Problem 88 - Master Resource Book in JEE Main Physics

Question

If a given mass of gas occupies a volume of \(100 \mathrm{cc}\) at 1 atm pressure and temperature of \(100^{\circ} \mathrm{C}(373.15 \mathrm{~K})\). What will be its volume at 4 atm pressure; the temperature being the same? (a) \(100 \mathrm{cc}\) (b) \(400 \mathrm{cc}\) [c) \(25 \mathrm{cc}\) (d) \(104 \mathrm{cc}\)

Step-by-Step Solution

Answer

The volume at 4 atm is 25 cc (option c).

Step 1: Understand the Given Conditions

We are given that a gas occupies a volume of \(100 \ \text{cc}\) at a pressure of \(1 \ \text{atm}\) and a temperature of \(373.15 \ \text{K}\). We need to find out its volume at a pressure of \(4 \ \text{atm}\) while maintaining the same temperature.

Step 2: Use Boyle's Law

Since the temperature remains constant, we apply Boyle's Law, which states \( P_1V_1 = P_2V_2 \). Here, \(P_1 = 1 \ \text{atm}\), \(V_1 = 100 \ \text{cc}\), and \(P_2 = 4 \ \text{atm}\). We need to find \(V_2\), the new volume.

Step 3: Substitute Values into the Boyle's Law Equation

Substitute the given values into the equation: \(1 \ \text{atm} \times 100 \ \text{cc} = 4 \ \text{atm} \times V_2\).

Step 4: Solve for V2

Rearrange the equation to solve for \(V_2\): \[ V_2 = \frac{1 \ \text{atm} \times 100 \ \text{cc}}{4 \ \text{atm}} = 25 \ \text{cc}\].

Step 5: Match the Solution with the Given Options

Compare the calculated volume of \(25 \ \text{cc}\) with the provided options. The correct answer is option (c) \(25 \ \text{cc}\).

Key Concepts

Gas Laws

Gas laws are fundamental principles in science that describe the behavior of gases. These laws help us understand how gases react under different conditions like pressure, volume, and temperature. The three primary gas laws are Boyle's Law, Charles's Law, and Avogadro's Law. Each of these laws highlights a distinct aspect of gas behavior.
Boyle's Law focuses on the relationship between pressure and volume when the temperature is held constant. Charles's Law deals with the volume-temperature relationship at constant pressure. Avogadro’s Law explains how the volume of a gas relates to the quantity of gas when the pressure and temperature are constant.
Understanding these gas laws elucidates the broader gas behaviors, providing predictions and calculations for real-world applications. Scientists and engineers rely on these laws for designing systems that involve gas, such as breathing apparatus or vehicle engines, ensuring efficiency and safety.

Pressure-Volume Relationship

The pressure-volume relationship, as described by Boyle's Law, is a key concept for understanding how gases behave. Boyle's Law states that the pressure of a gas is inversely proportional to its volume when temperature is kept constant. Mathematically, this can be represented as:\[ P_1 V_1 = P_2 V_2 \]where:

• \( P_1 \) and \( V_1 \) are the initial pressure and volume, respectively.
• \( P_2 \) and \( V_2 \) are the final pressure and volume, respectively.

This inverse relationship means that as one variable increases, the other decreases. For example, if you increase the pressure on a gas, its volume decreases, provided the temperature does not change.
This interdependence is particularly important in various technological and natural processes. For example, it explains how human lungs operate during breathing—by altering pressure to change lung volume—which is essential for gas exchange.

Constant Temperature Process

Constant temperature processes, or isothermal processes, happen when the temperature of a system remains unchanged. In thermodynamics, maintaining a constant temperature in a process allows us to apply certain gas laws, like Boyle's Law, to find relationships between other properties such as pressure and volume.
When dealing with a constant temperature, any change in pressure applied to gas results in a corresponding change in volume, and vice versa, without altering the internal energy due to heat. This simplification is valuable for calculations in physics and engineering as it isolates variables, making it easier to predict how gases will behave under specific conditions.
Recognizing when conditions allow for a constant temperature assumption helps solve problems accurately without unnecessary complexity. This discipline is crucial in many scientific fields, including chemistry, where reactions often require precise environmental control.