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 approximately 365 mL at 760 mm pressure.

Step 1: Identifying the required law

The exercise requires us to determine the volume of a gas under different conditions of pressure. Since the temperature is constant, we will use Boyle's Law which states that the volume of a given amount of gas held at constant temperature varies inversely with the pressure. The mathematical expression for Boyle's Law is \( P_1 V_1 = P_2 V_2 \), where \( P_1 \) and \( V_1 \) are the initial pressure and volume, \( P_2 \) and \( V_2 \) are the final pressure and volume.

Step 2: Setting up the equation

Based on Boyle's Law, we set up the equation using the known values. We have initial conditions: \( P_1 = 730 \, \mathrm{mm} \) and \( V_1 = 380 \, \mathrm{mL} \). The final pressure \( P_2 \) is given as \( 760 \, \mathrm{mm} \). The volume \( V_2 \) is what we need to find. Thus, the equation becomes \( 730 \, \mathrm{mm} \times 380 \, \mathrm{mL} = 760 \, \mathrm{mm} \times V_2 \).

Step 3: Solving for the unknown volume

Rearrange the equation to solve for \( V_2 \):\[V_2 = \frac{730 \, \mathrm{mm} \times 380 \, \mathrm{mL}}{760 \, \mathrm{mm}}\]Calculate the above expression to find \( V_2 \).\[V_2 = \frac{277400}{760} \approx 365 \, \mathrm{mL}\]

Key Concepts

Pressure-Volume Relationship

In the study of gases, one of the most fundamental concepts is the pressure-volume relationship, famously described by Boyle's Law. This states that for a given mass of gas at a constant temperature, the volume of the gas is inversely proportional to its pressure. To put it simply, if the pressure of a gas increases, its volume decreases, provided the temperature remains unchanged. Conversely, if the pressure decreases, the gas volume increases.

This relationship is expressed mathematically as:

• \( P_1 V_1 = P_2 V_2 \)

Here, \( P_1 \) and \( V_1 \) represent the initial pressure and volume, respectively, while \( P_2 \) and \( V_2 \) represent the pressure and volume after a change has occurred.

In practical situations, understanding this relationship helps us predict how gases will behave when subject to changing pressures. For instance, in our original exercise, by applying this knowledge, we were able to calculate how much space or volume the gas would need when subjected to a new pressure.

Gas Laws

Gas laws are a series of fundamental principles that describe the behavior of gases in response to different variables such as pressure, volume, temperature, and the number of moles. These laws form the very foundation of chemistry and physics related to gases and are crucial in understanding and predicting gas behavior in various scenarios.

The most notable laws include:

• **Boyle's Law**: As previously mentioned, it describes the inverse relationship between pressure and volume at constant temperature.
• **Charles's Law**: Observes the direct relationship between volume and temperature at constant pressure; when temperature increases, so does the volume.
• **Avogadro's Law**: States that the volume of gas is directly proportional to the number of moles at constant temperature and pressure.
• **Ideal Gas Law**: Combines all previous laws into one equation \( PV = nRT \), encompassing pressure, volume, temperature, and number of moles (\(n\)) into one relationship.

These gas laws are vital in studying and working with gases, whether it be in laboratories or industrial applications. They help simplify complex gas behaviors into manageable calculations.

Temperature Constant

When referring to a 'constant temperature' situation in gas behavior studies, it implies that the temperature does not change throughout the experiment or calculation. This constancy is crucial because temperature significantly influences the properties of gases.

Boyle's Law, which deals with the pressure-volume relationship, relies on the assumption that the temperature remains fixed. If the temperature were to vary, it would introduce additional variables and complexity into the equation, which are considered in other gas laws like Charles's Law.

Keeping temperature constant simplifies calculations and predictions, as it isolates pressure and volume as the only changing factors. This is particularly helpful when conducting experiments or solving problems where only limited information is available and allows for focus on the direct relationship between pressure and volume without the need to account for heat exchange or other temperature effects.

In practical applications, maintaining a constant temperature can involve using thermal control systems or performing calculations assuming no temperature fluctuation for the sake of simplicity and instructional clarity. Understanding the pivotal role of constant temperature helps in effectively applying Boyle's Law to solve real-world and theoretical gas problems.