The average molecular weight of a multi-component gas stream is calculated as a mole-fraction-weighted average of the component molecular weights. Choose a calculation basis, multiply each component’s mole fraction by its molecular weight, and sum the results:
[ \overline{M}=\sum_i y_iM_i ]
where (y_i) is the mole fraction and (M_i) is the molecular weight of component (i).
Use a 100 kmol basis when the composition is given in mole percent: the total mass of the mixture divided by 100 kmol is the average molecular weight.
Calculating Average Molecular Weight from Mole Percent
Select a calculation basis
A convenient basis is 100 kmol of total gas. On this basis, a component present at 40 mol% corresponds to 40 kmol.
The basis does not affect the final average molecular weight. It only makes the component calculations easier to organize.
Determine each component’s mass
For each gas component:
[ m_i=n_iM_i ]
where:
- (m_i) is the component mass
- (n_i) is the component amount in kmol
- (M_i) is the component molecular weight
The total mixture mass is:
[ m_{\text{total}}=\sum_i n_iM_i ]
Divide total mass by total moles
The average molecular weight is:
[ \overline{M}=\frac{m_{\text{total}}}{n_{\text{total}}} ]
Using mole fractions directly gives the equivalent expression:
[ \overline{M}=\sum_i y_iM_i ]
If molecular weights are expressed in (\text{kg/kmol}), the resulting average is in (\text{kg/kmol}). Numerically, this has the same value as (\text{g/mol}).
Worked Gas-Stream Example
Example composition
Consider a synthetic gas containing:
| Component | Mole percent | Molecular weight, kg/kmol |
|---|---|---|
| Hydrogen, (H_2) | 40% | 2.016 |
| Carbon monoxide, (CO) | 20% | 28.01 |
| Carbon dioxide, (CO_2) | 20% | 44.01 |
| Methane, (CH_4) | 20% | 16.04 |
Using a basis of 100 kmol:
Calculate component masses
[ m_{H_2}=40(2.016)=80.64\ \text{kg} ]
[ m_{CO}=20(28.01)=560.2\ \text{kg} ]
[ m_{CO_2}=20(44.01)=880.2\ \text{kg} ]
[ m_{CH_4}=20(16.04)=320.8\ \text{kg} ]
The total mass is:
[ m_{\text{total}}=80.64+560.2+880.2+320.8=1841.84\ \text{kg} ]
Calculate the mixture average
[ \overline{M}=\frac{1841.84\ \text{kg}}{100\ \text{kmol}} ]
[ \boxed{\overline{M}=18.42\ \text{kg/kmol}} ]
Therefore, the average molecular weight of the gas stream is approximately 18.42 g/mol.
When the Composition Is Given as Mass Fractions
Convert mass fractions to mole fractions
If an analyzer reports mass fractions instead of mole fractions, the direct mole-weighted formula cannot be applied immediately.
Choose a basis, such as 100 kg of gas. For each component:
[ m_i=x_i m_{\text{total}} ]
[ n_i=\frac{m_i}{M_i} ]
Then calculate the mole fraction:
[ y_i=\frac{n_i}{\sum_i n_i} ]
Finally:
[ \overline{M}=\sum_i y_iM_i ]
Use the reciprocal mass-fraction relationship
A direct relationship is also available:
[ \frac{1}{\overline{M}}=\sum_i\frac{x_i}{M_i} ]
Therefore:
[ \boxed{\overline{M}=\frac{1}{\sum_i(x_i/M_i)}} ]
Here, (x_i) must be expressed as a mass fraction, not a percentage. For example, 25 mass% must be entered as (0.25).
Why Average Molecular Weight Matters in Unit Operations
Convert gas flow rates
Experimental gas flows may be measured volumetrically, while mass balances require mass flow rates.
Using the ideal-gas relationship:
[ \dot{m}=\frac{P\dot{V}\overline{M}}{RT} ]
where (P) is pressure, (\dot{V}) is volumetric flow rate, (T) is absolute temperature, and (R) is the gas constant.
Estimate gas density
For an ideal gas:
[ \rho=\frac{P\overline{M}}{RT} ]
A heavier gas mixture has a greater density at the same temperature and pressure.
Support equipment calculations
Average molecular weight is used in gas absorption, scrubbing, combustion, and reactor experiments to help calculate gas density, convert between flow units, size equipment, and validate material balances.
The value is also important when the feed composition changes during an experimental run.
Understanding the Trade-offs
Mole fractions are usually preferred for gas calculations
Gas-law and stoichiometric calculations naturally use mole fractions because gas volumes are related to the number of moles under specified temperature and pressure conditions.
Mass fractions remain useful when the measurement system reports mass-based data or when the material balance is organized around mass flow.
The result depends on the composition basis
A composition must clearly state whether it is wet or dry, and whether it is measured at the inlet, outlet, or another location.
For example, including water vapor in a wet gas composition changes both the total composition and the calculated average molecular weight.
Consistent units are essential
Molecular weights, flow rates, pressure, temperature, and the gas constant must use compatible units.
Common errors include treating mole percentages as fractions without dividing by 100, mixing (\text{g/mol}) with (\text{kg/kmol}) inconsistently, and using gauge pressure instead of absolute pressure in gas-law calculations.
Ideal-gas behavior may be an approximation
The weighted-average molecular-weight equation is based on composition and remains the standard mixture calculation.
However, density and volumetric-flow calculations may require a compressibility factor (Z) at high pressures or when the gas is far from ideal:
[ \rho=\frac{P\overline{M}}{ZRT} ]
Applying This to Your Experiment
The correct procedure depends primarily on how the gas composition was measured.
- If your primary focus is a mole-percent gas analysis: Use (\overline{M}=\sum_i y_iM_i), or use a 100 kmol basis and divide the total calculated mass by 100 kmol.
- If your primary focus is a mass-fraction analysis: Use (\overline{M}=1/\sum_i(x_i/M_i)), after confirming that the reported percentages have been converted to fractions.
- If your primary focus is gas density or volumetric flow: Combine the calculated average molecular weight with absolute pressure, absolute temperature, and an appropriate compressibility factor when necessary.
- If your primary focus is a pilot-plant mass balance: Confirm that all component compositions use the same wet or dry basis and correspond to the same stream location.
Once the composition basis and units are consistent, the average molecular weight is a straightforward and reliable bridge between molar, mass, and volumetric gas-stream calculations.
Summary Table:
| Calculation need | Recommended method | Key formula or requirement |
|---|---|---|
| Mole-percent composition | Use a mole-fraction-weighted average | M̄ = Σ(yᵢMᵢ) |
| Mole-percent basis calculation | Choose 100 kmol of total gas | Total mixture mass divided by 100 kmol |
| Mass-fraction composition | Convert to mole fractions or use the reciprocal relationship | M̄ = 1 / Σ(xᵢ/Mᵢ) |
| Gas mass flow or density | Combine average molecular weight with gas conditions | Use absolute pressure and temperature; include Z when needed |
| Pilot-plant material balances | Confirm a consistent composition basis | Check wet/dry basis, stream location, units, and composition data |
| Worked example result | 40% H₂, 20% CO, 20% CO₂, 20% CH₄ | Average molecular weight: 18.42 kg/kmol (g/mol) |
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