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 factor is \( \frac{313}{293} \), corresponding to option (d).

Step 1: Understanding Kinetic Energy and Temperature Relationship

The average kinetic energy of a gas in an ideal gas scenario is directly proportional to its absolute temperature (Kelvin scale). The formula for kinetic energy is given by \( KE = \frac{3}{2}kT \) where \( k \) is Boltzmann's constant and \( T \) is the temperature in Kelvin. Thus, \( KE \propto T \) which implies that the kinetic energy changes in the same ratio as the temperature.

Step 2: Convert Temperatures to Kelvin

Since kinetic energy is dependent on temperature measured in Kelvin, convert the given Celsius temperatures to Kelvin:\[ T_1 = 20^{\circ}\text{C} = 20 + 273 = 293 \, \text{K} \]\[ T_2 = 40^{\circ}\text{C} = 40 + 273 = 313 \, \text{K} \]

Step 3: Determine Factor of Change in Temperature

Calculate the factor by which the temperature changes:\[ \text{Factor} = \frac{T_2}{T_1} = \frac{313}{293} \]This tells us by what factor the kinetic energy changes, because kinetic energy is directly proportional to temperature.

Step 4: Select Correct Option

From the calculation, the change in temperature factor which corresponds to the change in kinetic energy is \( \frac{313}{293} \). Therefore, the correct answer is option (d).

Key Concepts

Temperature Conversion

In many physics and chemistry problems, you'll often need to convert temperatures from Celsius to Kelvin. This is crucial when dealing with equations that require absolute temperature inputs, like those involving gases. The Kelvin scale starts at absolute zero, making it useful for scientific calculations. To convert from Celsius to Kelvin, simply add 273 to the Celsius temperature. For example:

• If you have a temperature of 20°C, the Kelvin equivalent would be 20 + 273 = 293 K.
• Similarly, for a temperature of 40°C, it becomes 40 + 273 = 313 K.

By using the Kelvin scale, you can more accurately calculate changes in physical properties, such as the kinetic energy of gases. Always remember: Kelvin is the baseline for thermodynamic calculations, so never forget that tiny +273 adjustment!

Ideal Gas

The ideal gas is a helpful model to understand behavior of gases under various conditions. It assumes that gases consist of a large number of small particles, which are constantly moving and colliding elastically. This model simplifies many real-world scenarios by neglecting intermolecular forces, making it very useful for calculations.Ideal gases follow several laws, one of which is that their average kinetic energy depends only on their temperature in Kelvin, not on their type or pressure. This makes calculations straightforward:

• Use the formula: \[ KE = \frac{3}{2}kT \]where KE is kinetic energy, k is Boltzmann's constant, and T is absolute temperature.
• Any changes in temperature directly affect kinetic energy according to these laws.

With ideal gases, you can predict how gases behave when temperature changes, aligning with the direct proportionality of kinetic energy to temperature.

Boltzmann's Constant

Boltzmann's constant, often represented by the symbol \( k \), is a fundamental constant in physics. It plays a key role in statistical mechanics by linking microscopic and macroscopic physical quantities. The constant is used in equations that relate the average kinetic energy of particles in a gas with the temperature of the gas.Its value is approximately \( 1.38 \times 10^{-23} \) joules per Kelvin (J/K). This small number represents the amount of energy per temperature interval for each gas particle.

• In the equation for kinetic energy, \( KE = \frac{3}{2}kT \), Boltzmann's constant helps calculate how energy fluctuates with temperature.
• It's vital for understanding how temperature affects particle motion within gases.

Having this constant bridges the gap between thermodynamic and molecular explanations of energy.

Proportionality Relationship

Understanding proportionality is crucial in physics, especially in dealing with gases. Proportionality refers to how two quantities change in relation to one another. If one quantity increases, the other does too, by the same factor.For ideal gases, this concept is seen in the relationship between temperature and kinetic energy:

• Kinetic energy is proportional to the gas’s absolute temperature, described by the equation \( KE \propto T \).
• This means that if the temperature of the gas doubles, its average kinetic energy also doubles.

In the problem given, as the temperature increases from 293 K to 313 K, the factor by which the temperature changes is \( \frac{313}{293} \).This same factor applies to the change in kinetic energy due to its direct proportionality to temperature, making understanding proportionality very useful in physics calculations.