Problem 31 - A Complete Resource Book in Chemistry for JEE Main

Question

At \(25^{\circ} \mathrm{C}\) and \(730 \mathrm{~mm}\) pressure, \(380 \mathrm{~mL}\) of dry oxygen was collected. If the temperature is constant, what volume will the oxygen occupy at \(760 \mathrm{~mm}\) pressure? (a) \(365 \mathrm{~mL}\) (b) \(2 \mathrm{~mL}\) (c) \(10 \mathrm{~mL}\) (d) \(20 \mathrm{~mL}\)

Step-by-Step Solution

Answer

The oxygen will occupy 365 mL at 760 mm pressure.

Step 1: Understand Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the volume of a gas is inversely proportional to its pressure. The formula is expressed as \( P_1 V_1 = P_2 V_2 \), where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume.

Step 2: Identify Known Values

From the problem statement, we know that the initial volume \(V_1\) is 380 mL, the initial pressure \(P_1\) is 730 mm, and the final pressure \(P_2\) is 760 mm. We need to find the final volume \(V_2\).

Step 3: Rearrange Boyle's Law Equation

To find \(V_2\), rearrange the equation \( P_1 V_1 = P_2 V_2 \) to solve for \(V_2\). The rearranged equation is \( V_2 = \frac{P_1 V_1}{P_2} \).

Step 4: Substitute Known Values into Equation

Substitute the known values into the equation: \( V_2 = \frac{730 \, \text{mmHg} \times 380 \, \text{mL}}{760 \, \text{mmHg}} \).

Step 5: Calculate Final Volume

Perform the calculation: \( V_2 = \frac{730 \, \text{mmHg} \times 380 \, \text{mL}}{760 \, \text{mmHg}} = 364.47 \, \text{mL} \). Round this value to the nearest whole number, which results in \(V_2 = 365 \, \text{mL}\).

Key Concepts

Gas Volume Calculation

Calculating the volume of a gas under different conditions is crucial when working with gases. It involves determining how much space a specific amount of gas will occupy at given pressure and temperature. In real-world applications, the number of gas molecules remains unchanged, but their volume can vary due to changes in pressure or temperature.

When calculating gas volume, it's important to understand the initial conditions, such as the beginning volume, pressure, and temperature. Using mathematical formulas like Boyle's Law can help us predict the behavior of gases under varying pressures when the temperature remains constant.

Pressure-Volume Relationship

In the study of gases, understanding the connection between pressure and volume is essential. This relationship is beautifully captured by Boyle's Law, which states that the volume of a gas is inversely proportional to its pressure when the temperature is held constant.

This means that as pressure increases, the volume decreases, and vice versa, as long as the amount of gas and temperature do not change.

• Boyle's Law formula: \( P_1 V_1 = P_2 V_2 \)
• If you increase the pressure on a gas, the volume will decrease.
• If the pressure decreases, the volume of the gas will increase.

The exercise we have here directly applies this principle to find the volume of gas when only the pressure changes. By keeping the temperature constant, it isolates pressure and volume, clearly showing how they influence each other.

Ideal Gas Law

The Ideal Gas Law offers a broader view of the behavior of gases by connecting pressure, volume, and temperature. The law is often represented by the equation \(PV = nRT\), where:

• \(P\) is the pressure
• \(V\) is the volume
• \(n\) is the number of moles of the gas
• \(R\) is the gas constant
• \(T\) is the temperature in Kelvin

The Ideal Gas Law helps us understand complex gas behaviors when all three factors—pressure, volume, and temperature—are in play. Unlike Boyle's Law, which only looks at pressure and volume at a constant temperature, the Ideal Gas Law allows us to examine scenarios where all parameters might change.Although Boyle's Law is a part of the collective understanding established by the Ideal Gas Law, it is primarily used when the number of moles and temperature remain constant. By looking at gas laws holistically, students can gain a comprehensive understanding of how gases operate under various conditions.