Problem 50 - A Complete Resource Book in Chemistry for JEE Main

Question

' \(x\) ' grams of water is mixed in 69 grams of ethanol Mole fraction of ethanol in the resultant solution is \(0.6\). what is the value of ' \(\mathrm{x}\) ' in grams? (a) 54 (b) 36 (c) 180 (d) 18

Step-by-Step Solution

Answer

The value of \( x \) is 18 grams.

Step 1: Understanding Mole Fraction

The mole fraction of a component is the ratio of the moles of that component to the total number of moles in the solution. In this problem, the mole fraction of ethanol is given as 0.6.

Step 2: Molar Masses Identification

Find the molar masses of the involved substances. The molar mass of ethanol (C\(_2\)H\(_5\)OH) is approximately 46 g/mol, and the molar mass of water (H\(_2\)O) is approximately 18 g/mol.

Step 3: Calculating Moles of Ethanol

Calculate the moles of ethanol using its mass and molar mass. \[ \text{Moles of ethanol} = \frac{69}{46} \approx 1.5 \]

Step 4: Setting Up the Mole Fraction Equation

Use the mole fraction formula to set up the equation. Let the moles of water be \( \frac{x}{18} \). According to the mole fraction of ethanol being 0.6, we have: \[0.6 = \frac{\frac{69}{46}}{\frac{69}{46} + \frac{x}{18}} \]

Step 5: Solving the Equation

Rearrange the equation to solve for \( x \): Multiply both sides by the denominator: \[0.6 \left( \frac{69}{46} + \frac{x}{18} \right) = \frac{69}{46} \]Simplify:\[0.6 \times 1.5 + 0.6 \times \frac{x}{18} = 1.5 \]Solving: \[ 0.9 + \frac{0.6x}{18} = 1.5 \]\[\frac{0.6x}{18} = 0.6 \]Multiply through by 18: \[0.6x = 10.8 \]Finally:\[x = 18 \]

Step 6: Verification Step

To ensure our calculation is correct, verify it by plugging \( x = 18 \) grams back into the mole fraction formula:For \( x = 18 \), moles of water = 18 / 18 = 1.Total moles = 1.5 + 1 = 2.5.Mole fraction of ethanol = 1.5 / 2.5 = 0.6, which confirms our solution.

Key Concepts

Ethanol

Ethanol is a simple alcohol with the chemical formula C\(_2\)H\(_5\)OH. It is widely known and used for various purposes, including as a solvent, in the production of alcoholic beverages, and as a fuel additive. Understanding ethanol is essential when dealing with chemistry problems involving solutions, like the problem at hand.

Some key characteristics of ethanol include:

• It is a polar compound, meaning it has a partial positive and negative charge, which allows it to form hydrogen bonds with water and other polar substances.
• Ethanol's molar mass is approximately 46 g/mol. This means that if you have 46 grams of ethanol, you have exactly one mole of ethanol molecules.
• Due to its chemical structure, ethanol is miscible with water, which means it can mix in all proportions, forming a homogeneous solution.

When calculating properties like mole fraction in a solution involving ethanol, it is crucial to start by determining its molar mass. This allows us to convert grams to moles, which are then used to find the mole fraction. The mole fraction is vital in understanding the composition of solutions, dictating the concentration of ethanol relative to other components.

Water

Water is a vital solvent in many chemical reactions due to its polar nature and ability to dissolve a wide range of substances. Its chemical formula is H\(_2\)O, and it plays an essential role in many chemistry problems, including those involving solutions.

Here are some fundamental properties of water:

• The molar mass of water is approximately 18 g/mol, meaning one mole of water weighs 18 grams.
• Water is a polar molecule, much like ethanol, which enables it to form hydrogen bonds. This property makes it an excellent solvent for many ionic and covalent substances, such as salts and alcohols like ethanol.
• The density of water is 1 g/mL, making it straightforward to convert between volume and mass measurements when working in laboratory settings.

In solution chemistry, water often acts as a solvent, but it can also be a solute, depending on the context, such as in the given problem. Understanding how to calculate the moles of water when given its mass is crucial in calculating mole fractions, which describe the concentration of components in a solution.

Molar Mass

Molar mass is a fundamental concept in chemistry that refers to the mass of one mole of a given substance. It is usually expressed in grams per mole (g/mol) and is derived from the atomic masses of the elements in a compound, as found on the periodic table.

The importance of molar mass in calculations includes:

• It allows us to convert between grams and moles, which is essential for applications like calculating the extent of reactions and the composition of solutions.
• For a compound like ethanol (C\(_2\)H\(_5\)OH), the molar mass is about 46 g/mol because it includes the combined atomic masses of two carbon atoms, six hydrogen atoms, and one oxygen atom.
• The molar mass of water (H\(_2\)O) is about 18 g/mol, derived from two hydrogen atoms and one oxygen atom.

Using the molar masses, you can calculate how many moles are present in a given mass of a substance, as was done in the problem's step-by-step solution to determine the mole ratio and therefore the mole fractions. Accurate understanding and usage of molar mass enable precise stoichiometric calculations and clear comprehension of reaction dynamics.