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 a volume of 365 mL at 760 mm pressure, matching option (a).

Step 1: Identify the Known Values

First, we note the values given: initial pressure \( P_1 = 730 \; \text{mmHg} \), initial volume \( V_1 = 380 \; \text{mL} \), and new pressure \( P_2 = 760 \; \text{mmHg} \). The temperature remains constant, so we do not need to consider it in our calculations.

Step 2: Use Boyle's Law

Since the temperature is constant, we can use Boyle’s Law which states that \( P_1 V_1 = P_2 V_2 \), where \( V_2 \) is the volume at the new pressure. Rearranging the formula gives \( V_2 = \frac{P_1 V_1}{P_2} \).

Step 3: Substitute the Known Values

Insert the known values into the equation: \[ V_2 = \frac{730 \times 380}{760} \] and calculate \( V_2 \).

Step 4: Perform the Calculation

Calculate the expression \( \frac{730 \times 380}{760} = 365 \; \text{mL} \). Thus, \( V_2 = 365 \; \text{mL} \).

Step 5: Confirm the Answer

The volume \( V_2 = 365 \; \text{mL} \) matches option (a) \(365 \text{ mL}\). Therefore, the answer is choice (a).

Key Concepts

Understanding the Ideal Gas Law

The Ideal Gas Law is a cornerstone concept in chemistry and physics that relates the four main properties of a gas: pressure, volume, temperature, and the amount of gas, denoted by the number of moles. Represented by the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.
In the context of the Ideal Gas Law, it's crucial to remember:

• The equation represents a hypothetical gas called an "ideal gas" that perfectly follows the predicted behavior of the law.
• Real gases behave similarly to ideal gases at ordinary temperatures and pressures.
• This law helps predict the behavior of gases in different conditions, whether that involves changing temperatures, pressures, or volumes.

In scenarios where temperature is held constant, and only pressure and volume are of interest, principles derived from the Ideal Gas Law come into play, such as Boyle's Law.

Exploring Boyle's Law and the Pressure-Volume Relationship

Boyle's Law describes the inverse relationship between pressure and volume in gases at a constant temperature. If pressure increases, the volume decreases, and vice versa, which can be mathematically expressed as \( P_1V_1 = P_2V_2 \).
This concept is fundamental because:

• It helps understand how gases compress and expand, which is essential in various scientific and industrial applications.
• Boyle's Law is integral to understanding how breathing works, as lungs expand and contract to let air in and out.
• It is critical for systems where the temperature remains constant, like the scenario in which oxygen was collected in the exercise example.

The calculation of the new volume of oxygen at a different pressure but the same temperature illustrates Boyle's Law in action. Adjusting the initial pressure-volume product allows finding the new conditions by maintaining the constancy portrayed by the law.

Diving Deeper into Gas Laws

Gas laws encompass various relationships that define how gases behave under different conditions. Each law simplifies the element of gas behavior it focuses on, whether that's pressure, volume, temperature, or the number of moles.
Key gas laws include:

• Charles's Law: Relates volume and temperature, stating that the volume of a gas is directly proportional to its temperature when pressure is held constant.
• Avogadro's Law: Relates volume and the number of moles, asserting volumes are proportional to the number of gas molecules at a fixed temperature and pressure.
• Gay-Lussac's Law: Relates pressure and temperature, noting pressure is directly proportional to temperature for a fixed volume of gas.

Gas laws are crucial for predicting how gases will respond to changes in environmental conditions and form the basis for deeper exploration of thermodynamics and kinetic theory of gases. With the Ideal Gas Law and laws like Boyle's serving as foundational knowledge, students can predict and calculate the effects seen when manipulating one property of a gas while keeping others constant.