A mixture containing 0.50 mol H2(g), 1.00 mol O2(g), and 3.50 mol N2(g) is confined in a 25.0-L vessel at 25 °C. Calculate the partial pressure of H2, O2, and N2.

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1. Step 1: Use the ideal gas law to find the total pressure of the gas mixture. The ideal gas law is given by the equation: \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L·atm/mol·K), and \( T \) is the temperature in Kelvin. First, convert the temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature.
2. Step 2: Calculate the total number of moles in the mixture by adding the moles of each gas: \( n_{\text{total}} = n_{\text{H}_2} + n_{\text{O}_2} + n_{\text{N}_2} \).
3. Step 3: Substitute the total number of moles, the volume of the vessel, the temperature in Kelvin, and the ideal gas constant into the ideal gas law equation to solve for the total pressure \( P_{\text{total}} \).
4. Step 4: Use Dalton's Law of Partial Pressures to find the partial pressure of each gas. According to Dalton's Law, the partial pressure of a gas in a mixture is equal to the mole fraction of that gas multiplied by the total pressure. Calculate the mole fraction of each gas: \( \text{Mole fraction of } H_2 = \frac{n_{\text{H}_2}}{n_{\text{total}}} \), \( \text{Mole fraction of } O_2 = \frac{n_{\text{O}_2}}{n_{\text{total}}} \), \( \text{Mole fraction of } N_2 = \frac{n_{\text{N}_2}}{n_{\text{total}}} \).
5. Step 5: Calculate the partial pressure of each gas by multiplying its mole fraction by the total pressure: \( P_{\text{H}_2} = \text{Mole fraction of } H_2 \times P_{\text{total}} \), \( P_{\text{O}_2} = \text{Mole fraction of } O_2 \times P_{\text{total}} \), \( P_{\text{N}_2} = \text{Mole fraction of } N_2 \times P_{\text{total}} \).