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Chapter 13: Problem 100

A solution contains \(0.115 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\) and an unknown number of moles of sodium chloride. The vapor pressure of the solution at \(30^{\circ} \mathrm{C}\) is 25.7 torr. The vapor pressure of pure water at this temperature is 31.8 torr. Calculate the number of moles of sodium chloride in the solution. (Hint: Remember that sodium chloride is a strong electrolyte.)

Short Answer

Expert verified
The number of moles of sodium chloride (NaCl) in the solution is approximately 0.0273 moles.

Step by step solution

01

Write down the given values.

The given values are: - Moles of water (H2O) = 0.115 mol - Vapor pressure of the solution (Psol) = 25.7 torr - Vapor pressure of pure water (Pwater) = 31.8 torr
02

Write down Raoult's Law equation for the solution.

Raoult's Law states that the partial vapor pressure of each component in an ideal solution is equal to the product of the mole fraction of the component and its vapor pressure in the pure state. Mathematically, it can be written as: \(P_{A} = X_{A} * P_{A}^{\circ}\) For water in the solution, we can write: \(P_{H2O} = X_{H2O} * P_{H2O}^{\circ}\) Where, \(P_{H2O}\) is the vapor pressure of water in the solution, \(X_{H2O}\) is the mole fraction of water in the solution, and \(P_{H2O}^{\circ} = 31.8\, torr\) is the vapor pressure of pure water.
03

Calculate the mole fraction of water in the solution.

As sodium chloride (NaCl) is a strong electrolyte, it dissociates completely into ions (Na+ and Cl-) in the solution. Let's represent the unknown number of moles of NaCl as 'x'. Then, the total moles present in the solution will be (0.115 + x). The mole fraction of water in the solution can then be calculated as: \(X_{H2O} = \frac{moles \, of \, H2O}{total \, moles \, in \, solution} = \frac{0.115}{0.115+x}\)
04

Substitute the given vapor pressure of the solution in Raoult's Law equation.

We have the vapor pressure of the solution (Psol) = 25.7 torr. According to Raoult's Law for water, we can write: \(P_{H2O} = X_{H2O} * P_{H2O}^{\circ}\) Substituting the values, we have: \(25.7 = \frac{0.115}{0.115+x} * 31.8\)
05

Solve for x (number of moles of NaCl).

We can now solve the above equation for the unknown number of moles of NaCl (x): \(25.7 = \frac{0.115}{0.115+x} * 31.8\) Cross-multiplying, we get: \(25.7(0.115+x) = 0.115 * 31.8\) Now, simplify and solve for x: \(25.7(0.115+x) = 3.657\) Expanding and simplifying, we get: \(2.9555 + 25.7x = 3.657\) Subtracting 2.9555 from both sides, we have: \(25.7x = 0.7015\) Now, dividing by 25.7: \(x = \frac{0.7015}{25.7} = 0.0273 \, moles\) Therefore, the number of moles of sodium chloride (NaCl) in the solution is approximately 0.0273 moles.

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Most popular questions from this chapter

Explain why pressure substantially affects the solubility of \(\mathrm{O}_{2}\) in water but has little effect on the solubility of \(\mathrm{NaCl}\) in water.

Caffeine \(\left(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{~N}_{4} \mathrm{O}_{2}\right)\) is a stimulant found in coffee and tea. If a solution of caffeine in chloroform \(\left(\mathrm{CHCl}_{3}\right)\) as a solvent has a concentration of \(0.0500 \mathrm{~m}\), calculate (a) the percent caffeine by mass, (b) the mole fraction of caffeine.

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)?

(a) Why is there no colloid in which both the dispersed substance and the dispersing substance are gases? (b) Michael Faraday first prepared ruby-red colloids of gold particles in water that were stable indefinitely. To the unaided eye these brightly colored colloids are not distinguishable from solutions. How could you determine whether a given colored preparation is a solution or colloid?

Which of the following in each pair is likely to be more soluble in water: (a) cyclohexane \(\left(\mathrm{C}_{6} \mathrm{H}_{12}\right)\) or glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) (Figure 13.12 ); (b) propionic acid \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{COOH}\right)\) or sodium propionate \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{COONa}\right) ;\) (c) HCl or ethyl chloride \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{Cl}\right) ?\) Explain in each case.
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