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Chapter 13: Problem 64

At \(20^{\circ} \mathrm{C}\) the vapor pressure of benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) is 75 torr, and that of toluene \(\left(\mathrm{C}_{7} \mathrm{H}_{8}\right)\) is 22 torr. Assume that benzene and toluene form an ideal solution. (a) What is the composition in mole fractions of a solution that has a vapor pressure of 35 torr at \(20^{\circ} \mathrm{C} ?\) (b) What is the mole fraction of benzene in the vapor above the solution described in part (a)?

Short Answer

Expert verified
The composition of an ideal solution with a vapor pressure of 35 torr at 20掳C has mole fractions of benzene (C鈧咹鈧�) and toluene (C鈧嘓鈧�) in the solution as 0.245 and 0.755, respectively. The mole fraction of benzene in the vapor above the solution is 0.721.

Step by step solution

01

(a) Calculate the mole fractions of benzene and toluene in the solution.

According to Raoult's law, the partial vapor pressure of a component in an ideal solution is equal to the product of its mole fraction in the solution and its vapor pressure. We are given the vapor pressures of benzene and toluene at 20掳C and the total vapor pressure of the solution, and asked to find the mole fractions of benzene and toluene in the solution. Let x鈧� and x鈧� represent the mole fractions of benzene and toluene in the solution, respectively. The total pressure above the solution is given by: \(P = x_1P_1^0 + x_2P_2^0\) where \(P_1^0\) and \(P_2^0\) are the vapor pressures of benzene and toluene, respectively. We also know that: \(x_1 + x_2 = 1\) Now, plugging in the given values: \(35\,\text{torr} = x_1(75\,\text{torr}) + x_2(22\,\text{torr})\) We can solve for x鈧� and x鈧� using these two equations.
02

(a) Solving for x鈧� and x鈧� using the equations.

We can rearrange the equation for mole fractions: \(x_2 = 1 - x_1\) Now substitute this expression of x鈧� into the equation for total pressure: \(35\,\text{torr} = x_1(75\,\text{torr}) + (1 - x_1)(22\,\text{torr})\) Expanding and simplifying this equation: \(35\,\text{torr} = 75x_1\,\text{torr} + 22\,\text{torr} - 22x_1\,\text{torr}\) \(13\,\text{torr} = 53x_1\,\text{torr}\) So, \(x_1= \frac{13\,\text{torr}}{53\,\text{torr}} = 0.245\) Now, we can find x鈧�: \(x_2 = 1-0.245=0.755\) Therefore, the mole fractions of benzene and toluene in the solution are 0.245 and 0.755, respectively.
03

(b) Calculate the mole fraction of benzene in the vapor.

To calculate the mole fraction of benzene in the vapor, we can use the following relation between the mole fractions in the solution and the vapor: \(\frac{y_1}{y_2} = \frac{x_1P_1^0}{x_2P_2^0}\) Plugging in the given values and the values of x鈧� and x鈧� that we just calculated: \(\frac{y_1}{y_2} = \frac{(0.245)(75\,\text{torr})}{(0.755)(22\,\text{torr})}\) \( \frac{y_1}{y_2} = 2.587 \) Now, we can find the mole fraction of benzene and toluene in the vapor using: \(y_1 + y_2 = 1\) \(y_1 = 1 - y_2\) Substitute this expression of y鈧� into the equation for the ratio of mole fractions: \(\frac{1 - y_2}{y_2} = 2.587\) Solve for y鈧�: \(y_2 = \frac{1}{3.587} = 0.279\) Now, find y鈧�: \(y_1 = 1 - 0.279 = 0.721\) Therefore, the mole fraction of benzene in the vapor above the solution is 0.721.

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Most popular questions from this chapter

At \(35^{\circ} \mathrm{C}\) the vapor pressure of acetone, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{CO}\), is 360 torr, and that of chloroform, \(\mathrm{CHCl}_{3}\), is 300 torr. Acetone and chloroform can form weak hydrogen bonds between one another as follows: A solution composed of an equal number of moles of acetone and chloroform has a vapor pressure of 250 torr at \(35^{\circ} \mathrm{C}\). (a) What would be the vapor pressure of the solution if it exhibited ideal behavior? (b) Use the existence of hydrogen bonds between acetone and chloroform molecules to explain the deviation from ideal behavior. (c) Based on the behavior of the solution, predict whether the mixing of acetone and chloroform is an exothermic \(\left(\Delta H_{\text {soln }}<0\right)\) or endothermic \(\left(\Delta H_{\text {soln }}>0\right.\) ) process.

Indicate whether each of the following is a hydrophilic or a hydrophobic colloid: (a) butterfat in homogenized milk, (b) hemoglobin in blood, (c) vegetable oil in a salad dressing, (d) colloidal gold particles in water.

A \(40.0 \%\) by weight solution of \(\mathrm{KSCN}\) in water at \(20^{\circ} \mathrm{C}\) has a density of \(1.22 \mathrm{~g} / \mathrm{mL}\). (a) What is the mole fraction of \(\mathrm{KSCN}\) in the solution, and what are the molarity and molality? (b) Given the calculated mole fraction of salt in the solution, comment on the total number of water molecules available to hydrate each anion and cation. What ion pairing (if any) would you expect to find in the solution? Would you expect the colligative properties of such a solution to be those predicted by the formulas given in this chapter? Explain.

The density of toluene \(\left(\mathrm{C}_{7} \mathrm{H}_{8}\right)\) is \(0.867 \mathrm{~g} / \mathrm{mL}\), and the density of thiophene \(\left(\mathrm{C}_{4} \mathrm{H}_{4} \mathrm{~S}\right)\) is \(1.065 \mathrm{~g} / \mathrm{mL}\). A solution is made by dissolving \(9.08 \mathrm{~g}\) of thiophene in \(250.0 \mathrm{~mL}\) of toluene. (a) Calculate the mole fraction of thiophene in the solution. (b) Calculate the molality of thiophene in the solution. (c) Assuming that the volumes of the solute and solvent are additive, what is the molarity of thiophene in the solution?

A sulfuric acid solution containing \(571.6 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) per Jiter of solution has a density of \(1.329 \mathrm{~g} / \mathrm{cm}^{3}\). Calculate (a) the mass percentage, (b) the mole fraction, (c) the molality, (d) the molarity of \(\mathrm{H}_{2} \mathrm{SO}_{4}\) in this solution.
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