Chapter 11, Problem 1P

A patient has a “cholesterol count” of 214. Like manyblood-chemistry measurements,this result is measured inunits of milligrams per deciliter $\left(\text{mg}{\text{dL}}^{-1}\right).$
Determine the molar concentration of cholesterol inthis patient’s blood, taking the molar mass of cholesterolto be $386.64\text{g}{\text{mol}}^{-1}.$

1. Estimate the molality of cholesterol in the patient’sblood.
2. If 214 is a typical cholesterol reading among men inthe United States, determine the volume of such bloodrequired to furnish 8.10 g of cholesterol.

Interpretation Introduction

Interpretation: The molar concentration of cholesterol in blood of patient needs to be calculated by taking molar mass of cholesterol equal to 386.64 g/mol.

Concept Introduction: The molar concentration or molarity of a substance is number of moles of solute in 1 L of the solution.

Explanation of Solution

The cholesterol count of patient is 214 mg dL-1.

Thus, molar concentration of cholesterol (mol/L) can be calculated as follows:

  $C=XM$

Here, C is molar concentration, X is cholesterol count and M is molar mass.

Substitute the values,

  $\begin{array}{c}C=\frac{X}{M}\\ =\frac{\left(214\cancel{\text{mg}}\text{/}\cancel{\text{dL}}\right)\left(\frac{1\cancel{\text{dL}}}{{10}^{-1}\text{L}}\right)\left(\frac{{10}^{-3}\cancel{\text{g}}}{1\cancel{\text{mg}}}\right)}{\left(386.64\cancel{\text{g}}\text{/mol}\right)}\\ =0.005535\text{ mol/L}\\ \text{=5}\text{.535}\times {\text{10}}^{-3}\text{ mol/L}\end{array}$

Therefore, molar concentration of cholesterol is $\text{5}\text{.535}\times {\text{10}}^{-3}\text{ mol/L}$ .

Interpretation Introduction

Interpretation: The molality of cholesterol in the patient’s blood needs to be estimated.

Concept Introduction: The molality of a substance is number of moles of a substance in 1 kg of solvent.

Explanation of Solution

It is related to molarity of solution or molar concentration of solution as follows:

  $m=\frac{1000\times M}{M\times M_b-1000\times d}$

Here, M is molarity, $M_b$ is molar mass of solute and d is density of solution.

In the given case, solute is cholesterol and solvent is blood. The relation between molarity (molar concentration) and molality of solution is as follows:

  $m=\frac{1000\times M}{M\times M_b-1000\times d}$

Here, molarity of solution is $\text{5}\text{.535}\times {\text{10}}^{-3}\text{ mol/L}$ . The density of solution is 0.994 g/mL (density of blood) and molar mass of cholesterol is 386.64 g/mol.

Substitute the values,

  $\begin{array}{l}m=\frac{\left(1000\right)\left(\text{5}\text{.535}\times {\text{10}}^{-3}\text{ mol/L}\right)}{\left(\text{5}\text{.535}\times {\text{10}}^{-3}\text{ mol/L}\right)\left(386.64\text{ g/mol}-1000\times 0.994\text{ g/mL}\right)}\\ =\frac{\text{5}\text{.535 }}{0.005535\times 607.36}\\ =1.65\text{ m}\end{array}$

Thus, the molality of solution is $1.65\text{ m}$ .

Interpretation Introduction

Interpretation: The volume of blood required to furnish 8.10 g of cholesterol needs to be calculated.

Concept Introduction: To calculate the volume of blood requited for given mass of cholesterol, unitary method can be used. Here, mass of cholesterol in 1 dL of blood is given thus, volume of blood required for given mass of cholesterol can be calculated.

Explanation of Solution

The cholesterol count of patient is 214 mg dL-1. Thus, in 1 dL of blood there is 214 mg of cholesterol.

The volume of blood required for 8.10 g of cholesterol will be:

  $\begin{array}{c}V=\frac{\left(8.10\cancel{\text{g}}\right)}{\left(214\frac{\cancel{\text{mg}}}{\text{dL}}\right)\left(\frac{{10}^{-3}\cancel{\text{g}}}{1\cancel{\text{mg}}}\right)}\\ =37.850\text{ dL}\left(\frac{{10}^{-1}\text{ L}}{1\text{ dL}}\right)\\ =3.785\text{ L}\end{array}$

Therefore, volume of blood required is $3.785\text{ L}$ .