equations and understand why gas molecules behave the way they do, which is key knowledge for the A Level paper.

15.1 The Mole and Avogadro Constant

The Concept of Amount of Substance (The Mole)

In Physics (and Chemistry!), dealing with individual atoms is impossible because the numbers are huge. We need a counting unit for large amounts of particles, and that unit is the mole (mol).

• The amount of substance is an SI base quantity, measured in moles (mol).
• One mole of any substance is defined as the amount containing the same number of particles as there are atoms in exactly 12 grams of Carbon-12.

The Avogadro Constant (\(N_A\))

The number of particles in one mole is constant, known as the Avogadro constant (\(N_A\)).

\(N_A \approx 6.02 \times 10^{23} \text{ mol}^{-1}\)

Analogy: Think of the mole as the "Physics Dozen." If a baker sells bread in dozens (12 units), a scientist measures particles in moles (\(6.02 \times 10^{23}\) units).

Molar Quantities

Molar quantities link the macroscopic (large-scale) world to the microscopic (atomic-scale) world.

• If a sample of gas contains \(n\) moles, the total number of molecules, \(N\), in the sample is:
  $$N = n \times N_A$$

Key Takeaway: The mole is simply a convenient unit for counting incredibly large numbers of particles, tied directly to the Avogadro constant.

15.2 The Equation of State for an Ideal Gas

Defining the Ideal Gas

An ideal gas is a theoretical model that perfectly obeys the gas laws under all conditions. Although no gas is truly ideal, real gases (like air, oxygen, nitrogen) behave very much like ideal gases at high temperatures and low pressures.

The defining characteristic of an ideal gas is that its pressure (\(p\)), volume (\(V\)), and thermodynamic temperature (\(T\)) are related by the proportionality:

$$p V \propto T$$

Remember: \(T\) MUST be the thermodynamic temperature, measured in Kelvin (K). Never use Celsius in gas equations!

The Ideal Gas Equation (Using Moles)

This is the standard form of the equation of state:

$$p V = n R T$$

Where:

• \(p\) = Pressure (Pa or N m\(^{-2}\))
• \(V\) = Volume (m\(^3\))
• \(n\) = Amount of substance (number of moles, mol)
• \(T\) = Thermodynamic Temperature (K)
• \(R\) = The Molar Gas Constant (or Universal Gas Constant).

Did you know? \(R\) is a constant for *any* ideal gas, linking the energy scale in Physics (Joules) to the temperature scale (Kelvin) for a given mole of substance.

The Ideal Gas Equation (Using Molecules)

Sometimes, you are given the total number of molecules (\(N\)) instead of the number of moles (\(n\)). We can adjust the equation using the Boltzmann constant (\(k\)).

Since \(n = N/N_A\), we substitute this into \(pV = nRT\):

$$p V = \left(\frac{N}{N_A}\right) R T$$

We define the Boltzmann constant as:

$$k = \frac{R}{N_A}$$

Substituting \(k\) gives the molecular form of the equation:

$$p V = N k T$$

• \(k\) = Boltzmann Constant (J K\(^{-1}\)). This constant relates the average translational kinetic energy of a single gas molecule to the temperature of the gas.

Key Takeaway: The ideal gas law is the master equation. It combines pressure, volume, and temperature, using either the molar gas constant \(R\) or the Boltzmann constant \(k\), depending on whether you are counting moles or individual molecules.

15.3 The Kinetic Theory of Gases

The kinetic theory explains the macroscopic properties of gases (like pressure and temperature) by considering the random motion of their microscopic molecules.

The Basic Assumptions

To derive the kinetic theory relationship, we must assume the gas is ideal, meaning its particles behave according to these rules. This simplifies the physics greatly!

1. Large Number of Molecules: The gas consists of a very large number of identical, perfectly spherical molecules.
2. Random Motion: Molecules are in constant, rapid, random motion. (They move in all directions with a wide range of speeds).
3. Negligible Volume: The total volume of the molecules themselves is negligible compared to the volume occupied by the gas container. (The molecules are mostly empty space).
4. Elastic Collisions: All collisions (between molecules and between molecules and walls) are perfectly elastic. No kinetic energy is lost during collisions.
5. Negligible Forces: Forces between molecules are negligible, except during collisions. This means molecules travel in straight lines at constant speed between collisions.
6. Duration of Collisions: The time duration of collisions is negligible compared to the time between collisions.

