Question

Equal masses of ethane and hydrogen are mixed in an empty container at \(25^{\circ} \mathrm{C}\). The fraction of the total pressure exerted by hydrogen is (a) \(1: 2\) (b) \(1: 1\) (c) \(1: 16\) (d) \(15: 16\)

Short answer

(d) 15: 16

Step-by-step solution

- Understand the Concept

The partial pressure of each gas in a mixture is directly proportional to its mole fraction according to Dalton's Law of partial pressures. When equal masses of different gases are mixed, the number of moles of each gas present will depend on their respective molar masses.

- Calculate Moles of Each Gas

To find the moles of each gas, we use the formula: number of moles (n) = mass (m) / molar mass (M). The molar mass of hydrogen (H2) is 2 g/mol, and ethane (C2H6) is 30 g/mol. Since the masses are equal and the number of moles is mass divided by molar mass, the number of moles of hydrogen will be greater because of its lower molar mass.

- Calculate Mole Fraction of Hydrogen

The mole fraction of hydrogen (X_H2) is calculated as the moles of hydrogen (n_H2) divided by the total moles of gas (n_total). Assuming the mass of each gas is 1 g for simplicity, n_H2 = 1 g / 2 g/mol = 0.5 moles and n_C2H6 = 1 g / 30 g/mol ≈ 0.033 moles. Therefore, n_total = n_H2 + n_C2H6 = 0.5 moles + 0.033 moles ≈ 0.533 moles, and X_H2 = n_H2 / n_total = 0.5 moles / 0.533 moles ≈ 0.938.

- Calculate Partial Pressure Contribution

The partial pressure of hydrogen (P_H2) would be its mole fraction times the total pressure. Since the total pressure will be the sum of the pressures of hydrogen and ethane, and we are looking for the fraction of pressure exerted by hydrogen, we have already calculated it in the form of its mole fraction since P_H2 = X_H2 * P_total.

- Determine the Correct Answer

The fraction of the total pressure exerted by hydrogen is approximately 0.938, which is very close to 15/16 when converted into a fraction. Thus, the closest answer choice is (d) 15:16.

Key concepts

Partial Pressure

Imagine you have a mixture of gases, each one exerting its own individual pressure as if it were alone in the container. This independent pressure is what we call partial pressure. According to Dalton's Law of partial pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. It's quite interesting because gases behave independently even when mixed, although they can also interact. When studying gas mixtures, recognizing how each component contributes to the overall pressure is crucial. This concept is not only fundamental in chemistry but also in fields like meteorology and respiratory physiology.

Understanding Dalton's Law in Everyday Life

Consider a balloon filled with helium and oxygen. The pressure inside keeps it inflated due to both gases exerting their pressures independently. If you could measure each gas's pressure separately, that would be their partial pressure. But what you feel as the balloon's firmness is the combined effect, the total pressure.

Mole Fraction

The mole fraction is a way of expressing the concentration of a component in a mixture. It's a ratio of the number of moles of a particular substance to the total number of moles in the mixture. We use the symbol X followed by a subscript to denote the substance we're referring to. In essence, the mole fraction is a dimensionless number that tells us what fraction of the whole mixture a particular component represents—like a slice of the pie in the grand dessert that is the chemical mixture. By knowing the mole fractions, you can calculate partial pressures and predict how a gas will behave under certain conditions.

Calculating Mole Fraction

To calculate the mole fraction, simply divide the number of moles of the component of interest by the total number of moles in the mixture. It's a straightforward calculation, but one that's extremely valuable for predicting how a gas mixture will respond to changes in conditions, such as temperature or volume.

Molar Mass

The molar mass is a property that tells us the mass of one mole of a substance. It's typically expressed in grams per mole (g/mol) and can be found on the periodic table for each element as its atomic weight. For compounds, you'd sum the atomic weights of each constituent element according to its molecular formula to find the molar mass. This concept is a bridge between the macroscopic and microscopic worlds, enabling us to count out atoms and molecules by weighing them. It's what allows chemists to make precise recipes for their reactions, ensuring that the proportions of reactants are just right for the desired product.

The Role in Calculating Moles

Molar mass becomes incredibly useful when you need to convert between mass and moles, a common task in chemistry. By dividing the total mass of substance by its molar mass, you find the number of moles present—like converting a number of apples to bushels, if apples were atoms and bushels were moles. This conversion is critical for applying Dalton's Law, as we need to know the mole fraction to determine each gas's partial pressure.