Question

A gaseous mixture contains \(70 \% \mathrm{~N}\), and \(30 \%\) unknown gas, by volume. If the average molecular mass of gaseous mixture is \(37.60\), the molecular mass of unknown gas is (a) \(42.2\) (b) 60 (c) 40 (d) 50

Short answer

The molecular mass of the unknown gas is 60.

Step-by-step solution

Understanding the problem

The problem provides the percentage composition by volume of a gaseous mixture and the average molecular mass of the mixture. We are asked to calculate the molecular mass of the unknown gas in the mixture.

Using the weighted average formula

To find the molecular mass of the unknown gas, we need to use the weighted average formula for the average molecular mass of the mixture: \[ M_{avg} = \frac{\sum (X_i \cdot M_i)}{100} \] where \( M_{avg} \) is the average molecular mass of the mixture, \( X_i \) is the percentage of each gas by volume, and \( M_i \) is the molecular mass of each gas. In our case, we have Nitrogen (N) and the unknown gas. We let \( M_{unknown} \) be the molecular mass of the unknown gas.

Assigning values to known variables

For Nitrogen (N), which has a molecular mass of \( 28 \) (since it is diatomic, and its atomic mass is 14), and occupies 70% of the volume, its part of the average is \( 70 \% \times 28 \). The unknown gas occupies the remaining 30% by volume. The average molecular mass of the mixture is given as \( 37.60 \). So, we have the equation \[ 37.60 = \frac{70 \% \times 28 + 30 \% \times M_{unknown}}{100} \].

Solving for the molecular mass of the unknown gas

Now we can solve for \( M_{unknown} \) from the above equation by first simplifying the percentages: \[ 37.60 = \frac{70 \times 28 + 30 \times M_{unknown}}{100} \] which can be further simplified to: \[ 37.60 = 0.7 \times 28 + 0.3 \times M_{unknown} \] Then we solve for \( M_{unknown} \): \[ 37.60 = 19.6 + 0.3 \times M_{unknown} \] \[ 37.60 - 19.6 = 0.3 \times M_{unknown} \] \[ 18 = 0.3 \times M_{unknown} \] \[ M_{unknown} = \frac{18}{0.3} \] \[ M_{unknown} = 60 \]

Key concepts

Average Molecular Mass

Understanding the average molecular mass is crucial when dealing with mixtures of gases, as it represents the average mass of all the molecules in a given volume of gas. It takes into account the proportion of each type of molecule present. To visualize this, imagine having a bag of mixed candies, each type with a different weight. If you were to pick out one candy at random, the 'average weight' would reflect the typical weight you'd expect based on the variety of candies and their individual weights.

To calculate the average molecular mass of a gas mixture, you would need to know the percentage composition of each gas in the mixture and the individual molecular masses of the gases involved. This concept becomes particularly important in situations where the properties of a gas mixture, such as its behavior under changing pressure and temperature conditions, are evaluated in chemical engineering and material sciences.

Weighted Average Formula

The weighted average formula is a mathematical way to take into account the different proportions of components when calculating an average. It multiplies each component's value by its weight, sums up these products, and then divides by the sum of the weights. In the context of physical chemistry, this means that you can determine the average molecular mass of a mixture by multiplying the molecular mass of each individual gas by the percentage of that gas in the mixture (its 'weight'), adding these values together, and dividing by 100 (since percentages sum up to 100%).

When using the weighted average formula, it is important to convert the percentage values into decimal form (for example, 70% becomes 0.7) before performing the calculations. This approach ensures that the relative proportions of each gas are correctly factored into the average, providing an accurate measure of the mixture’s overall molecular mass. For students working through physical chemistry problems, mastering this formula is essential for accurately describing mixtures of gases and understanding their properties.

Molecular Mass Calculation

Molecular mass calculation plays a pivotal role in many areas of chemistry, particularly when balancing chemical equations or determining the stoichiometry of reactions. The molecular mass of a substance is the sum of the atomic masses of all atoms in a molecule. For elemental gases, such as hydrogen (H₂), oxygen (O₂), and nitrogen (N₂), the molecular mass is simply the atomic mass from the periodic table multiplied by the number of atoms per molecule.

For mixtures, the process encompasses finding the molecular mass calculation for each gas and then employing the weighted average formula to obtain the average molecular mass of the mixture—much like what is shown in the provided exercise. This process involves not only comprehending the concept of molecular mass but also being proficient with algebraic manipulation. The ability to calculate molecular mass is crucial for predicting the behavior of gases using the ideal gas law and related equations, making it a fundamental skill for students to master in physical chemistry.