B.3.1 Kinetic theory and gas behavior

Pressure, Amount of Substance, and the Ideal Gas Law

Pressure: Force Distributed Over Area

Definition

Pressure

Pressure is the force applied per unit area.

Mathematically, it is expressed as:

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

where:

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

Tip

Pressure is a scalar quantity.

Hint

The unit of pressure, the pascal (Pa), is equivalent to one newton per square meter (N/m²).

Example

Imagine a block weighing 100 N resting on a surface with an area of 0.5 m². To find the pressure it exerts:

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

$= \frac{100 N}{0.5 m^{2}} = 200 Pa$

Example

A cylinder with a weight of 500 N and a base area of 0.2 m² exerts a pressure of:

$P = \frac{500 N}{0.2 m^{2}} = 2500 Pa$

Tip

Always ensure the force is perpendicular to the surface when calculating pressure.

Amount of Substance: The Mole and Avogadro’s Constant

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

Definition

Mole

A mole is defined as the amount of substance containing as many particles (atoms, molecules, etc.) as there are atoms in 12 grams of carbon-12.

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:

$n$ is the number of moles
$N$ is the total number of particles
$N_{A}$ is the Avogadro constant ( $6.022 \times 10^{23}$ particles/mol)

Note

The molar mass of a substance (in grams per mole) tells you the mass of one mole of that substance. For example, water (H₂O) has a molar mass of 18 g/mol.

Example question

Calculating moles

How many moles are in $1.2 \times 10^{24}$ molecules of water?

Solution

Using the formula:

$n = \frac{N}{N_{A}}$

$= \frac{1.2 \times 10^{24}}{6.022 \times 10^{23}}$

$\approx 2 mol$

The Ideal Gas Law: A Universal Equation

Definition

Ideal gas law

The ideal gas law is the equation of state of a hypothetical ideal gas which relates the pressure, volume, temperature, and amount of substance in a gas.

It is expressed as:

$P V = n R T$

where:

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

Example question

Calculations using the ideal gas law

Calculate the pressure of 0.5 moles of an ideal gas in a 0.02 m³ container at 300 K.

Solution

Using the ideal gas law:

$P V = n R T$

$P = \frac{n R T}{V} = \frac{0.5 \times 8.31 \times 300}{0.02} = 62, 325 Pa$

Tip

Always convert temperature to kelvin when using the ideal gas law.

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 $k_{B}$ :

$P V = N k_{B} T$

where:

$N$ is the number of molecules
$k_{B}$ is the Boltzmann constant ( $1.38 \times 10^{- 23} J K^{- 1}$ )

Hint

The Boltzmann constant $k_{B}$ is related to the universal gas constant $R$ by the equation $R = N_{A} k_{B}$ .

Deriving 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 ( $P \propto T$ ).

$\frac{P_{1}}{T_{1}} = \frac{P_{2}}{T_{2}}$

Combining these relationships, we find:

$\frac{P V}{T} = constant$

This constant is proportional to the number of moles $n$ , leading to the ideal gas law:

$P V = n R T$

Example question

A gas at 2 atm and 3 L is compressed to 1.5 L at a constant temperature. What is the new pressure?

Solution

Using Boyle’s Law ( $P_{1} V_{1} = P_{2} V_{2}$ ):

$P_{2} = \frac{P_{1} V_{1}}{V_{2}}$

$= \frac{2 atm \times 3 L}{1.5, L} = 4 atm$

Applications and Limitations of the Ideal Gas Law

The ideal gas law is a powerful tool for understanding gas behavior, but it has limitations.
It assumes:
- No intermolecular forces between gas particles.
- Negligible volume of gas particles compared to the container.

Note

The ideal gas law breaks down under conditions of high pressure or low temperature, where intermolecular forces and particle volume become significant.

Reflection and Connections

Theory of Knowledge

How do the assumptions of the ideal gas model reflect the broader scientific principle of simplification in modeling?
Can you think of other models in physics or other sciences that make similar assumptions?

The ideal gas law connects microscopic properties (like the number of molecules) with macroscopic properties (like pressure and volume).

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B.3.1 Kinetic theory and gas behavior

Pressure, Amount of Substance, and the Ideal Gas Law

Pressure: Force Distributed Over Area

Amount of Substance: The Mole and Avogadro’s Constant

Calculating Moles

Calculating moles

The Ideal Gas Law: A Universal Equation

Calculations using the ideal gas law

Alternative Form: Using the Boltzmann Constant

Deriving the Ideal Gas Law: Empirical Observations

Boyle’s Law

Charles’s Law

Gay-Lussac’s Law

Applications and Limitations of the Ideal Gas Law

Reflection and Connections

You've read 2/2 free chapters this week.

Introduction to Kinetic Theory and Gas Behavior