Problem 15 - A Complete Resource Book in Chemistry for JEE Main

Question

At what Centigrade temperature will be the volume of a gas at \(0^{\circ} \mathrm{C}\) double of itself, when pressure remains constant? (a) \(0^{\circ} \mathrm{C}\) (b) \(273^{\circ} \mathrm{C}\) (c) \(273 \mathrm{~K}\) (d) \(546 \mathrm{~K}\)

Step-by-Step Solution

Answer

The temperature at which the volume of the gas doubles is \(273^{\circ} \mathrm{C}\).

Step 1: Identify the Problem

The problem asks us to find the temperature in degrees Celsius at which the volume of a gas doubles, given that the starting temperature is \(0^{\circ} \mathrm{C}\) and the pressure remains constant.

Step 2: Apply Charles's Law

According to Charles's Law, the volume of a gas is directly proportional to its temperature when the pressure is held constant. Mathematically, this can be expressed as \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\).

Step 3: Insert Known Values

Based on the given problem, \(V_2 = 2V_1\), \(T_1 = 0^{\circ} \mathrm{C} = 273 \mathrm{~K}\). Thus, the equation becomes \(\frac{V_1}{273} = \frac{2V_1}{T_2}\).

Step 4: Solve for \(T_2\)

Simplify the equation to solve for \(T_2\). Cancel \(V_1\) and multiply both sides by \(T_2\), giving \(T_2 = 2 \times 273 \). So, \(T_2 = 546 \mathrm{~K}\).

Step 5: Convert K to °C

Subtract 273 from \(546 \mathrm{~K}\) to convert the temperature to degrees Celsius: \(546 - 273 = 273^{\circ} \mathrm{C}\).

Key Concepts

Understanding Gas Laws

Gas laws describe the behavior of gases under varying conditions of pressure, volume, and temperature. One of the key insights offered by gas laws is how gases expand and contract with changes in temperature when pressure is kept constant. Charles's Law, named after the French scientist Jacques Charles, is particularly important because it helps us understand the direct relationship between the volume of a gas and its temperature.

• Charles's Law: This law states that the volume of a gas is directly proportional to the absolute temperature (in Kelvin), provided the pressure remains unchanged. The mathematical expression for Charles's Law is \(rac{V_1}{T_1} = rac{V_2}{T_2}\), where \(V_1\) and \(V_2\) are the initial and final volumes, and \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin respectively.
• Application: Charles's Law is useful in predicting how gases will behave under specific thermal conditions, which is crucial in many areas such as chemistry, physics, and engineering. The law assists in calculating one of the four major variables (pressure, volume, temperature, or amount), given the other three.

Temperature Conversion Made Easy

When working with gases, understanding how to convert temperatures between different scales is essential. The most common temperature scales used in scientific contexts are Celsius and Kelvin. Each of these scales has specific applications based on the scenarios.

• Celsius to Kelvin: The Celsius scale (°C) is often used in everyday contexts, but in scientific calculations involving gases, Kelvin (K) is preferred because it is an absolute temperature scale. To convert temperatures from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature: \(T(K) = T(°C) + 273.15\).
• Kelvin to Celsius: Similarly, converting Kelvin back to Celsius involves subtracting 273.15 from the Kelvin temperature: \(T(°C) = T(K) - 273.15\).
• Why Kelvin? The Kelvin scale is vital for gas law calculations because it starts at absolute zero, where all molecular motion ceases. This starting point makes calculations involving gases much simpler and avoids negative temperatures which could complicate mathematical relationships.

Exploring the Volume-Temperature Relationship

The relationship between volume and temperature is foundational in understanding how gases behave under various thermal conditions. According to Charles's Law, the volume of a gas changes proportionally with its temperature, assuming constant pressure.

• Direct Proportionality: As the temperature of a gas increases, its kinetic energy boosts, causing particles to move more vigorously and occupy more space, thus increasing the gas volume. The volume's increase is directly proportional to the increase in temperature measured in Kelvins.
• Example Calculation: In the scenario where a gas volume doubles, we apply Charles's Law. If a gas initially at a temperature of \(0^{\circ}C\) (or 273 K) has its volume doubled, we use the relationship \(T_2 = 2 \times 273\), resulting in \(T_2 = 546K\), which equates to \(273^{\circ}C\) upon converting back.
• Real-Life Applications: This relationship is crucial in various applications, such as designing airbags for vehicles, where understanding how gases expand with heat is essential for functionality. It also applies in meteorology for predicting weather patterns based on temperature changes.