Problem 59 - Chemistry and Chemical Reactivity

Question

You have a sample of helium gas at \(-33^{\circ} \mathrm{C}\) and you want to increase the rms speed of helium atoms by \(10.0 \% .\) To what temperature should the gas be heated to accomplish this?

Step-by-Step Solution

Answer

Heat the helium gas to approximately 17.43°C.

Step 1: Understand Root Mean Square (RMS) Speed

The root mean square speed \( v_{rms} \) of gas molecules is given by the formula: \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( m \) is the mass of a gas molecule. We need to find the new temperature \( T_2 \) for which the rms speed is increased by 10%.

Step 2: Calculate Initial RMS Speed

Start with the initial temperature \( T_1 = -33^{\circ}C = 240.15 \, K \). Use the RMS speed formula in terms of \( T_1 \):\[ v_{rms1} = \sqrt{\frac{3kT_1}{m}}. \]

Step 3: Determine New RMS Speed

To achieve a 10% increase in the RMS speed:\[ v_{rms2} = 1.1 \times v_{rms1} = \sqrt{\frac{3kT_2}{m}}. \]

Step 4: Set Up the Equation for RMS Speed Increase

Substitute \( v_{rms2} \) into the equation:\[ 1.1 \times \sqrt{\frac{3kT_1}{m}} = \sqrt{\frac{3kT_2}{m}}. \]

Step 5: Solve for New Temperature \( T_2 \)

Square both sides to eliminate the square roots:\[ (1.1)^2 \times \frac{3kT_1}{m} = \frac{3kT_2}{m}. \]Simplify the equation by canceling \( \frac{3k}{m} \):\[ (1.1)^2 \times T_1 = T_2. \]Calculate \( (1.1)^2 \approx 1.21 \), then:\[ T_2 = 1.21 \times T_1 = 1.21 \times 240.15 \, K \approx 290.58 \, K. \]

Step 6: Convert the Temperature Back to Celsius

To convert from Kelvin to Celsius:\[ T_{2(°C)} = T_2 - 273.15. \]\[ T_{2(°C)} = 290.58 \, K - 273.15 = 17.43 \, ^{\circ}C. \]

Key Concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory explains the behavior of gases, allowing us to predict properties like pressure, volume, and temperature. At its core, this theory states that gas molecules are in constant, random motion and that their speed changes with temperature shifts.
When a gas is heated, its molecules gain kinetic energy. This means their velocity, including the root mean square (RMS) speed, increases. The equation for RMS speed, \( v_{rms} = \sqrt{\frac{3kT}{m}} \), reveals that higher temperatures result in faster-moving gas particles, if the mass \( m \) remains constant.

Key points of the Kinetic Molecular Theory include:

• Gas molecules are in ceaseless, straight-line motion.
• Collisions between molecules and the walls of the container cause pressure.
• The average kinetic energy of gas molecules is directly proportional to the absolute temperature.

This understanding helps us predict how heating helium gas will affect its molecular motion and RMS speed.

Temperature Conversion

Temperature is a fundamental measure in physics that influences the behavior and properties of gases. In scientific calculations, it's usually measured in Kelvin because it starts at absolute zero, providing a true measure of thermal motion.
Converting between Celsius and Kelvin is straightforward:

• To convert Celsius to Kelvin, add 273.15.
• To convert Kelvin to Celsius, subtract 273.15.

For example, in the exercise, the initial temperature of helium gas is provided in Celsius. Converting \(-33^{\circ} \mathrm{C}\) to Kelvin, we get \( 240.15 \, K \). After computing for a new temperature in Kelvin, we can convert back if needed to understand it in everyday terms.
This conversion is crucial, especially when calculating gas behaviors, due to the temperature's direct relationship with energy levels.

Helium Gas Properties

Helium, a noble gas, is colorless, odorless, and tasteless. Its molecules are small and light, contributing to helium's characteristic low density and high diffusion rate. Unlike many gases, helium remains chemically inert, not easily reacting with other elements.
These properties affect its diffusion speed and, consequently, its RMS speed. Helium's low atomic mass makes it an excellent example when studying molecular speeds, as its atoms can reach relatively high velocities at the same temperature compared to heavier gases.
The behavior of helium under different temperatures follows the principles of the Kinetic Molecular Theory, allowing students to precisely calculate changes in speed, like the 10% increase required in the given exercise. Understanding helium's unique traits enhances our comprehension of gas behavior in practical and experimental settings.