9.3 Stoichiometry of Gaseous Substances, Mixtures, and Reactions

Gas Laws and Stoichiometry

Ideal gas law calculations

The ideal gas law ties together the four key variables that describe a gas sample into one equation:

$PV = nRT$

where $P$ is pressure, $V$ is volume, $n$ is moles of gas, $R$ is the ideal gas constant, and $T$ is temperature in Kelvin. If you know any three of these variables, you can solve for the fourth.

Two useful rearrangements of this law come up frequently:

• Gas density can be calculated from $d = \frac{PM}{RT}$, where $M$ is the molar mass of the gas. This is handy because it means you don't need to know the actual mass and volume separately. For example, you can compare the densities of air (average $M \approx 29 \text{ g/mol}$) and helium ($M = 4 \text{ g/mol}$) at the same conditions to see why helium balloons float.
• Molar mass can be determined experimentally by rearranging to $M = \frac{dRT}{P}$. If you measure the density of an unknown gas at a known temperature and pressure, you can figure out its molar mass.

Molar volume is the volume one mole of an ideal gas occupies at a given temperature and pressure. At STP (0°C and 1 atm), this value is approximately 22.4 L/mol. This serves as a useful conversion factor in many problems.

Gas stoichiometry in reactions

Balanced chemical equations give you mole ratios between reactants and products. When gases are involved, you can use the ideal gas law to convert between moles and measurable quantities like volume and pressure.

Here's how to solve a gas stoichiometry problem:

1. Balance the chemical equation so you have correct mole ratios.
2. Convert given quantities to moles. If you're given a gas volume, pressure, and temperature, use $PV = nRT$ to find moles. If you're given mass, divide by molar mass.
3. Use the mole ratio from the balanced equation to find moles of the unknown substance.
4. Convert moles of the unknown to the desired unit. Need volume? Use $PV = nRT$ again. Need mass? Multiply by molar mass.

For example, in the combustion of methane:

$CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O$

The equation tells you that 1 mole of $CH_4$ reacts with 2 moles of $O_2$ and produces 1 mole of $CO_2$. If you know the volume of $CH_4$ at a certain temperature and pressure, you can find moles of $CH_4$, then use the 1:1 ratio to find moles of $CO_2$, then convert back to volume.

At STP, you can skip the full ideal gas law and use the shortcut that 1 mole of any ideal gas = 22.4 L.

Gas Mixtures

Dalton's law for mixtures

When multiple non-reacting gases share a container, each gas contributes to the total pressure independently. Dalton's law says the total pressure is simply the sum of each gas's partial pressure:

$P_{total} = P_1 + P_2 + ... + P_n$

A gas's partial pressure ($P_n$) is the pressure it would exert if it alone occupied the entire container. You calculate it using the mole fraction:

$P_n = X_n \times P_{total}$

The mole fraction ($X_n$) is the ratio of moles of one gas to the total moles in the mixture:

$X_n = \frac{n_n}{n_{total}}$

For example, air is roughly 78% $N_2$ and 21% $O_2$ by moles. At 1 atm total pressure, the partial pressure of $N_2$ is about 0.78 atm and $O_2$ is about 0.21 atm. This concept matters in real applications like calculating the gas mix in scuba tanks or determining the concentration of $CO_2$ in exhaled breath.

Note that all mole fractions in a mixture must add up to 1.

Gas Laws

Three earlier gas laws describe what happens when you hold one variable constant. Each is really a special case of the ideal gas law:

• Charles's law: At constant pressure, volume is directly proportional to temperature. Heat a gas up, and it expands. ($V \propto T$)
• Boyle's law: At constant temperature, pressure and volume are inversely proportional. Compress a gas into a smaller space, and its pressure increases. ($P \propto \frac{1}{V}$)
• Gay-Lussac's law: At constant volume, pressure is directly proportional to temperature. Heat a sealed container, and the pressure rises. ($P \propto T$)

These relationships all assume temperature is measured in Kelvin. Using Celsius in these equations will give you wrong answers.