Question

At what temperature pressure remaining constant will the \(\mathrm{rms}\) speed of a gas molecules increase by \(10 \%\) of the \(\mathrm{rms}\) speed at NTP? (A) \(57.3 \mathrm{~K}\) (B) \(57.3^{\circ} \mathrm{C}\) (C) \(557.3 \mathrm{~K}\) (D) \(-57.3^{\circ} \mathrm{C}\)

Short answer

The temperature at which the rms speed of gas molecules increases by 10% of the rms speed at NTP, with pressure remaining constant, is approximately \(57.3^{\circ}\mathrm{C}\) (Option (B)).

Step-by-step solution

Write down the rms speed formula

The formula for the root-mean-square (rms) speed of a gas molecule is given by: \[v_{rms} = \sqrt{\frac{3kT}{m}}\] where \(v_{rms}\) is the rms speed, \(k\) is the Boltzmann constant, \(T\) is the temperature in Kelvin, and \(m\) is the molar mass of a gas molecule.

Find the expression relating rms speeds at different temperatures

We need to find the temperature at which the rms speed increases by 10% compared to its value at NTP. Let \(T_1\) be the NTP temperature (273.15 K), and \(T_2\) be the temperature where the rms speed increases by 10%. We can write the rms speed at \(T_1\) and \(T_2\) as: \[v_{1} = \sqrt{\frac{3kT_1}{m}}\] \[v_{2} = \sqrt{\frac{3kT_2}{m}}\] Since \(v_2 = 1.1 \times v_1\), we can divide the above two equations to eliminate \(m\) and \(k\): \[\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}}\]

Solve for the temperature T2

Now, plug in the value \(v_2 = 1.1 \times v_1\) into the equation from step 2 and solve for \(T_2\): \[1.1 = \sqrt{\frac{T_2}{273.15}}\] Square both sides of the equation to get rid of the square root: \[(1.1)^2 = \frac{T_2}{273.15}\] Now, solve for \(T_2\): \(T_2 = (1.1)^2 \times 273.15\) \(T_2 \approx 330.15 \mathrm{~K}\)

Convert the temperature to Celsius

To find the temperature in Celsius, subtract 273.15 from the Kelvin temperature: \(T_2^{\circ}\mathrm{C} = T_2 \mathrm{~K} - 273.15\) \(T_2^{\circ}\mathrm{C} \approx 57 \degree \mathrm{C}\)

Choose the correct answer

With the temperature at which the rms speed increases by 10% calculated, we can compare it to the given options and find that the correct answer is: (B) \(57.3^{\circ} \mathrm{C}\)

Key concepts

Root-Mean-Square Speed

The Root-Mean-Square Speed (often abbreviated as RMS speed) is a key concept in the Kinetic Theory of Gases. It provides a measure of the speed of particles in a gas. This speed is crucial because it is directly related to the energy that gas molecules inherit from their thermal environment.
In mathematical terms, the RMS speed is defined by the equation:\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]where:

• \(v_{rms}\) is the root-mean-square speed.
• \(k\) is Boltzmann's constant.
• \(T\) is the absolute temperature in Kelvin.
• \(m\) is the molar mass of the gas molecule.

This formula shows that RMS speed is directly proportional to the square root of the temperature of the gas. Hence, if the temperature increases, the RMS speed increases as well. Understanding this relationship is essential to interpreting how gases behave under varying thermal conditions.

Kinetic Theory of Gases

The Kinetic Theory of Gases provides a comprehensive model for predicting the behavior of gases. It is based on several key assumptions that help simplify the complex nature of gas molecules.

• Gas molecules are considered to be point particles that are constantly moving and colliding with each other and the walls of the container.
• The collisions are elastic, meaning no energy is lost; rather, energy is exchanged between the molecules or conserved within the system.
• The average kinetic energy of gas molecules is directly proportional to the absolute temperature of the gas.
• There are no attractive or repulsive forces between the molecules, simplifying the interactions to ideal conditions.

These assumptions help derive crucial equations for gas behavior, such as the formula for RMS speed. They also form the basis for understanding other phenomena in gases, such as pressure and temperature changes. In reality, not all gases perfectly adhere to these assumptions, but the Kinetic Theory provides an excellent approximation for gases under many conditions.

Temperature and Gas Laws

Temperature is a fundamental variable in the study of gases, closely tied to gas laws such as Boyle's Law, Charles's Law, and Avogadro's Law. Understanding these relationships is pivotal for predicting how a gas will behave under various conditions.
**Temperature and Energy**: The temperature of a gas is a direct indicator of the average kinetic energy of its molecules. Higher temperatures equate to higher energies and, consequently, faster moving particles. This energy translation underpins the RMS speed equation. **Gas Laws Overview**:

• Boyle's Law describes how gas pressure decreases as volume increases when temperature is held constant.
• Charles's Law states that gas volume expands with increased temperature at constant pressure.
• Avogadro's Law explains how equal volumes of gases at the same temperature and pressure contain the same number of molecules.

Recognizing the interplay between these principles helps in solving real-world problems, such as determining the temperature required to achieve a certain change in pressure or volume, as seen in the problem of increasing RMS speed by a specific percentage. By applying these laws, we can gain a deeper understanding of gas behavior and its thermodynamic properties.