Question

An automobile antifreeze mixture is made by mixing equal volumes of ethylene glycol \((d=1.114 \mathrm{~g} / \mathrm{mL} ; \mathscr{M}=62.07 \mathrm{~g} / \mathrm{mol})\) and water \((d=1.00 \mathrm{~g} / \mathrm{mL})\) at \(20^{\circ} \mathrm{C}\). The density of the mixture is \(1.070 \mathrm{~g} / \mathrm{mL}\). Express the concentration of ethylene glycol as (a) Volume percent (b) Mass percent (c) Molarity (d) Molality (e) Mole fraction

Short answer

a) 50.6% b) 52.7% c) 9.09 M d) 17.96 m e) 0.244

Step-by-step solution

- Determine volumes

Since equal volumes of ethylene glycol and water are mixed, assume 100 mL of each. Therefore, volume of ethylene glycol, \(V_{EG}\), and volume of water, \(V_{H2O}\) are both 100 mL.

- Calculate the mass of each component

Using the density formula \(d = \frac{m}{V}\), calculate the mass of ethylene glycol and water. Mass of ethylene glycol: \(m_{EG} = d_{EG} \times V_{EG} = 1.114 \, \mathrm{g/mL} \times 100 \, \mathrm{mL} = 111.4 \, \mathrm{g}\). Mass of water: \(m_{H2O} = d_{H2O} \times V_{H2O} = 1.00 \, \mathrm{g/mL} \times 100 \, \mathrm{mL} = 100 \, \mathrm{g}\).

- Calculate the mass of the mixture

The total mass of the mixture is the sum of the masses of ethylene glycol and water. Mass of mixture \(m_{mixture} = m_{EG} + m_{H2O} = 111.4 \, \mathrm{g} + 100 \, \mathrm{g} = 211.4 \, \mathrm{g}\).

- Calculate the volume of the mixture

Using the density of the mixture \(d_{mixture} = 1.070 \, \mathrm{g/mL}\), and the mass of the mixture, calculate the volume. Volume of mixture, \(V_{mixture} = \frac{m_{mixture}}{d_{mixture}} = \frac{211.4 \, \mathrm{g}}{1.070 \, \mathrm{g/mL}} = 197.57 \, \mathrm{mL}\).

- Calculate volume percent of ethylene glycol

Volume percent \(\% V/V\) is given by \(\frac{V_{solute}}{V_{solution}} \times 100\). \(\% V/V = \frac{100 \mathrm{~mL}}{197.57 \mathrm{~mL}} \times 100 \approx 50.6\%\).

- Calculate the mass percent of ethylene glycol

Mass percent \(\% m/m\) is given by \(\frac{m_{solute}}{m_{solution}} \times 100\). \(\% m/m = \frac{111.4 \mathrm{~g}}{211.4 \mathrm{~g}} \times 100 \approx 52.7\%\).

- Calculate the molarity of ethylene glycol

Molarity \(M\) is given by \(\frac{moles \; solute}{volume \; solution \; (L)}\). Moles of ethylene glycol \(n_{EG} = \frac{m_{EG}}{M_{EG}} = \frac{111.4 \mathrm{~g}}{62.07 \mathrm{~g/mol}} \approx 1.796 \mathrm{~mol}\). Volume of the mixture in liters: \(V_{mixture} = 197.57 \mathrm{~mL} \times \frac{1 \mathrm{~L}}{1000 \mathrm{~mL}} \approx 0.19757 \mathrm{~L}\). Molarity \(M = \frac{1.796 \mathrm{~mol}}{0.19757 \mathrm{~L}} \approx 9.09 \mathrm{~M}\).

- Calculate the molality of ethylene glycol

Molality \(m\) is given by \(\frac{moles \; solute}{mass \; solvent \; (kg)}\). Note: the solvent here is water. Mass of water in kg: \(m_{H2O(kg)} = \frac{100 \mathrm{~g}}{1000 \mathrm{~g/kg}} = 0.100 \mathrm{~kg}\). Molality \(m = \frac{1.796 \mathrm{~mol}}{0.100 \mathrm{~kg}} \approx 17.96 \mathrm{~m}\).

- Calculate the mole fraction of ethylene glycol

Mole fraction \(X_{EG}\) is given by \(\frac{moles \; solute}{total \; moles}\). Moles of water \(n_{H2O} = \frac{m_{H2O}}{M_{H2O}} = \frac{100 \mathrm{~g}}{18.015 \mathrm{~g/mol}} \approx 5.55 \mathrm{~mol}\). Total moles \(n_{total} = n_{EG} + n_{H2O} = 1.796 \mathrm{~mol} + 5.55 \mathrm{~mol} = 7.346 \mathrm{~mol}\). Mole fraction \(X_{EG} = \frac{1.796 \mathrm{~mol}}{7.346 \mathrm{~mol}} \approx 0.244\).

Key concepts

Volume Percent

Volume percent, often noted as \(\% V/V\), measures the volume of a solute in relation to the total volume of the solution. To calculate it, divide the volume of ethylene glycol by the total volume of the mixture and multiply by 100. For example, if you have 100 mL of ethylene glycol in a 197.57 mL solution, the volume percent is \( \left( \frac{100 \, \mathrm{mL}}{197.57 \, \mathrm{mL}} \right) \times 100 \approx 50.6\% \). This tells you that 50.6% of the solution's volume is ethylene glycol. This concept is very handy when dealing with liquid-liquid mixtures and is quite common in chemistry and various industrial processes.

Molality

Molality, denoted by \(m\)