Problem 107 - Chemistry and Chemical Reactivity

Question

If the absolute temperature of a gas doubles, by how much does the rms speed of the gaseous molecules increase?

Step-by-Step Solution

Answer

The rms speed increases by a factor of \( \sqrt{2} \) when the temperature doubles.

Step 1: Understanding the Formula

The root-mean-square (rms) speed of gas molecules is given by the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( v_{rms} \) is the rms speed, \( k \) is the Boltzmann constant, \( T \) is the absolute temperature, and \( m \) is the mass of a gas molecule. We are asked to find how much the rms speed increases when the temperature doubles.

Step 2: Initial Situation: Original RMS Speed

Let's denote the initial temperature as \( T_1 = T \). The rms speed at this temperature is \( v_{rms_1} = \sqrt{\frac{3kT_1}{m}} = \sqrt{\frac{3kT}{m}} \).

Step 3: New Situation: Temperature Doubled

Now, let the temperature be doubled, so \( T_2 = 2T_1 = 2T \). The new rms speed is given by the formula \( v_{rms_2} = \sqrt{\frac{3kT_2}{m}} = \sqrt{\frac{3k(2T)}{m}} \).

Step 4: Calculate the Increase in RMS Speed

Plug the new temperature into the rms speed formula: \( v_{rms_2} = \sqrt{\frac{6kT}{m}} \). Now, simplify this: \( v_{rms_2} = \sqrt{2} \times \sqrt{\frac{3kT}{m}} = \sqrt{2} \times v_{rms_1} \).

Step 5: Conclusion: Comparison Between Speeds

Hence, when the temperature doubles, the rms speed of the gas molecules increases by a factor of \( \sqrt{2} \) compared to the original speed.

Key Concepts

Boltzmann constant

The Boltzmann constant is vital in the study of thermodynamics and statistical mechanics. It's a bridge between macroscopic and microscopic sciences, relating the average kinetic energy of particles in a gas to temperature.

• Denoted by \( k \), it is a fundamental physical constant.
• Its value is approximately \(1.38 \times 10^{-23} \text{ J/K} \).
• It appears in several important equations, including the ideal gas law and the root-mean-square speed of gas molecules.

This constant aids in understanding how temperature at a macroscopic level impacts the energy of individual gas molecules. By using the Boltzmann constant, we connect thermal energy on a large scale to the energy of individual atoms or molecules.

Absolute Temperature

Absolute temperature is a temperature measurement that is taken using the Kelvin scale. This scale is based on absolute zero, the theoretical point where particles have minimum thermal motion.

• It is symbolized by \( T \) in equations.
• The absolute temperature is essential for accurate scientific calculations and thermodynamic processes.

When studying gases, absolute temperature allows for a precise understanding of how energy is distributed among molecules. In the context of the root-mean-square speed, absolute temperature directly influences the speed of gas molecules.

Gas Molecules

Gas molecules are small, freely moving particles that make up gases. Their movement is random and chaotic, and they occupy the container they are in.

• The speed of gas molecules is related to their kinetic energy and temperature.
• Increased temperature generally leads to increased molecular speed.

The root-mean-square (rms) speed calculated for gas molecules represents an average speed considering their kinetic energy. Understanding gas molecules' behavior helps explain various phenomena, like pressure and diffusion.

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental theory in physics that explains gas properties based on the motion of its molecules.

• Assumes that gas consists of a large number of small particles in constant random motion.
• Helps to link macroscopic properties like pressure and temperature to microscopic phenomena.

According to this theory, the temperature of a gas is a measure of the average kinetic energy of its molecules. This idea is crucial when examining how changes in temperature, like doubling it, affect the speed of gas molecules, as seen in the root-mean-square speed formula.