Q. how to calculate partial pressure

Definition and fundamental law. Dalton’s law states that the total pressure of a mixture of ideal gases equals the sum of the partial pressures of each gas. A compact form is

\[ P_{\text{total}} \;=\; \sum_{i} P_{i} \]

Meaning of partial pressure. The partial pressure of a gas in a mixture is the pressure that gas would exert if it alone occupied the whole volume at the same temperature. Two common ways to calculate a partial pressure are by using mole fraction or by using the ideal gas law for that component.

Mole fraction method. First compute the mole fraction of species i using

\[ x_{i} \;=\; \frac{n_{i}}{n_{\text{total}}} \]

Then obtain the partial pressure from the total pressure by

\[ P_{i} \;=\; x_{i}\,P_{\text{total}} \]

Ideal gas method for a component. If you know the number of moles of species i, the temperature and the common volume, you can use the ideal gas law for that component directly:

\[ P_{i} \;=\; \frac{n_{i} R T}{V} \]

Because the same \( R \), \( T \), and \( V \) apply to every component, summing the component pressures reproduces the total pressure. That is,

\[ P_{\text{total}} \;=\; \frac{n_{\text{total}} R T}{V} \;=\; \sum_{i} \frac{n_{i} R T}{V} \;=\; \sum_{i} P_{i} \]

Step by step procedure summary. Step 1. Determine the number of moles \( n_{i} \) of each gas present. Step 2. Compute total moles \( n_{\text{total}}=\sum_{i} n_{i} \). Step 3. If total pressure is known, compute mole fractions \( x_{i}=n_{i}/n_{\text{total}} \) and then \( P_{i}=x_{i}P_{\text{total}} \). Step 4. If total pressure is not given but temperature and volume are given, compute total pressure using \( P_{\text{total}}=n_{\text{total}}RT/V \) and then use step 3, or compute each \( P_{i}=n_{i}RT/V \) directly. Step 5. Check that the sum of calculated partial pressures equals the total pressure within rounding error.

Worked example A known total pressure. Suppose a mixture contains 1.00 mol of gas A and 3.00 mol of gas B and the measured total pressure is 4.00 atm. Step A1 compute total moles

\[ n_{\text{total}}=1.00+3.00=4.00\ \text{mol} \]

Step A2 compute mole fractions

\[ x_{A}=\frac{1.00}{4.00}=0.250 \qquad x_{B}=\frac{3.00}{4.00}=0.750 \]

Step A3 compute partial pressures

\[ P_{A}=x_{A}P_{\text{total}}=0.250\times 4.00\ \text{atm}=1.00\ \text{atm} \]

\[ P_{B}=x_{B}P_{\text{total}}=0.750\times 4.00\ \text{atm}=3.00\ \text{atm} \]

Worked example B using ideal gas law. Suppose a 10.0 L container at 298 K contains 2.00 mol O2 and 3.00 mol N2. Use \( R=0.082057\ \text{L atm mol}^{-1}\text{K}^{-1} \). Step B1 total moles

\[ n_{\text{total}}=2.00+3.00=5.00\ \text{mol} \]

Step B2 compute total pressure

\[ P_{\text{total}}=\frac{n_{\text{total}} R T}{V}=\frac{5.00\times 0.082057\times 298}{10.0}=12.2265\ \text{atm (approximately)} \]

Step B3 compute partial pressures directly or via mole fraction. Mole fraction of O2 is \( x_{\text{O}_{2}}=2.00/5.00=0.400 \). Then

\[ P_{\text{O}_{2}}=x_{\text{O}_{2}}P_{\text{total}}=0.400\times 12.2265=4.8906\ \text{atm (approximately)} \]

Alternatively direct ideal gas calculation for O2 gives the same result

\[ P_{\text{O}_{2}}=\frac{n_{\text{O}_{2}} R T}{V}=\frac{2.00\times 0.082057\times 298}{10.0}=4.8906\ \text{atm} \]

Step B4 for N2

\[ P_{\text{N}_{2}}=\frac{3.00\times 0.082057\times 298}{10.0}=7.3359\ \text{atm} \]

Consistency check

\[ P_{\text{O}_{2}}+P_{\text{N}_{2}}=4.8906+7.3359=12.2265\ \text{atm}=P_{\text{total}}\ ]

Units and common pitfalls. Ensure consistent units for R, volume, pressure, and temperature. Use R appropriate to the pressure units desired. When gases are not ideal at high pressures or low temperatures, corrections are needed but Dalton’s law for partial pressures still applies to partial pressures of each species in the real mixture if measured experimentally.

Quick checklist for solving a partial pressure problem. 1 Identify known quantities. 2 Choose mole fraction method if total pressure known. 3 Choose ideal gas method if n, T, V known. 4 Compute partial pressures. 5 Verify sum equals total pressure.