Question

A gaseous mixture contains three gases \(A, B\) and \(C\) with a total number of moles of 10 and total pressure of 10 atm. The partial pressure of \(A\) and \(B\) are 3 atm and 1 atm respectively and if \(C\) has molecular weight of \(2 \mathrm{~g} / \mathrm{mol}\). Then, the weight of \(C\) present in the mixture will be : (a) \(8 \mathrm{~g}\) (b) \(12 \mathrm{~g}\) (c) \(3 \mathrm{~g}\) (d) \(6 \mathrm{~g}\)

Short answer

The weight of C in the mixture is 12 g.

Step-by-step solution

Determine the partial pressure of gas C

To find the partial pressure of gas C, subtract the known partial pressures of gases A and B from the total pressure. Use the equation, total pressure = partial pressure of A + partial pressure of B + partial pressure of C.

Calculate the moles of gas C

Using the partial pressures and Dalton's law of partial pressures, calculate the moles of gas C in the mixture. The ratio of the partial pressure of gas C to the total pressure is equal to the ratio of moles of gas C to the total moles.

Calculate the weight of gas C

To find the weight of gas C, multiply the number of moles of C by its molecular weight. The molecular weight of gas C is given as 2 g/mol.

Key concepts

Dalton's Law of Partial Pressures

Understanding the behavior of gas mixtures can be simplified using Dalton's Law of Partial Pressures. This principle states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases.

A partial pressure is the pressure that a gas would exert if it alone occupied the volume of the mixture at the same temperature. This concept becomes particularly useful when analyzing gas mixtures where each gas contributes to the total pressure in proportion to its fraction of the total number of moles of gas present.

For example, for a gas mixture containing gases A, B, and C, with the total pressure being the sum of the pressures these gases would exert independently, we can represent this as: \[ P_{total} = P_A + P_B + P_C \] Here, \(P_{total}\) is the total pressure of the mixture, while \(P_A\), \(P_B\), and \(P_C\) are the partial pressures of gases A, B, and C respectively.

In solving problems relating to gas mixtures, Dalton’s law helps us relate the known pressures of some gases to uncover unknown values for others within the same system. This is the first step in our example problem.

Calculating Moles

When dealing with gases, the mole concept is essential. A mole is a unit that measures the amount of substance and it's vital when considering chemical reactions and the properties of gases. In the context of Dalton’s law, once we have the partial pressure of a gas, we can find out how many moles of that gas we have in a mixture.

Here’s the relationship given by Dalton's law for moles: \[ \frac{n_C}{n_{total}} = \frac{P_C}{P_{total}} \] Where \(n_C\) is the moles of gas C, \(n_{total}\) is the total moles of the gas mixture, \(P_C\) is the partial pressure of gas C, and \(P_{total}\) is the total pressure.

By understanding the proportion of pressures, we can derive the proportion of moles for any gas in a mixture. The ultimate goal is to equate the fraction of the partial pressure of the one gas to the fraction of moles of that same gas. By rearranging the above equation, we can calculate the unknown number of moles of a specific gas when given its partial pressure and the total moles in the mixture. This step is critical in our step-by-step problem solution.

Molecular Weight

Molecular weight is a measure of the total mass of atoms in a molecule and is typically expressed in terms of grams per mole (g/mol). It can be calculated by summing the atomic masses of all atoms within a molecule, as found on the periodic table.

This measure is fundamental when converting between the number of moles of a substance and its mass in grams. The formula to find the mass (m) of a gas is: \[ m = n \times MW \] Where \(m\) is the mass of the gas, \(n\) is the number of moles of the gas, and \(MW\) is the molecular weight of the gas.

In our example problem, the molecular weight of gas C is given as 2 g/mol. Once we’ve calculated the moles of gas C, as shown in the previous sections, we simply multiply this value by the molecular weight of C to obtain the total mass. Accurate calculation of molecular weight and understanding its application is the final step to solving the weight of gas C in the mixture.