Problem 148 - A Complete Resource Book in Chemistry for JEE Main

Question

As the temperature is raised from \(20^{\circ} \mathrm{C}\) to \(40^{\circ} \mathrm{C}\), the average kinetic energy of neon atoms changes by a factor of which of the following? [2004] (a) \(1 / 2\) (b) 2 (c) \(\sqrt{3} 13 / 293\) (d) \(313 / 293\)

Step-by-Step Solution

Answer

The average kinetic energy changes by a factor of \( \frac{313}{293} \) (option d).

Step 1: Understanding Kinetic Energy and Temperature Relation

The average kinetic energy of a gas is directly proportional to the temperature of the gas in Kelvin. The formula for the average kinetic energy is given by \( KE = \frac{3}{2} k T \), where \( k \) is the Boltzmann constant and \( T \) is the temperature in Kelvin.

Step 2: Convert Celsius to Kelvin

To convert the given temperatures from Celsius to Kelvin, use the conversion formula: \( T(K) = T(^{\circ}C) + 273.15 \). Thus, \( T_1 = 20^{\circ}C = 20 + 273.15 = 293.15K \) and \( T_2 = 40^{\circ}C = 40 + 273.15 = 313.15K \).

Step 3: Calculate the Initial and Final Kinetic Energy

Since the kinetic energy \( KE \) is proportional to the temperature \( T \), we can express the ratios of the respective energies as \( \frac{KE_2}{KE_1} = \frac{T_2}{T_1} \). Substitute the values: \( \frac{KE_2}{KE_1} = \frac{313.15}{293.15} \).

Step 4: Simplify the Expression

The ratio simplifies to approximately \( \frac{313}{293} \) assuming rounding to integer values in the options corresponds to the correct choice. Thus, the change in kinetic energy is by a factor of \( \frac{313}{293} \), matching the option provided.

Key Concepts

Gas Laws

Gas laws help us understand the behavior of gas particles in different conditions, such as temperature, pressure, and volume. They form the foundation of concepts related to gases. Two important gas laws relevant here are:

• Charles's Law: This law states that the volume of a gas is directly proportional to its temperature when the pressure is held constant. Mathematically, it's expressed as \( V_1/T_1 = V_2/T_2 \). This relation highlights the effect temperature changes have on gas behavior, supporting the connection between temperature and kinetic energy.
• Gay-Lussac's Law: This law shows how pressure changes with temperature at constant volume. Expressed as \( P_1/T_1 = P_2/T_2 \), it helps us see the proportional relationship between pressure and temperature. Both Charles's and Gay-Lussac's Laws rely on temperatures being in Kelvin for calculations to maintain direct proportionality.

Understanding these relationships is key in physics and chemistry as it helps predict how changes in one property (like temperature) will affect others.

Temperature Conversion

Temperature conversion is essential when dealing with scientific formulas, especially when involving gas calculations. In scientific equations, temperature must always be in Kelvin. To convert temperatures from Celsius to Kelvin, the formula is: \[T(K) = T(\degree C) + 273.15\]
This ensures precision needed for scientific work. For example, converting \(20^{\circ} C\) to Kelvin gives us \(293.15\, K\), while \(40^{\circ} C\) gives us \(313.15\, K\). Kelvin is used because it starts at absolute zero, which is considered the lowest temperature attainable. This starting point provides a true zero point relative to kinetic energy, making it more suitable for formulas working with molecular energy.

Boltzmann Constant

The Boltzmann constant \( k \) is a fundamental physical constant that plays a critical role in statistical mechanics and thermodynamics. It relates the average kinetic energy of particles in a gas with the temperature of the gas. Expressed in Joules per Kelvin (J/K), it acts as a bridge between the macroscopic and microscopic worlds.
The formula for average kinetic energy is: \[KE = \frac{3}{2} k T\] Here, \( k \) takes its value approximately as \(1.38 \times 10^{-23} \, \text{J/K}\). This constant is crucial when calculating energy distributions, particularly when temperature changes. When you look at particles like neon atoms in a gas, knowing \( k \) helps determine the energy dynamics as you adjust the temperature, illuminating the changes in kinetic energy which mirrors changes in temperature. Thus, it serves an invaluable role in understanding the physical behavior of gas particles.