Step 1: Calculate the number of moles of each gas.
The molar masses of He, Ne, and Ar are 4.003 g, 20.18 g, and 39.95 g, respectively.
[tex]\begin{aligned} & n_{\text{He}} = \text{15.50 g} \times \frac{\text{1 mol}}{\text{4.003 g}} = \text{3.872 mol} \\ & n_{\text{Ne}} = \text{25.08 g} \times \frac{\text{1 mol}}{\text{20.18 g}} = \text{1.243 mol} \\ & n_{\text{Ar}} = \text{25.90 g} \times \frac{\text{1 mol}}{\text{39.95 g}} = \text{0.6483 mol}\end{aligned}[/tex]
Step 2: Calculate the mole fraction of each gas.
• For He
[tex]\begin{aligned} X_{\text{He}} & = \frac{n_{\text{He}}}{n_{\text{He}} + n_{\text{Ne}} + n_{\text{Ar}}} \\ & = \frac{\text{3.872 mol}}{\text{3.872 mol + 1.243 mol + 0.6483 mol}} \\ & = 0.6718 \end{aligned}[/tex]
• For Ne
[tex]\begin{aligned} X_{\text{Ne}} & = \frac{n_{\text{Ne}}}{n_{\text{He}} + n_{\text{Ne}} + n_{\text{Ar}}} \\ & = \frac{\text{1.243 mol}}{\text{3.872 mol + 1.243 mol + 0.6483 mol}} \\ & = 0.2157 \end{aligned}[/tex]
• For Ar
[tex]\begin{aligned} X_{\text{Ar}} & = \frac{n_{\text{Ar}}}{n_{\text{He}} + n_{\text{Ne}} + n_{\text{Ar}}} \\ & = \frac{\text{0.6483 mol}}{\text{3.872 mol + 1.243 mol + 0.6483 mol}} \\ & = 0.1125 \end{aligned}[/tex]
Step 3: Calculate the partial pressures of each gas.
The total pressure at STP is 1.000 atm.
• For He
[tex]\begin{aligned} P_{\text{He}} & = X_{\text{He}}P_{\text{T}} \\ & = \text{(0.6718)(1.000 atm)} \\ & = \boxed{\text{0.6718 atm}} \end{aligned}[/tex]
• For Ne
[tex]\begin{aligned} P_{\text{Ne}} & = X_{\text{Ne}}P_{\text{T}} \\ & = \text{(0.2157)(1.000 atm)} \\ & = \boxed{\text{0.2157 atm}} \end{aligned}[/tex]
• For Ar
[tex]\begin{aligned} P_{\text{Ar}} & = X_{\text{Ar}}P_{\text{T}} \\ & = \text{(0.1125)(1.000 atm)} \\ & = \boxed{\text{0.1125 atm}} \end{aligned}[/tex]
Hence, the partial pressures of He, Ne, and Ar are 0.6718 atm, 0.2157 atm, and 0.1125 atm, respectively.
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