Calculate the molarity of a solution of ethanol in water in which the mole fraction of ethanol is 0.040 (assume the density of water to be one).

The correct option is 2.314.

To calculate the molarity of the ethanol solution, we can use the definition of mole fraction and molarity. Let's break down the derivation step-by-step.

Step 1: Understand the Given Data
Mole fraction of ethanol in water ($X_{\text{ethanol}}$) is given as:
$X_{\text{ethanol}}=0.040$
The density of water is assumed to be $1{\text{ g mL}}^{-1}$.

Step 2: Relate Mole Fraction to Moles of Components
Let us assume that the total number of moles in the solution is 1 mole.
The mole fraction of a component is the ratio of the number of moles of that component to the total number of moles in the solution:
$X_{\text{ethanol}}=\frac{n_{\text{ethanol}}}{n_{\text{ethanol}}+n_{\text{water}}}$
Since the total number of moles ($n_{\text{ethanol}}+n_{\text{water}}$) is 1 mole:
$n_{\text{ethanol}}=0.040\text{ mol}$
Therefore, the number of moles of water is:
$n_{\text{water}}=1-0.040=0.960\text{ mol}$

Step 3: Calculate the Mass and Volume of Water
The molar mass of water (${\text{H}}_2\text{O}$) is approximately $18{\text{ g mol}}^{-1}$.
The mass of water ($m_{\text{water}}$) in the solution is:
$m_{\text{water}}=n_{\text{water}}\times \text{Molar Mass of Water}$
$m_{\text{water}}=0.960\text{ mol}\times 18{\text{ g mol}}^{-1}=17.28\text{ g}$
Using the density of water ($1{\text{ g mL}}^{-1}$), the volume of water is:
$\text{Volume of water}=\frac{\text{Mass}}{\text{Density}}=\frac{17.28\text{ g}}{1{\text{ g mL}}^{-1}}=17.28\text{ mL}$
Converting the volume of the solvent (which approximates the total volume of this dilute solution) to liters:
$\text{Volume of solution}\approx 17.28\times {10}^{-3}\text{ L}=0.01728\text{ L}$

Step 4: Calculate Molarity
Molarity ($M$) is defined as the number of moles of solute per liter of solution:
$M=\frac{n_{\text{ethanol}}}{\text{Volume of solution in Liters}}$
$M=\frac{0.040}{0.01728}\approx 2.3148\text{ M}$
Rounding to three decimal places matches the option 2.314.