Chapter 7, Problem 7.71E

Interpretation Introduction

Interpretation:

The reason as to why people who live at high altitudes are advised to add salt to water when boiling food like pasta is to be explained. The mole fraction of $\text{NaCl}$ needed to raise the boiling point of ${\text{H}}_{\text{2}}\text{O}$ by $3°\text{C}$ is to be calculated. Whether the given amount of salt added to water substantially changes the boiling point or not is to be identified.

Concept introduction:

When a non-volatile solute is added to a solvent, the boiling point of the solution gets raised in comparison to that of the pure solvent. The formula for elevation in boiling point is given as,

$\Delta T_{\text{b}}=m{K}_{\text{b}}$

Where,

• $m$ represents the molality of the solute.

• $K_{\text{b}}$ represents the boiling point elevation constant.

Answer to Problem 7.71E

People who live at high altitudes are advised to add salt to water when boiling food like pasta because the pressure at high altitudes is low and the water starts boiling below its normal boiling point. Therefore, salt is added to increase the boiling point of the water. The mole fraction of $\text{NaCl}$ needed to raise the boiling point of ${\text{H}}_{\text{2}}\text{O}$ by $3°\text{C}$ is $0.0957$.The given amount of salt added to water does not substantially change the boiling point.

Explanation of Solution

The pressure at high altitudes is low and the water starts boiling below its normal boiling point. Therefore, the water starts boiling before the food cooks and it is advised to add salt to water when boiling food like pasta.

The boiling point elevation constant of water is $0.51°\text{C/molal}$.

The required elevation in the boiling temperature of the water is $3°\text{C}$.

The formula for elevation in boiling point is given as,

$\Delta T_{\text{b}}=m{K}_{\text{b}}\cdot \cdot \cdot (1)$

Where,

• $m$ represents the molality of the solute.

• $K_{\text{b}}$ represents the boiling point elevation constant.

Rearrange the above equation for the value of $m$.

Substitute the value of $\Delta T_{\text{b}}$ and $K_{\text{b}}$ in the above equation.

$\begin{array}{rll}m& =& \frac{3°\text{C}}{0.51°\text{C}/\text{molal}}\\ & =& \left(5.8834\text{ molal}\right)\left(\frac{1\text{mol}/\text{kg}}{1\text{ molal}}\right)\\ & =& 5.8824\text{mol}/\text{kg}\end{array}$

The molality of the salt solution is $5.8824\text{mol/kg}$. Therefore, $5.8824\text{mol}$ of the $\text{NaCl}$ is dissolved in $1\text{ kg}$ of water.

The molar mass of the water is $18\text{g/mol}$.

The number of mole of a substance is given as:

$n=\frac{M}{M_{\text{m}}}$

Where,

• $M$ represents the mass of the substance.

• $M_{\text{m}}$ represents the molar mass of the substance.

Substitute the value of the mass and molar mass of water in the above equation.

$\begin{array}{rll}n& =& \frac{\left(1\text{ kg}\right)\left(\frac{1000\text{ g}}{1\text{ kg}}\right)}{18\text{g}/\text{mol}}\\ & =& 55.55\text{mol}\end{array}$

The number of moles of water is $55.55\text{mol}$.

The mole fraction of a substance present in two-component system is given as,

$x_{\text{a}}=\frac{n_{\text{a}}}{n_{\text{a}}+n_{\text{b}}}$

Where,

• $n_{\text{a}}$ represents the number of moles of substance $\text{a}$.

• $n_{\text{b}}$ represents the number of moles of substance $\text{b}$.

Substitute the value of a number of moles of water and the number of moles of sodium chloride in the above equation for the value of mole fraction of sodium chloride.

$\begin{array}{rll}{x}_{\text{a}}& =& \frac{5.8824\text{mol}}{5.8824\text{mol}+55.5555\text{mol}}\\ & =& \frac{5.8824\text{mol}}{61.4379\text{mol}}\\ & =& 0.0957\end{array}$

The mole fraction of $\text{NaCl}$ is needed to raise the boiling point of ${\text{H}}_{\text{2}}\text{O}$ by $3°\text{C}$ is $0.0957$.

One teaspoon of salt is dissolved into four quarts of water.

The molar mass of sodium chloride is $58.44\text{g/mol}$.

The standard amount present in one teaspoon is $5\text{mL}$.

One quart has $\text{946}\text{.353}\text{mL}$.

The density of water is $1\text{g/mL}$.

The density of sodium chloride is $2.16\text{g/mL}$.

The relation between mass, volume and density is given as,

$M=DV$ (2)

Where,

• $M$ represents the mass of the substance.

• $D$ represents the density of the substance.

• $V$ represents the volume of the substance.

Substitute the value of density and volume of water in the equation (2).

$\begin{array}{rll}M& =& \left(1\text{g/mL}\right)\left(4\text{quarts}\right)\left(\frac{\text{946}\text{.353}\text{mL}}{1\text{quart}}\right)\\ & =& 3785.412\text{g}\end{array}$

Substitute the value of density and volume of sodium chloride in the equation (2).

$\begin{array}{rll}M& =& \left(2.16\text{g/mL}\right)\left(1\text{teaspoon}\right)\left(\frac{5\text{mL}}{1\text{teaspoon}}\right)\\ & =& 10.8\text{g}\end{array}$

The molality of a salt solution is given as,

$m=\frac{M_{\text{s}}}{M_{\text{m}}{M}_{\text{W}}}$

Where,

• $M_{\text{s}}$ represents the mass of the solute.

• $M_{\text{m}}$ represents the molar mass of the solute.

• $M_{\text{w}}$ represents the mass of the solvent.

Substitute the molar mass of sodium chloride, the mass of the sodium chloride and mass of the water in the above equation.

$\begin{array}{rll}m& =& \frac{10.8\text{g}}{\left(58.44\text{g/mol}\right)\left(3785.412\text{g}\right)\left(\frac{1\text{ kg}}{1000\text{ g}}\right)}\\ & =& \left(0.0488\text{ mol/kg}\right)\left(\frac{1\text{molal}}{1\text{ mol/kg }}\right)\\ & =& 0.0488\text{ molal}\end{array}$

Therefore, the molality of the solution that is formed by dissolving one teaspoon of salt into four quarts of water is $0.0488\text{ molal}$.

Substitute the value of molality of salt solution and $K_{\text{b}}$ in the equation (1).

$\begin{array}{rll}\Delta T_{\text{b}}& =& \left(0.0488\text{ molal}\right)\left(0.51°\text{C/molal}\right)\\ & =& 0.0249°\text{C}\end{array}$

The elevation in the boiling point of water is very low. Therefore, the given amount of salt added to water does not substantially change the boiling point.

Conclusion

People who live at high altitudes are advised to add salt to water when boiling food like pasta to increase the boiling point of water The mole fraction of $\text{NaCl}$ is needed to raise the boiling point of ${\text{H}}_{\text{2}}\text{O}$ by $3°\text{C}$ is $0.0957$.The given amount of salt added to water does not substantially change the boiling point.