Problem 106 - A Complete Resource Book in Chemistry for JEE Main

Question

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

Step-by-Step Solution

Answer

The fraction of total pressure exerted by hydrogen is (d) 15:16.

Step 1: Determine Molar Masses

First, let's determine the molar masses of ethane (C2H6) and hydrogen (H2). Ethane (C2H6) has a molar mass of approximately 30 g/mol, and hydrogen (H2) has a molar mass of approximately 2 g/mol.

Step 2: Calculate Moles Present

Given equal weights, say 2 grams each, to calculate moles (moles = mass/molar mass), - Moles of ethane: \[ \frac{2}{30} = \frac{1}{15} \] mole - Moles of hydrogen: \[ \frac{2}{2} = 1 \] mole.

Step 3: Calculate Total Moles

Add the moles of both substances together:- Total moles = Moles of ethane + Moles of hydrogen = \( \frac{1}{15} + 1 = \frac{16}{15} \) moles.

Step 4: Determine Mole Fraction of Hydrogen

The mole fraction (X) of a component is the number of moles of that component divided by the total moles:- Mole fraction of hydrogen: \[ X_{H_2} = \frac{1}{\frac{16}{15}} = \frac{15}{16} \]

Step 5: Determine Fraction of Total Pressure Exerted by Hydrogen

Since the pressure is directly proportional to the mole fraction, the fraction of the total pressure exerted by hydrogen will also be \( \frac{15}{16} \).

Step 6: Choose the Correct Option

The fraction of the total pressure exerted by hydrogen is \( \frac{15}{16} \), which corresponds to option (d).

Key Concepts

Mole Fraction

The concept of mole fraction is a way to express the concentration of a component in a mixture. It tells us what portion of the entire mixture is made up of one particular substance. To calculate the mole fraction, you divide the number of moles of a particular substance by the total number of moles in the mixture. For instance, in our exercise, we have moles of both ethane and hydrogen. Once we know these values, we use them to calculate how much of the whole mixture is just hydrogen, for instance, by using the formula:

• Mole fraction \( X_{i} = \frac{n_{i}}{n_{\text{total}}} \)

Here, \( n_{i} \) is the moles of the component, and \( n_{\text{total}} \) is the total moles of all substances in the mixture. Understanding mole fraction helps us to figure out other things like partial pressures in gas mixtures.

Molar Mass

Molar mass is an important concept that tells us how many grams one mole of a substance weighs. Knowing this allows us to relate mass to number of moles, which we use to make calculations about reactions and concentrations in chemistry. For instance, ethane (\( \text{C}_2\text{H}_6 \)) has a molar mass of 30 g/mol, while hydrogen (\( \text{H}_2 \)) has a molar mass of about 2 g/mol. With this information, we can convert the mass of a gas to moles.

• To calculate moles when you have mass: \( \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \)

In our example, this is essential for determining how much of each gas contributes to the mixture. Without understanding molar masses, we wouldn't be able to move on to calculating things like mole fractions or pressures.

Total Pressure

When we talk about total pressure in a gas mixture, we mean the sum of pressures that all gases in the mixture exert. This is described by Dalton's Law of Partial Pressures. Each gas in a mixture behaves independently, but when we add them together, we get the total pressure inside the container. Dalton's Law states:

• \( P_{\text{total}} = P_{1} + P_{2} + \cdots + P_{n} \)

Where each \( P \) is the partial pressure of a gas in the mixture. The total pressure helps us figure out the behavior of the mixture as a whole and is often used with mole fractions to identify the contribution of each gas in the container. If you know the total pressure and the mole fraction of a gas, you can also find out that gas's partial pressure, further highlighting the interconnectedness of these properties.

Partial Pressure

Partial pressure is a key concept in understanding gas mixtures. It refers to the pressure that a single type of gas would exert if it were alone in a container. The idea is based on the fact that gases with different properties still share space and contribute to the overall behavior of a gas mixture.

• Formula for partial pressure: \( P_{i} = X_{i} \cdot P_{\text{total}} \)

Here, \( P_{i} \) represents the partial pressure of gas \( i \), \( X_{i} \) is its mole fraction, and \( P_{\text{total}} \) is the total pressure. In essence, the partial pressure tells us the individual pressure contribution of a gas based on its proportion in the mixture. For hydrogen in our example, knowing its mole fraction allowed us to deduce its partial pressure as part of understanding the overall gas mixture.