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Chapter 13: Problem 49

The density of acetonitrile \(\left(\mathrm{CH}_{3} \mathrm{CN}\right)\) is \(0.786 \mathrm{~g} / \mathrm{mL}\) and the density of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\) is $0.791 \mathrm{~g} / \mathrm{mL}\(. A solution is made by dissolving \)25.0 \mathrm{~mL}$ of \(\mathrm{CH}_{3} \mathrm{OH}\) in \(100 \mathrm{~mL}\) of $\mathrm{CH}_{3} \mathrm{CN}$ (a) What is the mole fraction of methanol in the solution? (b) What is the molality of the solution? (c) Assuming that the volumes are additive, what is the molarity of \(\mathrm{CH}_{3} \mathrm{OH}\) in the solution?

Short Answer

Expert verified
In conclusion, the mole fraction of methanol in the solution is 0.244, the molality of the solution is 7.84 mol/kg, and the molarity of methanol in the solution is 4.94 mol/L, assuming that the volumes are additive.

Step by step solution

01

Find the mass of acetonitrile and methanol

First, we need to find the mass of each substance in the solution. To do this, we will use the densities and volumes of each substance. Mass of acetonitrile (CH₃CN): Density = 0.786 g/mL Volume = 100 mL Mass = Density × Volume \(Mass_{CH_3CN} = 0.786 \ g/mL \times 100 \ mL = 78.6 \ g\) Mass of methanol (CH₃OH): Density = 0.791 g/mL Volume = 25 mL Mass = Density × Volume \(Mass_{CH_3OH} = 0.791 \ g/mL \times 25 \ mL = 19.775 \ g\)
02

Convert mass to moles

To calculate the mole fraction and molality, we need to know the number of moles of each substance. To do this, we will use the molar mass of each substance to convert the mass to moles. Molar mass of acetonitrile (CH₃CN) = 12.01 g/mol (C) + 3 × 1.01 g/mol (H) + 14.01 g/mol (N) = 41.05 g/mol Molar mass of methanol (CH₃OH) = 12.01 g/mol (C) + 4 × 1.01 g/mol (H) + 16.00 g/mol (O) = 32.04 g/mol Number of moles of acetonitrile: \(n_{CH_3CN} = \frac{Mass_{CH_3CN}}{Molar \ Mass_{CH_3CN}}\) \(n_{CH_3CN} = \frac{78.6 \ g}{41.05 \ g/mol} = 1.914 \ mol\) Number of moles of methanol: \(n_{CH_3OH} = \frac{Mass_{CH_3OH}}{Molar \ Mass_{CH_3OH}}\) \(n_{CH_3OH} = \frac{19.775 \ g}{32.04 \ g/mol} = 0.617 \ mol\)
03

Calculate the mole fraction of methanol

To find the mole fraction of methanol in the solution, we can use the following equation: Mole fraction of methanol (Xₘ) = \(\frac{n_{CH_3OH}}{n_{CH_3OH} + n_{CH_3CN}}\) \(X_m = \frac{0.617 \ mol}{0.617 \ mol + 1.914 \ mol} = 0.244\) The mole fraction of methanol in the solution is 0.244.
04

Calculate the molality of the solution

To find the molality of the solution, we can use the following equation: Molality (m) = \(\frac{n_{CH_3OH}}{Mass_{CH_3CN} (kg)}\) \(m = \frac{0.617 \ mol}{0.0786 \ kg} = 7.84 \ mol/kg\) The molality of the solution is 7.84 mol/kg.
05

Calculate the molarity of methanol in the solution

Assuming that the volumes are additive, we know that the total volume of the solution is 100 mL (CH₃CN) + 25 mL (CH₃OH) = 125 mL or 0.125 L. To find the molarity of methanol in the solution, we can use the following equation: Molarity (M) = \(\frac{n_{CH_3OH}}{Volume_{solution} (L)}\) \(M = \frac{0.617 \ mol}{0.125 \ L} = 4.94 \ mol/L\) The molarity of methanol in the solution is 4.94 mol/L. In conclusion, the mole fraction of methanol in the solution is 0.244, the molality of the solution is 7.84 mol/kg, and the molarity of methanol in the solution is 4.94 mol/L, assuming that the volumes are additive.

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Most popular questions from this chapter

Compounds like sodium stearate, called "surfactants" in general, can form structures known as micelles in water, once the solution concentration reaches the value known as the critical micelle concentration (cmc). Micelles contain dozens to hundreds of molecules. The cmc depends on the substance, the solvent, and the temperature. At and above the \(\mathrm{cmc}\), the properties of the solution vary drastically. (a) The turbidity (the amount of light scattering) of solutions increases dramatically at the \(\mathrm{cmc}\). Suggest an explanation. (b) The ionic conductivity of the solution dramatically changes at the \(\mathrm{cmc}\). Suggest an explanation. (c) Chemists have developed fluorescent dyes that glow brightly only when the dye molecules are in a hydrophobic environment. Predict how the intensity of such fluorescence would relate to the concentration of sodium stearate as the sodium stearate concentration approaches and then increases past the \(\mathrm{cmc}\).

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