Analogy: Imagine a swarm of angry, tiny, super-bouncy billiard balls trapped in a room. They don't stick to each other, and they never stop moving!

Pressure and Molecular Movement

The most crucial outcome of the kinetic theory is explaining where pressure comes from.

How is Pressure Caused?

Pressure is caused by the bombardment of the container walls by the rapidly moving gas molecules.

1. A gas molecule hits the wall. Because the collision is elastic, it bounces off with the same speed, but the direction (and thus velocity) changes.
2. This change in velocity means there is a change in momentum (\(\Delta p\)).
3. According to Newton’s Second Law (\(F = \Delta p / \Delta t\)), a force is exerted on the molecule by the wall.
4. By Newton’s Third Law, the molecule exerts an equal and opposite force on the wall.
5. Since billions of molecules hit the walls every second, this continuous bombardment creates a constant outward force.
6. Pressure is simply this force distributed over the area of the wall (\(P = F/A\)).

The Kinetic Theory Equation for Pressure

By combining the laws of momentum and Newton’s laws, and considering motion in three dimensions, the derivation leads to the fundamental kinetic theory relationship:

$$p V = \frac{1}{3} N m \langle c^2 \rangle$$

Where:

• \(p\) = Pressure (Pa)
• \(V\) = Volume (m\(^3\))
• \(N\) = Total number of molecules
• \(m\) = Mass of one molecule (kg)
• \(\langle c^2 \rangle\) = The mean-square speed (m\(^2\) s\(^{-2}\)).

Wait, what is \(\langle c^2 \rangle\)? Because molecules move at different speeds, we cannot use a single speed \(c\). We have to average the square of their speeds.

$$\langle c^2 \rangle = \frac{c_1^2 + c_2^2 + c_3^2 + ... + c_N^2}{N}$$

Key Takeaway: Pressure is directly related to the mass of the particles, the number of particles, and, most importantly, the average squared speed of those particles.

15.3 Molecular Speed and Kinetic Energy

The Root-Mean-Square Speed (\(c_{r.m.s.}\))

While \(\langle c^2 \rangle\) is useful in the kinetic theory equation, it has squared units. To find a typical speed in m s\(^{-1}\), we take the square root of the mean-square speed. This is called the root-mean-square speed, \(c_{r.m.s.}\):

$$c_{r.m.s.} = \sqrt{\langle c^2 \rangle}$$

Relating Kinetic Energy to Temperature

This is the most fundamental link between the microscopic world and temperature. We compare the two key gas equations:

1. Ideal Gas Equation (Molecular form):
$$p V = N k T$$

2. Kinetic Theory Equation:
$$p V = \frac{1}{3} N m \langle c^2 \rangle$$

Since both expressions equal \(pV\), we can equate them:

$$\require{cancel} \cancel{N} k T = \frac{1}{3} \cancel{N} m \langle c^2 \rangle$$

Now, recall the formula for the kinetic energy (\(E_K\)) of a single particle: \(E_K = \frac{1}{2} m c^2\).

Let's rearrange our combined equation to isolate the kinetic energy term:

$$k T = \frac{1}{3} m \langle c^2 \rangle$$

Multiply both sides by \(\frac{3}{2}\):

$$\frac{3}{2} k T = \frac{1}{2} m \langle c^2 \rangle$$

Since the average translational kinetic energy, $\langle E_K \rangle$, is given by $\frac{1}{2} m \langle c^2 \rangle$, we deduce:

Average Translational Kinetic Energy of a Molecule

$$\langle E_K \rangle = \frac{3}{2} k T$$

This is a critical result!

• The average translational kinetic energy of a molecule in an ideal gas is directly proportional to the thermodynamic temperature \(T\).
• This energy is independent of the type of gas or the mass of the molecules. At the same temperature, hydrogen molecules and oxygen molecules have the same average kinetic energy!

This explains why temperature is such a fundamental measure in physics: it is essentially a measure of the average random kinetic energy of the particles.