IB Physics B.3.1 Kinetic Theory and Gas Behavior Notes

Pressure, Amount of Substance, and the Ideal Gas Law

Pressure: Force Distributed Over Area

Mathematically, it is expressed as:

$$
P = \frac{F}{A}
$$

where:

1. $P$ is the pressure (in pascals, Pa)
2. $F$ is the force applied (in newtons, N)
3. $A$ is the area over which the force is distributed (in square meters, m²)

Amount of Substance: The Mole and Avogadro’s Constant

The amount of substance is measured in moles, a fundamental concept in chemistry and physics.

This number is the Avogadro constant,$N_A$, approximately $6.022 \times 10^{23}$ particles per mole.

Calculating Moles

If a substance contains $N$ particles, the number of moles $n$ is given by:

$$
n = \frac{N}{N_A}
$$

where:

1. $n$ is the number of moles
2. $N$ is the total number of particles
3. $N_A$ is the Avogadro constant ($6.022 \times 10^{23}$ particles/mol)

The Ideal Gas Law: A Universal Equation

It is expressed as:

$$
PV = nRT
$$

where:

1. $P$ is the pressure (in pascals, Pa)
2. $V$ is the volume (in cubic meters, m³)
3. $n$ is the number of moles
4. $R$ is the universal gas constant ($8.31 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1}$)
5. $T$ is the absolute temperature (in kelvin, K)

Alternative Form: Using the Boltzmann Constant

The ideal gas law can also be expressed in terms of the number of molecules $N$ and the Boltzmann constant $k_B$:

$$
PV = Nk_BT
$$

where:

1. $N$ is the number of molecules
2. $k_B$ is the Boltzmann constant ($1.38 \times 10^{-23} \, \text{J} \, \text{K}^{-1}$)

Observing the Ideal Gas Law: Empirical Observations

The ideal gas law is derived from three fundamental gas laws:

Boyle’s Law

At constant temperature, the pressure of a gas is inversely proportional to its volume ($P \propto \frac{1}{V}$).

$$P_1 V_1 = P_2 V_2$$

Charles’s Law

At constant pressure, the volume of a gas is directly proportional to its absolute temperature ($V \propto T$).

$$\frac{V_1}{T_1}=\frac{V_2}{T_2}$$

Gay-Lussac’s Law

At constant volume, the pressure of a gas is directly proportional to its absolute temperature ($P \propto T$).

$$\frac{P_1}{T_1}=\frac{P_2}{T_2}$$

1. Combining these relationships, we find: $$
   \frac{PV}{T} = \text{constant}
   $$
2. This constant is proportional to the number of moles $n$, leading to the ideal gas law: $$
   PV = nRT
   $$

Representing Gas Laws on Pressure-Volume Diagrams

A pressure-volume (PV) diagram plots the pressure of a gas against its volume, and each gas law traces a characteristic shape on it.

1. An isotherm shows Boyle's law at constant temperature, curving so that $P \propto \frac{1}{V}$.
2. Raising the temperature shifts the isotherm further from the origin.
3. A change of state appears where an isotherm flattens, as the gas condenses into a liquid at nearly constant pressure while its volume falls sharply.

Applications and Limitations of the Ideal Gas Law

1. The ideal gas law is a powerful tool for understanding gas behavior, but it has limitations.
2. It assumes molecules have negligible volume and exert no intermolecular forces.
3. It breaks down at high pressure or low temperature, where real gases deviate (see B.3.2).