The partial pressure of gas X is 15 atm, gas Y is 6 atm, and gas Z is 3 atm.
To calculate the partial pressure of each gas in a mixture, you can use Dalton's Law of Partial Pressures. According to Dalton's Law:
\[ P_{\text{total}} = P_X + P_Y + P_Z \]
where \( P_{\text{total}} \) is the total pressure of the mixture, and \( P_X, P_Y, P_Z \) are the partial pressures of gases X, Y, and Z, respectively.
Given that the total pressure (\( P_{\text{total}} \)) is 24 atm and the moles of each gas, you can calculate the partial pressures using the mole fraction of each gas:
\[ P_X = \frac{n_X}{n_{\text{total}}} \times P_{\text{total}} \]
\[ P_Y = \frac{n_Y}{n_{\text{total}}} \times P_{\text{total}} \]
\[ P_Z = \frac{n_Z}{n_{\text{total}}} \times P_{\text{total}} \]
where \( n_X, n_Y, n_Z \) are the moles of gases X, Y, and Z, and \( n_{\text{total}} \) is the total number of moles in the mixture.
Given:
\[ n_X = 5 \, \text{moles} \]
\[ n_Y = 2 \, \text{moles} \]
\[ n_Z = 1 \, \text{mole} \]
\[ n_{\text{total}} = n_X + n_Y + n_Z = 5 + 2 + 1 = 8 \, \text{moles} \]
Calculate partial pressures:
\[ P_X = \frac{5}{8} \times 24 \, \text{atm} \]
\[ P_Y = \frac{2}{8} \times 24 \, \text{atm} \]
\[ P_Z = \frac{1}{8} \times 24 \, \text{atm} \]
Now, calculate the values:
\[ P_X = \frac{5}{8} \times 24 \, \text{atm} = 15 \, \text{atm} \]
\[ P_Y = \frac{2}{8} \times 24 \, \text{atm} = 6 \, \text{atm} \]
\[ P_Z = \frac{1}{8} \times 24 \, \text{atm} = 3 \, \text{atm} \]
Therefore, the partial pressure of gas X is 15 atm, gas Y is 6 atm, and gas Z is 3 atm.