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

Question

\({ }^{\prime} \mathrm{x}\) ' 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 ' \(x\) ' in grams? (a) 54 (b) 36 (c) 180 (d) 18

Step-by-Step Solution

Answer

The value of 'x' is 18 grams.

Step 1: Understand Mole Fraction

The mole fraction of a component in a solution is the ratio of the number of moles of that component to the total number of moles of all components in the solution. Mathematically, the mole fraction of ethanol can be represented as: \[ X_{ethanol} = \frac{n_{ethanol}}{n_{ethanol} + n_{water}} \]Where \(n_{ethanol}\) is the number of moles of ethanol and \(n_{water}\) is the number of moles of water.

Step 2: Calculate Moles of Ethanol

First, calculate the moles of ethanol. The molecular weight of ethanol (C₂H₅OH) is 46 g/mol. Therefore, the number of moles of ethanol is:\[ n_{ethanol} = \frac{69}{46} \approx 1.5 \text{ moles} \]

Step 3: Set Up the Mole Fraction Equation

We know the mole fraction \(X_{ethanol} = 0.6\). Substitute the values for the total moles into the equation:\[ 0.6 = \frac{1.5}{1.5 + n_{water}} \]

Step 4: Solve for Moles of Water

Rearrange the equation to solve for \(n_{water}\):\[ 0.6(1.5 + n_{water}) = 1.5 \]\[ 0.9 + 0.6n_{water} = 1.5 \]\[ 0.6n_{water} = 0.6 \]\[ n_{water} = 1 \text{ mole} \]

Step 5: Convert Moles of Water to Grams

The molecular weight of water (H₂O) is 18 g/mol. Therefore, 1 mole of water has a mass of:\[ 1 \times 18 = 18 \text{ grams} \]

Step 6: Result

Thus, the required value of \(x\), the grams of water added, is 18 grams.

Key Concepts

Ethanol and Water Solution

In many chemistry problems, we come across solutions where two or more substances are mixed. An ethanol and water solution is a common example of a liquid mixture. Ethanol is an alcohol often found in alcoholic beverages, whereas water is, of course, a universal solvent. In this context, a solution means a homogeneous mixture where the two liquids are mixed uniformly together.
When mixing ethanol and water, it's important to understand that the physical properties, like volume and boiling point, can change in non-linear ways due to intermolecular interactions. But for the mole fraction calculations, those complexities aren't directly involved. Instead, we focus on how many moles of each substance are present.
By exploring this solution further, you can understand how the concept of mole fraction helps us to precisely describe the composition of the solution. It's a way to quantify the presence of each component in the mixture.

Calculating Moles

To solve many chemistry problems, knowing how to calculate moles is crucial. A mole is a basic unit in chemistry that represents an amount of substance. It effectively counts how many molecules or atoms are in a given mass of a substance.
For ethanol (with a molecular weight of 46 g/mol), you calculate the moles by dividing the mass of ethanol by its molecular weight. For example, 69 grams of ethanol divided by 46 g/mol equals approximately 1.5 moles.
Similarly, for water (with a molecular weight of 18 g/mol), if you are given the moles, you multiply by the molecular weight to find the mass. Understanding these calculations lets you translate between mass and amount of substance, which is a common challenge in chemistry.

Chemical Calculations

Chemical calculations involve using equations and formulas to find the unknowns in a chemical context. With the problem involving ethanol and water, we use the mole fraction equation to find how much of each substance is in the solution.
The mole fraction equation is: \[ X_{ethanol} = \frac{n_{ethanol}}{n_{ethanol} + n_{water}} \]In this equation, \(n_{ethanol}\) represents the number of moles of ethanol, and \(n_{water}\) is for water. Understanding this equation helps us solve how much water is present, given other known variables like the mole fraction of ethanol. This is a core part of many chemical problem-solving applications.

Solution Concentration

Solution concentration is about measuring how much of a solute is present in a given solution. Concentration can be expressed in various ways including mole fraction, as seen in this exercise, where the mole fraction gives the proportion of each substance in relation to the total amount.
In the context of this solution, the mole fraction of ethanol was 0.6, meaning ethanol makes up 60% of the moles in the total solution. Understanding this allows us to set up an equation to solve for unknowns.

• With these calculations, you'd also recognize changes in concentration can impact solution properties like reaction rates and physical behavior.
• Learning to calculate and interpret concentration prepares you to tackle diverse problem scenarios in chemistry.

Recognizing these concepts and their practical use is essential in mastering chemical solutions.