6 Gases and the Gas Laws

Pressure and gas-state variables

Gas particles move constantly and collide with the walls of their container. The collisions exert force over an area, producing pressure. In equation form, pressure is $P = \frac{F}{A}$, where $F$ is force and $A$ is area.

A gas sample is described by pressure $P$, volume $V$, temperature $T$, and amount $n$, measured in moles. Pressure units include pascals ($\mathrm{Pa}$), kilopascals ($\mathrm{kPa}$), atmospheres ($\mathrm{atm}$), and torr. A useful conversion is

Gas-law temperature calculations use the Kelvin scale, not Celsius degrees:

Use a consistent set of units, including a gas constant that matches the pressure and volume units.

Simple gas laws

Simple gas laws describe how two gas variables change while the other relevant variables are held constant. The conditions matter: a relationship applies only when its stated quantities remain fixed.

• Boyle’s law: At constant temperature and amount of gas, pressure and volume vary inversely: $P_1V_1 = P_2V_2$. Compressing a gas at constant temperature therefore raises its pressure.

• Charles’s law: At constant pressure and amount of gas, volume is proportional to absolute temperature: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$. A balloon can expand as its gas warms if the external pressure remains approximately constant.

• Amontons’s (Gay-Lussac’s) law: At constant volume and amount of gas, pressure is proportional to absolute temperature: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$.

• Avogadro’s law: At constant pressure and temperature, volume is proportional to the amount of gas: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$.

Always use kelvins in relationships involving temperature. For example, a gas occupies $2.0\ \mathrm{L}$ at $300\ \mathrm{K}$. If it warms to $450\ \mathrm{K}$ at constant pressure, Charles’s law gives

Takeaway: Identify which quantities are constant before choosing a simple gas law.

Combining gas variables

When the amount of gas is fixed but pressure, volume, and temperature can all change, combine the pairwise relationships into the combined gas law:

For a broader relationship that includes the amount of gas, use the ideal gas equation:

The ideal gas constant $R$ has several possible values. Two common choices are $0.08206\ \mathrm{L\cdot atm\cdot mol^{-1}\cdot K^{-1}}$ and $8.314\ \mathrm{kPa\cdot L\cdot mol^{-1}\cdot K^{-1}}$. Choose the value that matches the pressure and volume units in the calculation. The ideal-gas model is a useful approximation for many gases, especially at relatively low pressure and high temperature.

Example: Find the volume of $0.500\ \mathrm{mol}$ of gas at $298\ \mathrm{K}$ and $1.00\ \mathrm{atm}$, using $R = 0.08206\ \mathrm{L\cdot atm\cdot mol^{-1}\cdot K^{-1}}$:

Takeaway: Use the combined gas law when the amount is fixed; use the ideal gas equation when the amount is part of the problem.

Partial pressures in gas mixtures

For a mixture of gases that do not react, Dalton’s law of partial pressures states that the total pressure is the sum of the pressures each component would exert on its own in the same container:

A component’s mole fraction, $X_i$, is the fraction of the mixture’s total gas amount contributed by that component. Its partial pressure is found from

For example, a mixture containing $25\%$ nitrogen and $75\%$ oxygen by moles has a total pressure of $2.0\ \mathrm{atm}$. The nitrogen partial pressure is $0.25 \times 2.0\ \mathrm{atm} = 0.50\ \mathrm{atm}$, and the oxygen partial pressure is $0.75 \times 2.0\ \mathrm{atm} = 1.5\ \mathrm{atm}$. Their sum is the total pressure.

When a gas is collected over water, the sample includes water vapor as well as the gas of interest. The total pressure is the sum of the dry gas pressure and the water vapor pressure, so

Takeaway: For a gas mixture, add component partial pressures to find the total; subtract water vapor pressure when determining the pressure of a dry gas collected over water.