Question

At what temperature is the kinetic energy of a gas molecule double that of its value at \(27^{\circ} \mathrm{C}\) (A) \(54^{\circ} \mathrm{C}\) (B) \(108^{\circ} \mathrm{C}\) (C) \(327^{\circ} \mathrm{C}\) (D) \(300^{\circ} \mathrm{C}\)

Short answer

The temperature at which the kinetic energy of a gas molecule is double that of its value at 27°C is approximately 327°C. The correct answer is (C) \(327^{\circ} \mathrm{C}\).

Step-by-step solution

Understand the relationship between kinetic energy and temperature

A gas molecule's kinetic energy (represented by K) is directly proportional to its temperature (represented by T), which is given by the following formula: \[ K = \frac{3}{2} kT \] where k is the Boltzmann constant.

Set up an equation for the doubled kinetic energy

We are asked to find the temperature when the kinetic energy is double that at 27°C. Let's denote the initial temperature (27°C) as \(T_1\) and the final temperature (which we need to find) as \(T_2\). First, we must convert the given temperature to Kelvin (K) by adding 273.15: \[T_1 = 27^{\circ} C + 273.15 \Rightarrow T_1 = 300.15 K\] Since we know that the kinetic energy at the final temperature, let's call it \(K_2\), is double the kinetic energy at the initial temperature, \(K_1\), we can write that as: \[K_2 = 2 \times K_1\] Now substituting the formula of kinetic energy from step 1: \[\frac{3}{2} kT_2 = 2 \times \frac{3}{2} kT_1\]

Solve for the final temperature

Now, we can cancel out the constants in the equation, which leaves us with: \[T_2 = 2 \times T_1\] Substitute the value of \(T_1\) we found earlier: \[T_2 = 2 \times 300.15\] \[T_2 = 600.3 K\] To convert the temperature back to Celsius, subtract 273.15: \[T_2 = 600.3 - 273.15 = 327.15^{\circ} C\] Thus, the temperature at which the kinetic energy of a gas molecule is double that of its value at 27°C is approximately 327°C. So, the correct answer is: (C) \(327^{\circ} \mathrm{C}\)

Key concepts

Temperature and Kinetic Energy Relationship

The kinetic energy of gas molecules is intimately tied to temperature, forming the basis of many thermodynamic principles. In particular, kinetic energy, symbolized as \( K \), is directly proportional to temperature, \( T \). This relationship is formalized in the equation:\[ K = \frac{3}{2} kT \]where \( k \) is the Boltzmann constant, a fundamental physical constant. In simpler terms:- As temperature increases, the average kinetic energy of gas molecules increases.
- This means molecules move faster at higher temperatures.
The equation shows that kinetic energy scales directly with temperature; double the temperature means double the average kinetic energy of the molecules. Whether dealing with a scientific experiment or a practical problem, understanding this relationship helps predict how gases will behave under different thermal conditions.

Boltzmann Constant

The Boltzmann constant, represented by \( k \), is a bridge between macroscopic and microscopic physics. It connects the average kinetic energy of particles in a gas with the temperature of the gas. In the equation \( K = \frac{3}{2} kT \), the Boltzmann constant helps provide a quantitative measure of this relationship.
Some key points about the Boltzmann constant include:- Its value is approximately \(1.38 \times 10^{-23} \text{J/K} \).
- It allows us to translate between the kinetic energy of individual molecules and the temperature of the gas.
- It’s crucial in statistical mechanics, helping us understand systems in terms of probability distributions of microscopic states.
Overall, the Boltzmann constant is vital for understanding how molecular motion translates into observable thermal properties.

Celsius to Kelvin Conversion

Temperature is commonly measured in Celsius, but in the world of physics, Kelvin is often used because it is directly related to thermodynamic principles without negative temperatures. Converting between these two scales is straightforward:- To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature, since absolute zero is at \(-273.15\, ^{\circ} C\): \[ T_K = T_{°C} + 273.15 \]- For instance, when converting \(27^{\circ} C\) to Kelvin, add 273.15 to get \(300.15 K\).
Understanding this conversion is crucial in scientific calculations where absolute temperatures without negative values are required.The Kelvin scale's zero point represents absolute zero, the theoretical point where molecular motion ceases. This scale is fundamental in thermodynamics and helps simplify many equations, such as those involving kinetic energy.