In reactions involving gases, it is often much easier to measure the pressure of the gases rather than their concentrations. To study gas equilibria, we use a constant called \( K_p \), which is expressed in terms of partial pressures. Before calculating \( K_p \), we must understand how to determine the mole fraction and partial pressure of each gas in a mixture.
🔑 Key Principle
The total pressure of a gas mixture is the sum of the individual pressures exerted by each gas. The contribution of each gas to the total pressure depends entirely on the relative number of gas particles (moles) present in the mixture, regardless of the chemical identity of the gas.
Mole Fractions
The mole fraction of a gas is the fraction of the total moles of gas in a mixture that is made up of that specific gas. It is a ratio and has no units.
The number of moles of a specific component in a mixture divided by the total number of moles of all components in the mixture:
\( \chi_A = \frac{\text{moles of gas } A}{\text{total moles of gas in mixture}} = \frac{n_A}{n_{\text{total}}} \)
Because it is a fraction of a whole, the sum of all the mole fractions in a mixture must always equal exactly 1:
\( \chi_A + \chi_B + \chi_C + \dots = 1 \)
Partial Pressures
In a mixture of gases, each individual gas exerts its own pressure. This is its partial pressure.
The pressure that an individual gas in a mixture would exert if it alone occupied the same volume at the same temperature.
The partial pressure of a gas \( A \) is calculated by multiplying its mole fraction by the total pressure of the system:
Where:
- \( p_A \) is the partial pressure of gas \( A \) (units match the total pressure, e.g., \( \text{kPa} \), \( \text{atm} \), or \( \text{Pa} \)).
- \( \dots \) lowercase \( p \) is used for partial pressures, while uppercase \( P_{\text{total}} \) represents the total pressure.
The total pressure of a mixture of non-reacting gases is equal to the sum of the partial pressures of the individual gases:
\[ P_{\text{total}} = p_A + p_B + p_C + \dots \]
Always use a lowercase \( p \) when writing partial pressures in equilibrium expressions or calculations, for example: \( p(\text{N}_2) \) or \( p_{\text{N}_2} \). Writing uppercase \( P \) for partial pressures can cost you marks on your exam papers because uppercase is reserved for the total pressure of the system.
Worked Examples
Step 1: Calculate the total number of moles of gas:
\( n_{\text{total}} = 2.00 + 4.00 + 1.50 = 7.50\text{ moles} \)
Step 2: Calculate the mole fraction (\( \chi \)) of each gas:
- \( \chi_{\text{N}_2} = \frac{2.00}{7.50} = 0.267 \)
- \( \chi_{\text{H}_2} = \frac{4.00}{7.50} = 0.533 \)
- \( \chi_{\text{NH}_3} = \frac{1.50}{7.50} = 0.200 \)
Step 3: Check your answers:
\( 0.267 + 0.533 + 0.200 = 1.000 \)
The mole fractions sum to exactly 1.00, confirming the calculations are correct.
Step 1: Apply the partial pressure formula \( p_A = \chi_A \times P_{\text{total}} \):
- \( p_{\text{N}_2} = 0.267 \times 150\text{ kPa} = 40.05\text{ kPa} \) (or 40.1 kPa to 3 sig figs)
- \( p_{\text{H}_2} = 0.533 \times 150\text{ kPa} = 79.95\text{ kPa} \) (or 80.0 kPa to 3 sig figs)
- \( p_{\text{NH}_3} = 0.200 \times 150\text{ kPa} = 30.00\text{ kPa} \) (or 30.0 kPa to 3 sig figs)
Step 2: Verify using Dalton's Law:
\( P_{\text{total}} = p_{\text{N}_2} + p_{\text{H}_2} + p_{\text{NH}_3} \)
\( 40.05 + 79.95 + 30.00 = 150.00\text{ kPa} \)
The sum of the calculated partial pressures matches the total pressure exactly.
If you round your mole fractions early, your partial pressures may not add up exactly to the total pressure due to rounding differences. To prevent this, always use the unrounded values in your calculator during calculations, and only round your final answers to the appropriate number of significant figures (usually 3, or matching the data given in the question).
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