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Step-by-Step Solution
Step 1: Understand the Concept
We use Raoult’s Law here, which states that, for an ideal solution, the partial vapor pressure of each component is equal to the vapor pressure of the pure component multiplied by its mole fraction in the solution.
Step 2: List the Given Data
Vapor pressure of pure benzene, $P^\circ_\text{benzene} = 75\,\text{torr}$
Vapor pressure of pure toluene, $P^\circ_\text{toluene} = 22\,\text{torr}$
Mass of benzene, $m_{\text{benzene}} = 78 \,\text{g}$
Mass of toluene, $m_{\text{toluene}} = 46 \,\text{g}$
Molar mass of benzene = $78 \,\text{g mol}^{-1}$
Molar mass of toluene = $92 \,\text{g mol}^{-1}$
Step 3: Calculate the Number of Moles
Number of moles of benzene, $n_{\text{benzene}}$:
$$
n_{\text{benzene}} = \frac{m_{\text{benzene}}}{\text{Molar mass of benzene}}
= \frac{78}{78} = 1 \,\text{mol}
$$
Number of moles of toluene, $n_{\text{toluene}}$:
$$
n_{\text{toluene}} = \frac{m_{\text{toluene}}}{\text{Molar mass of toluene}}
= \frac{46}{92} = 0.5 \,\text{mol}
$$
Step 4: Compute the Mole Fractions
Total number of moles, $n_\text{total} = n_{\text{benzene}} + n_{\text{toluene}}$:
$$
n_\text{total} = 1 + 0.5 = 1.5 \,\text{mol}
$$
Mole fraction of benzene, $X_{\text{benzene}}$:
$$
X_{\text{benzene}} = \frac{n_{\text{benzene}}}{n_\text{total}}
= \frac{1}{1.5} = \frac{2}{3}
$$
Step 5: Calculate Partial Vapor Pressure of Benzene
According to Raoult’s Law:
$$
P_{\text{benzene}} = P^\circ_{\text{benzene}} \times X_{\text{benzene}}
$$
Substituting the values:
$$
P_{\text{benzene}} = 75 \times \frac{2}{3} = 50 \,\text{torr}
$$
Step 6: Final Answer
Hence, the partial vapor pressure of benzene in the solution at 20 °C is
50 torr.
