Understanding and Calculating the Density of a Gas Mixture in Microsoft Excel

Density of a gas mixture is the mass per unit volume of the mixture under specific conditions of temperature and pressure. It depends on the composition and properties of the individual gases that make up the mixture. One way to calculate the density of a gas mixture is to use the ideal gas law, which relates the pressure, volume, temperature and amount of gas. However, the ideal gas law assumes that the gas molecules do not interact with each other and have negligible volume, which is not always true for real gases. Therefore, some corrections may be needed to account for the non-ideal behavior of gases, such as the van der Waals equation. Another way to calculate the density of a gas mixture is to use the mole fraction and the molar mass of each gas. The mole fraction is the ratio of the number of moles of a gas to the total number of moles of the mixture. The molar mass is the mass of one mole of a gas. The density of a gas mixture can be found by multiplying the mole fraction and the molar mass of each gas, and then adding them up. This method does not require the pressure and temperature of the mixture, but it assumes that the volume percent is equivalent to the mole percent, which is only true for ideal gases. Therefore, some errors may occur if the gases are not ideal.

Basic Theory

The density (ρ) of a gas mixture is defined as the mass of the mixture per unit volume. Mathematically, it can be expressed as:

$ρ_{\text{mixture}} = \frac{(\sum_{i} m_i)}{V_{\text{total}}}$

where $m_i$ is the mass of each individual gas component and $V_{\text{total}}$ is the total volume of the gas mixture.

Procedures in Microsoft Excel

Gather Data: Create a table in Excel to organize the data. Include columns for gas components, molecular weights ( $MW_i$ ), molar fractions ( $x_i$ ), and any other relevant parameters.
Calculate Mass Fractions: Calculate the mass fraction ( $w_i$ ) of each gas component using the formula:
$w_i = x_i \times MW_i$
Sum the Mass Fractions: Sum the mass fractions to obtain the total mass of the gas mixture:
$M_{\text{total}} = \sum_{i} w_i$
Determine Total Volume: Enter the total volume of the gas mixture ( $V_{\text{total}}$ ).
Calculate Gas Mixture Density: Use the formula to find the density of the gas mixture:
$ρ_{\text{mixture}} = \frac{M_{\text{total}}}{V_{\text{total}}}$

Scenario: Gas Mixture Calculation

Consider a gas mixture containing nitrogen (N2) and carbon dioxide (CO2) with molar fractions of 0.8 and 0.2, respectively. The molecular weights of N2 and CO2 are 28.02 g/mol and 44.01 g/mol, respectively. Assume a total volume of 10 m³.

Excel Calculation

Gas Component	$MW_i$ (g/mol)	$x_i$	$w_i$
N2	28.02	0.8	=B2*C2
CO2	44.01	0.2	=B3*C3
Total			=SUM(D2:D3)

$ρ_{\text{mixture}} = \frac{D4}{10}$

MATLAB Comparison


% MATLAB Calculation
molecularWeights = [28.02, 44.01];
molarFractions = [0.8, 0.2];
totalVolume = 10;

massFractions = molarFractions .* molecularWeights;
totalMass = sum(massFractions);
densityMixture = totalMass / totalVolume;

Results

After performing the calculations in Excel and MATLAB, the density of the gas mixture is found to be:

$ρ_{\text{mixture}} \approx 3.384 \, \text{g/mol}$

Remember to adapt these steps to your specific gas mixture and always double-check your data and formulas for accuracy. Excel provides a versatile and user-friendly platform for such calculations, while MATLAB offers a scripting environment with powerful mathematical capabilities for comparison purposes.