How to Find Mole Fraction from Mass Fraction?

To find the mole fraction of a component in a mixture from its mass fraction, you need to know the molar masses of all components in the mixture. The process involves calculating the mixture's molar mass and then applying a simple formula.

Here's a step-by-step guide:

1. Understand the Definitions

• Mass Fraction (w_i): The mass of a component i divided by the total mass of the mixture.

  • w<sub>i</sub> = mass of component i / total mass of mixture

• Mole Fraction (x_i): The number of moles of a component i divided by the total number of moles in the mixture.

  • x<sub>i</sub> = moles of component i / total moles of mixture

• Molar Mass (M_i): The mass of one mole of a substance, usually expressed in grams per mole (g/mol).

2. Calculate the Average Molar Mass of the Mixture (M_mix)

If you have the mass fractions (w_i) and molar masses (M_i) of all components, you can calculate the average molar mass of the mixture using the following formula:

M<sub>mix</sub> = 1 / Σ(w<sub>i</sub> / M<sub>i</sub>)

Where:

• M<sub>mix</sub> is the average molar mass of the mixture.
• w<sub>i</sub> is the mass fraction of component i.
• M<sub>i</sub> is the molar mass of component i.
• Σ represents the summation over all components in the mixture.

3. Calculate the Mole Fraction of Each Component (x_i)

Once you have the average molar mass of the mixture, you can calculate the mole fraction of each component using the following formula:

x<sub>i</sub> = (w<sub>i</sub> / M<sub>i</sub>) * M<sub>mix</sub>

Where:

• x<sub>i</sub> is the mole fraction of component i.
• w<sub>i</sub> is the mass fraction of component i.
• M<sub>i</sub> is the molar mass of component i.
• M<sub>mix</sub> is the average molar mass of the mixture.

Example:

Let's say we have a mixture of two components: A and B.

• Component A: w_A = 0.6 (mass fraction), M_A = 28 g/mol (molar mass)
• Component B: w_B = 0.4 (mass fraction), M_B = 32 g/mol (molar mass)

1. Calculate the average molar mass of the mixture (M_mix):

   M<sub>mix</sub> = 1 / ( (0.6 / 28) + (0.4 / 32) )
M<sub>mix</sub> = 1 / (0.0214 + 0.0125)
M<sub>mix</sub> = 1 / 0.0339
M<sub>mix</sub> = 29.49 g/mol

2. Calculate the mole fraction of component A (x_A):

   x<sub>A</sub> = (0.6 / 28) * 29.49
x<sub>A</sub> = 0.0214 * 29.49
x<sub>A</sub> = 0.631

3. Calculate the mole fraction of component B (x_B):

   x<sub>B</sub> = (0.4 / 32) * 29.49
x<sub>B</sub> = 0.0125 * 29.49
x<sub>B</sub> = 0.369

Therefore, in this mixture, the mole fraction of component A is approximately 0.631, and the mole fraction of component B is approximately 0.369. Note that x_A + x_B = 1.0.

*In summary, to convert from mass fraction to mole fraction, determine the molar masses of all components, calculate the mixture's average molar mass, and then apply the formula x_i = (w_i / M_i) M_mix for each component.**