Question

A 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?

Short answer

Heat the gas to approximately 17.43°C.

Step-by-step solution

Finding initial RMS speed

The root mean square (RMS) speed of gas molecules is given by the formula: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \]where \(k\) is Boltzmann's constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of a gas molecule. First, we need to convert the initial temperature from Celsius to Kelvin: \( T_{initial} = -33 + 273.15 = 240.15 \, K \).

Calculate desired RMS speed

We want the final RMS speed to be 10% more than the initial RMS speed. If \( v_{initial} \) is the initial RMS speed, the final RMS speed \( v_{final} \) is:\[ v_{final} = 1.10 \times v_{initial} \]

Derive the relation between temperatures

Using the proportionality of RMS speed and the square root of temperature, we have:\[ \frac{v_{final}}{v_{initial}} = \sqrt{\frac{T_{final}}{T_{initial}}} \]Squaring both sides gives:\[ \left( \frac{v_{final}}{v_{initial}} \right)^2 = \frac{T_{final}}{T_{initial}} \]Substitute \( v_{final} = 1.10 \times v_{initial} \):\[ (1.10)^2 = \frac{T_{final}}{240.15} \]

Solve for final temperature

Now solve for \( T_{final} \):\[ 1.21 = \frac{T_{final}}{240.15} \]Thus:\[ T_{final} = 1.21 \times 240.15 \approx 290.58 \, K \].Finally, convert \( T_{final} \) back to Celsius: \( T_{final} = 290.58 - 273.15 = 17.43^{\circ}C \).

Key concepts

Boltzmann's constant

Boltzmann's constant, denoted as \( k \), is a fundamental constant which plays a crucial role in the field of statistical mechanics. Its value is approximately \( 1.38 \times 10^{-23} \) J/K. Boltzmann's constant links the microscopic scale of atoms and molecules to macroscopic physical quantities like temperature and energy.

In the context of the exercise, Boltzmann's constant appears in the formula for calculating the root mean square (RMS) speed of gas molecules: \[ v_{rms} = \sqrt{\frac{3kT}{m}} \]Here, \( k \) acts as a bridge connecting the temperature \( T \) to the kinetic energy of the gas particles. It explains how an increase in temperature leads to an increase in the particle speed.

This constant is key in understanding the energy distribution among particles within a gas, and how changes in temperature affect their motion.

Temperature conversion

Temperature conversion is a fundamental skill in solving problems that involve gases, as temperature directly affects the speed and kinetic energy of gas particles.

In the exercise, the temperature of the helium gas is initially given in Celsius, but needs to be converted to Kelvin to work with equations in thermodynamics. The conversion is simple: you add 273.15 to the Celsius temperature to get Kelvin:

• For the initial temperature: \(-33^{\circ}C + 273.15 = 240.15 \, K\)
• For the final temperature, after calculation: \(290.58 \, K - 273.15 = 17.43^{\circ}C\)

Kelvin is used in scientific formulas because it is an absolute scale, starting from absolute zero, the point where all molecular motion stops. This makes it a universal and practical scale for scientific calculations.

Kinetic theory of gases

The kinetic theory of gases is a model that helps explain the behaviors and properties of gases by assuming that gases are made up of a large number of small particles (atoms or molecules) in constant motion. This theory provides a comprehensive framework for understanding how gas particles interact with each other and their environment.

Some key postulates include:

• Gas particles are in constant, random motion.
• The volume of individual gas particles is negligible compared to the volume of the container.
• There are no forces between the particles except during collisions.
• Collisions between gas particles and with the walls of the container are perfectly elastic, meaning there is no loss of kinetic energy during collisions.

In the problem, this theory explains why the rms speed of gas particles is linked to temperature. As the temperature of a gas increases, so does the average kinetic energy of its particles, which in turn increases their speed. This relationship is mathematically embedded in the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), underscoring the theory's assumption that temperature is a measure of the average kinetic energy of the gas particles.

Understanding this theory is fundamental for predicting and explaining gas behavior in various conditions, such as changes in temperature or pressure.