Question

In many states, a person is legally intoxicated if his or her blood has a concentration of$\mathbf{0}\mathbf{.10}\mathbf{}\mathbf{\text{g}}$ (or more) of ethyl alcohol$\mathbf{\left(}{\mathbf{\text{C}}}_{\mathbf{2}}{\mathbf{\text{H}}}_{\mathbf{5}}\mathbf{\text{OH}}\mathbf{\right)}$ per deciliter. Express this "threshold concentration"in.data-custom-editor="chemistry" $\mathbf{\text{mol}}\mathbf{/}\mathbf{\text{L}}$

Short answer

The threshold concentration in$\text{mol}/\text{L}$ is.$0.022\text{mol}/\text{L}$

Step-by-step solution

 Step 1: Define mass concentration and molar concentration

Molar concentration is described as the total number of moles of solute dissolved in per liter of solution, i.e.,$\mathbf{M}\mathbf{=}\mathit{m}\mathit{o}\mathit{l}\mathbf{/}\mathit{L}$ .

The molar concentration formula is given by:$\mathbf{M}\mathbf{=}\frac{\mathbf{A}\mathbf{m}\mathbf{o}\mathbf{u}\mathbf{n}\mathbf{t}\mathbf{i}\mathbf{n}\mathbf{m}\mathbf{o}\mathbf{l}\mathbf{e}\mathbf{s}}{\mathbf{V}\mathbf{o}\mathbf{l}\mathbf{u}\mathbf{m}\mathbf{e}\mathbf{o}\mathbf{f}\mathbf{S}\mathbf{o}\mathbf{l}\mathbf{u}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}}$ .

The mass concentration of a solution is the amount of solute present in a given amount of solution. It is also the amount of solute that is dissolved in given mass of the solvent.

Derive the relation between mass concentration and molar concentration

Mass v/s molar concentration is given as:

Mass concentration :$\gamma =\frac{m}{V}$ .

Molar concentration:$c=\frac{n}{V}$ .

The number of moles of a solute is calculated by the formula,$\mathrm{n}=\frac{m}{M}$ where $M$is the molar mass.

Substitute in the molar concentration formula:$\mathrm{c}=\frac{m}{M\times V}$ .

Thus, the final relation between the two will be:$c=\frac{\gamma }{M}$ .

Calculate the threshold concentration in mol/L

The value of mass concentration$\gamma$ is.$1\text{g}/\text{L}$

The molar mass of ethyl alcohol is:$\mathrm{M}\left(C_2{H}_{5}OH\right)=46.08g/mol$

Substitute these values in the relation derived in the previous step to find$c$:

$\begin{array}{l}c=\frac{1}{46.08}\\ c=0.022\text{mol}/\text{L}\end{array}$

Thus, the threshold concentration in$\text{mol}/\text{L}$ is.$0.022\text{mol}/\text{L}$