Calculate Partial Pressure of Nitrogen ($p_{N_2}$): According to Dalton's Law, the partial pressure of a component in a gas mixture is the product of its mole fraction in the gas phase and the total pressure. $p_{N_2} = \text{Mole fraction in air} \times P_{\text{total}}$ $p_{N_2} = 0.8 \times 5 \text{ atm} = 4.0 \text{ atm}$

Calculate Mole Fraction of Nitrogen in Solution ($x_{N_2}$): Using Henry's Law formula: $p = K_H \times x$ $4.0 \text{ atm} = (1.0 \times 10^5 \text{ atm}) \times x_{N_2}$ $x_{N_2} = \frac{4.0}{1.0 \times 10^5} = 4.0 \times 10^{-5}$

Calculate Moles of Nitrogen ($n_{N_2}$): The mole fraction $x_{N_2}$ is defined as $\frac{n_{N_2}}{n_{N_2} + n_{H_2O}}$. For dilute solutions, $n_{N_2} \ll n_{H_2O}$, so we approximate $x_{N_2} \approx \frac{n_{N_2}}{n_{H_2O}}$. $4.0 \times 10^{-5} = \frac{n_{N_2}}{10}$ $n_{N_2} = 4.0 \times 10^{-5} \times 10 = 4.0 \times 10^{-4} \text{ mol}$