Q. weight fraction to mole fraction

We are given mass (weight) fractions and want to convert them to mole fractions. I will give a clear derivation, a stepwise procedure you can follow, and a worked numerical example. Be careful to note the relationship between mass, molar mass, and number of moles.

Step 1. Definitions. Let the mixture contain species indexed by \(i\). Let \(m_i\) be the mass of species \(i\). Let \(M_i\) be the molar mass of species \(i\). Let \(n_i\) be the number of moles of species \(i\). Let \(w_i\) be the mass (weight) fraction of species \(i\), and let \(x_i\) be the mole fraction of species \(i\).

By definition, the mass fraction is the mass of the component divided by the total mass:

\[ w_i \;=\; \frac{m_i}{\sum_j m_j} \]

By definition, the number of moles is mass divided by molar mass:

\[ n_i \;=\; \frac{m_i}{M_i} \]

The mole fraction is the moles of the component divided by the total moles:

\[ x_i \;=\; \frac{n_i}{\sum_j n_j} \;=\; \frac{\dfrac{m_i}{M_i}}{\sum_j \dfrac{m_j}{M_j}} \]

Step 2. Express the mole fraction in terms of mass fractions. Substitute \(m_i = w_i \, m_{\text{tot}}\), where \(m_{\text{tot}}=\sum_j m_j\) is the total mass of the sample. Then

\[ x_i \;=\; \frac{\dfrac{w_i \, m_{\text{tot}}}{M_i}}{\sum_j \dfrac{w_j \, m_{\text{tot}}}{M_j}} \]

The total mass \(m_{\text{tot}}\) cancels from numerator and denominator, giving the convenient formula:

\[ x_i \;=\; \frac{\dfrac{w_i}{M_i}}{\sum_j \dfrac{w_j}{M_j}} \]

This is the general conversion formula: the mole fraction of species \(i\) equals its mass fraction divided by its molar mass, normalized by the sum of mass fraction over molar mass for all species.

Step 3. Practical step-by-step procedure to compute mole fractions from mass fractions (or mass percent):

1. If mass fractions are given as percents, convert them to fractions by dividing by 100. Optionally, assume a convenient total mass, e.g., \(100\) g, so that the mass of each component equals its mass percent in grams.

2. For each component, compute moles: \(n_i = \dfrac{m_i}{M_i}\).

3. Compute total moles: \(n_{\text{tot}}=\sum_j n_j\).

4. Compute mole fraction: \(x_i = \dfrac{n_i}{n_{\text{tot}}}\).

Step 4. Worked numerical example. Suppose a binary mixture has mass fractions \(w_A=0.30\) and \(w_B=0.70\). Let molar masses be \(M_A=18\) g mol^{-1} and \(M_B=28\) g mol^{-1}. Compute mole fractions.

Using the formula, first compute moles per unit total mass (or assume 100 g):

\[ n_A \;=\; \frac{w_A}{M_A} \;=\; \frac{0.30}{18} \;=\; 0.0166667 \text{ mol per g of mixture} \]

\[ n_B \;=\; \frac{w_B}{M_B} \;=\; \frac{0.70}{28} \;=\; 0.0250000 \text{ mol per g of mixture} \]

Sum of these (per g of mixture) is

\[ n_{\text{tot}} \;=\; 0.0166667 + 0.0250000 \;=\; 0.0416667 \text{ mol per g} \]

Thus the mole fractions are

\[ x_A \;=\; \frac{0.0166667}{0.0416667} \;=\; 0.40 \]

\[ x_B \;=\; \frac{0.0250000}{0.0416667} \;=\; 0.60 \]

Check: \(x_A + x_B = 1.00\).

Step 5. Final summary. The conversion formula you should remember is

\[ x_i \;=\; \frac{\dfrac{w_i}{M_i}}{\sum_j \dfrac{w_j}{M_j}} \]

Or equivalently, follow the four practical steps above: convert masses, compute moles, sum moles, divide to obtain mole fractions.