A 5.2 molal aqueous solution of methyl alcohol, CH3OH, is supplied. What is the mole fraction of met

Q: Why do we assume exactly 1000 g of water?

We assume 1000 g of water because molality is defined as moles of solute per 1 kg (1000 g) of solvent. Assuming 1000 g allows us to directly take the molality value as the number of moles of the solute, making the calculation straightforward.

Q: Can we assume any other mass of water, like 500 g?

Yes, absolutely! If you assume 500 g of water, the moles of methyl alcohol would be half of 5.2 (which is 2.6 moles). The ratio of moles will remain exactly the same, and you will get the same mole fraction. Assuming 1000 g just saves you a calculation step.

Q: Does temperature affect the mole fraction or molality?

No. Both molality and mole fraction are based entirely on mass and moles, neither of which change with temperature. This is unlike molarity, which depends on volume and therefore changes as the liquid expands or contracts with temperature.

Q: What if the solvent was not water?

If the solvent were something else, say benzene, you would still assume 1000 g of benzene. However, you would divide that 1000 g by the molar mass of benzene (78 g/mol) instead of 18 g/mol to find the moles of the solvent.

Q: Why does the mole fraction not have any units?

Mole fraction is a ratio of moles divided by total moles (mol / mol). The units cancel out completely, leaving it as a dimensionless pure number.

Question

A $5.2$ molal aqueous solution of methyl alcohol, $	ext{CH}_3	ext{OH}$, is supplied. What is the mole fraction of methyl alcohol in the solution ?

Options:

(A) $0.100$

(B) $0.190$

(C) $0.086$

(D) $0.050$

Sigma Percentile · Accepted Answer

### Decoding the Concentration Terms

When dealing with solutions in chemistry, we often encounter various ways to express concentration. Two of the most fundamental and temperature-independent terms are **molality** and **mole fraction**.

In this problem, we are given a $5.2$ molal aqueous solution of methyl alcohol ($	ext{CH}_3	ext{OH}$) and asked to find its mole fraction. The key to solving this lies in understanding exactly what "molal" means.

### The Power of Molality

Molality ($m$) is defined as the number of moles of solute dissolved per $1000	ext{ g}$ ($1	ext{ kg}$) of the solvent.

$$m = \frac{n_{	ext{solute}}}{W_{	ext{solvent}} 	ext{ (in kg)}}$$

By stating that the solution is $5.2$ molal, the problem is secretly handing us two crucial pieces of information on a silver platter:
1. The number of moles of our solute, methyl alcohol ($n_1$), is exactly **$5.2	ext{ mol}$**.
2. The mass of our solvent, water ($W_2$), is exactly **$1000	ext{ g}$**.

### Finding the Moles of Solvent

To calculate the mole fraction, we need the moles of *every* component in the mixture. We already have the moles of methyl alcohol. Now, we must convert the mass of our solvent (water) into moles.

The molar mass of water ($	ext{H}_2	ext{O}$) is $18	ext{ g/mol}$. Using the formula for moles:

$$n_2 = \frac{	ext{Given Mass}}{	ext{Molar Mass}} = \frac{1000}{18}$$

$$n_2 = 55.56	ext{ mol}$$

Now we know that our solution consists of $5.2	ext{ moles}$ of methyl alcohol swimming among $55.56	ext{ moles}$ of water.

### The Final Calculation: Mole Fraction

The mole fraction ($X$) of a component is simply the ratio of its moles to the total moles in the solution. First, let's find the total moles:

$$n_{	ext{total}} = n_1 + n_2 = 5.2 + 55.56 = 60.76	ext{ mol}$$

Finally, we calculate the mole fraction of methyl alcohol:

$$X_{	ext{CH}_3	ext{OH}} = \frac{n_1}{n_{	ext{total}}} = \frac{5.2}{60.76}$$

$$X_{	ext{CH}_3	ext{OH}} \approx 0.086$$

And there we have it! The mole fraction of methyl alcohol is **$0.086$**.

*Pro Tip:* If you ever need to find the mole fraction of the solvent (water) after this, you don't need to repeat the division. Since the sum of all mole fractions in a mixture is always exactly $1$, you can simply subtract: $X_{	ext{H}_2	ext{O}} = 1 - 0.086 = 0.914$.

Animated Solution for Chemistry - Basic Concepts in Chemistry: A $5.2$ molal aqueous solution of methyl alcohol, $CH_{3} OH$ , is supplied. What is the mole fraction of methyl alcohol in the solution ?

Select Answer:

Visualized Solution

\text{Definition of Molality}

\text{Extracting Values}

\text{Moles of Solvent}

\text{Total Moles}

\text{Mole Fraction of } \text{CH}_3\text{OH}

\text{The Way Forward}

The Sigma Insight: Molecular Mass, Mole Concept and Concentration

Decoding the Concentration Terms

The Power of Molality

Finding the Moles of Solvent

The Final Calculation

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