Question

\({ }^{4} \mathbf{x}\) ' 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

Short answer

The value of \(x\) is 18 grams, corresponding to option (d).

Step-by-step solution

Understand the Problem

We need to find the mass of water, represented by \(^{4} x\), required to mix with 69 grams of ethanol to create a solution where the mole fraction of ethanol is 0.6.

Define Mole Fraction

The mole fraction of ethanol, \( X_{ethanol} \), is given by \( X_{ethanol} = \frac{n_{ethanol}}{n_{ethanol} + n_{water}} = 0.6 \), where \( n \) represents the number of moles.

Calculate Moles of Ethanol

Ethanol has a molar mass of approximately 46 g/mol. Therefore, the moles of ethanol, \( n_{ethanol} \), are calculated as: \( n_{ethanol} = \frac{69}{46} \approx 1.5 \) moles.

Set Up the Equation for Mole Fraction

Using the mole fraction equation: \( 0.6 = \frac{1.5}{1.5 + n_{water}} \). Solve for \( n_{water} \) using algebra.

Solve for Moles of Water

Rearrange the equation: \( 0.6(1.5 + n_{water}) = 1.5 \). This simplifies to \( 0.9 + 0.6n_{water} = 1.5 \). Solving gives \( n_{water} = 1 \) mole.

Convert Moles of Water to Grams

The molar mass of water is approximately 18 g/mol. Therefore, \( 1 \) mole of water is \( 1 \times 18 = 18 \) grams.

Conclusion

Thus, the value of \( 'x' \) in grams is 18, which is answer (d).

Key concepts

Ethanol

Ethanol is an organic compound often used as a solvent and found in alcoholic beverages. Its molecular formula is \(C_2H_5OH\), indicating that it contains carbon, hydrogen, and oxygen.
Ethanol is a clear, colorless liquid with a distinct, pleasant odor. This compound is versatile and is used in numerous applications, including fuel, personal care products, and pharmaceuticals. Due to its ability to dissolve a wide range of chemical substances, ethanol is also a popular choice in scientific research and industrial processes. The molar mass of ethanol is approximately 46 g/mol, a crucial factor when calculating its molar concentration or mole fraction in solutions.

Water

Water, the universal solvent, has the chemical formula \(H_2O\). As a crucial component of life, it's the most abundant molecule on Earth's surface. Water has a unique structure, with two hydrogen atoms covalently bonded to one oxygen atom, which gives it its characteristic bent shape. This polarity allows it to dissolve many substances, making it an excellent medium for chemical reactions.
In the context of solutions, water is often the solvent, meaning other substances (solutes) are dissolved in it. The understanding of water's mass and its molar mass, which is approximately 18 g/mol, is essential for calculating the number of moles of water in chemical problems, such as determining the mole fraction in a mixture.

Moles and Molarity

The mole is a fundamental unit in chemistry, used to express amounts of a chemical substance. One mole contains exactly \(6.022 \times 10^{23}\) entities (usually atoms or molecules), known as Avogadro's number. Molarity, on the other hand, refers to the concentration of a solution, defined as the number of moles of solute per liter of solution.
The concept of moles and molarity is central to stoichiometry and allows chemists to calculate how much of one substance reacts with another. In the given problem, we use moles to convert between mass and molecules of ethanol and water, ultimately helping us find the solution's composition in terms of mole fraction. Understanding how to manipulate these calculations is crucial for anyone studying chemistry.

Molar Mass

Molar mass is an important concept in chemistry which refers to the mass of one mole of a given substance, typically expressed in grams per mole (g/mol). It allows us to convert between the amount of substance (in moles) and its mass (in grams).
Calculating molar mass involves summing up the atomic masses of all atoms in a molecule. For instance, the molar mass of water (\(H_2O\)) is calculated by adding the mass of two hydrogen atoms (approximately \(1\) g/mol each) and one oxygen atom (approximately \(16\) g/mol), resulting in a total molar mass of \(18\) g/mol.
The molar mass of ethanol (\(C_2H_5OH\)) is calculated by summing the masses of 2 carbon atoms (\(12.01\) g/mol each), 6 hydrogen atoms (\(1.01\) g/mol each), and 1 oxygen atom (\(16.00\) g/mol). This comes out to approximately \(46\) g/mol. Understanding molar mass is key to solving many chemical problems, including those involving mole fraction and mixing solutions.