8.12 The Gas Laws

The gas laws are a set of fundamental principles that describe the relationship between the macroscopic properties of a gas: pressure ($P$), volume ($V$), and temperature ($T$), for a given amount of gas ($n$). These laws were developed through experimental observations and form the basis of our understanding of gas behavior. They are combined in the Ideal Gas Law.

Pressure is a Derived Unit→ measured in Pascals (Pa).

Boyle's Law (Constant Temperature)

Boyle's Law describes the relationship between pressure and volume when the temperature and the amount of gas are held constant.

Statement: For a fixed mass of gas at constant temperature, the volume is inversely proportional to the pressure.

This means that if you increase the pressure on a gas, its volume will decrease proportionally, and vice versa.

Mathematical Formulation: $V\propto \frac{1}{P}\text{or}PV=\text{constant}$ For a gas changing from an initial state (1) to a final state (2): $P_1{V}_1=P_2{V}_2$

Graphical Representation: The graph of pressure versus volume for an isothermal (constant temperature) process is a hyperbola. The graph of $P$ vs $\frac{1}{V}$ is a straight line through the origin.

Charles's Law (Constant Pressure)

Charles's Law describes the relationship between volume and temperature when the pressure and the amount of gas are held constant.

Statement: For a fixed mass of gas at constant pressure, the volume is directly proportional to its absolute temperature (in Kelvin).

This means that heating a gas will cause it to expand, and cooling it will cause it to contract.

Mathematical Formulation: $V\propto T\text{or}\frac{V}{T}=\text{constant}$ For a gas changing from an initial state (1) to a final state (2): $\frac{V_1}{T_1}=\frac{V_2}{T_2}$

Graphical Representation: The graph of volume versus absolute temperature is a straight line passing through the origin, confirming the direct proportionality $V\propto T$.

Gay-Lussac's Law (Constant Volume)

Gay-Lussac's Law (also known as Amontons's Law) describes the relationship between pressure and temperature when the volume and the amount of gas are held constant.

Statement: For a fixed mass of gas at constant volume, the pressure is directly proportional to its absolute temperature (in Kelvin).

This explains why the pressure inside a sealed container of gas increases when it is heated.

Mathematical Formulation: $P\propto T\text{or}\frac{P}{T}=\text{constant}$ For a gas changing from an initial state (1) to a final state (2): $\frac{P_1}{T_1}=\frac{P_2}{T_2}$

Ideal Gas Equation

By combining Boyle's Law, Charles's Law, and Avogadro's Law, we derive the General Gas Equation (Equation of State for an Ideal Gas):

$PV=nRT$

Where:

• $P$ = pressure (Pa)
• $V$ = volume (m³)
• $n$ = number of moles (mol)
• $R$ = Universal Gas Constant $=8.314{\text{ J mol}}^{-1}{\text{ K}}^{-1}$
• $T$ = absolute temperature (K)

An equivalent form using the number of molecules $N$ and the Boltzmann constant $k_B=\frac{R}{N_A}=1.38\times 10^{-23}{\text{ J K}}^{-1}$:

$PV=N{k}_{B}T$

For an ideal gas, the internal energy depends solely on temperature.

Worked Example: A gas occupies $2.0{\text{ m}}^3$ at a pressure of $1.0\times 10^5\text{ Pa}$ and temperature $300\text{ K}$. Find the number of moles.

$n=\frac{PV}{RT}=\frac{(1.0\times 10^5)(2.0)}{(8.314)(300)}=\frac{2.0\times 10^5}{2494.2}\approx 80.2\text{ mol}$

Summary Table