Question

If the rms speed of He atoms in the exosphere (highest region of the atmosphere) is \(3.53 \times 10^{3} \mathrm{~m} / \mathrm{s}\), what is the temperature (in kelvins)?

Short answer

The temperature of the helium in the exosphere is approximately 1110 K.

Step-by-step solution

Understanding RMS Speed

RMS speed or Root Mean Square 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 \( (1.38 \times 10^{-23} \text{ J/K}) \), \( T \) is the temperature in Kelvin, and \( m \) is the mass of a single molecule in kilograms.

Converting Helium Atomic Mass

The atomic mass of helium is given in atomic mass units (amu). The atomic mass of helium is approximately \( 4 \text{ amu} \). To convert this to kilograms, use the relation: \( 1 \text{ amu} = 1.66 \times 10^{-27} \text{ kg} \). Therefore, the mass of one helium atom \( m \) in kilograms is: \[ m = 4 \times 1.66 \times 10^{-27} = 6.64 \times 10^{-27} \text{ kg} \]

Rearranging the RMS Speed Equation to Find Temperature

Re-arrange the RMS speed formula to solve for temperature \( T \): \[ T = \frac{m v_{rms}^2}{3k} \]

Substituting the Known Values

Substitute the given RMS speed \( v_{rms} = 3.53 \times 10^3 \text{ m/s} \), the mass of helium \( m = 6.64 \times 10^{-27} \text{ kg} \), and Boltzmann's constant \( k = 1.38 \times 10^{-23} \text{ J/K} \) into the equation: \[ T = \frac{(6.64 \times 10^{-27}) \times (3.53 \times 10^3)^2}{3 \times 1.38 \times 10^{-23}} \]

Calculating Temperature

Calculate \( T \) using the values substituted: \[ T = \frac{6.64 \times 10^{-27} \times 12.4609 \times 10^6}{4.14 \times 10^{-23}} \] Simplifying gives: \( T \approx 1.11 \times 10^3 \text{ K} \) or \( T \approx 1110 \text{ K} \).

Key concepts

Boltzmann constant

The Boltzmann constant, denoted as \( k \), is a fundamental, physical constant that bridges the gap between macroscopic and microscopic physics. It connects the average kinetic energy of particles in a gas with the temperature of the gas in kelvins. This constant has a value of \( 1.38 \times 10^{-23} \text{ J/K} \).

• Important for calculating properties of particles at different temperatures.
• Used in various equations in thermodynamics and statistical mechanics.

Without the Boltzmann constant, it would be challenging to understand how atomic-scale movements translate into measurable properties like temperature.

temperature calculation

Temperature calculation, especially in the context of gases, often requires using the root mean square (RMS) speed equation. In the case of the exercise, the temperature calculation involves determining the average speed of helium atoms at a given energy level. To find the temperature \( T \), the equation \( T = \frac{m v_{rms}^2}{3k} \) is used:

• \( v_{rms} \): RMS speed of the gas molecules.
• \( m \): Mass of a single molecule.
• \( k \): Boltzmann constant.

This formula helps prove that temperature is proportional to the average kinetic energy. So, by knowing the speed, you can backtrack to find how hot the gas might be.

helium atomic mass

The atomic mass is vital for determining many properties of an element, including its behavior as a gas. For helium, the atomic mass is approximately \( 4 \) atomic mass units (amu). However, to use it in equations like the RMS speed formula, it needs to be converted into kilograms:\[ m = 4 \times 1.66 \times 10^{-27} = 6.64 \times 10^{-27} \text{ kg} \]

• This step ensures the units align with the Boltzmann constant, which uses kilograms.
• It highlights the importance of unit conversion in scientific calculations.

Understanding atomic mass conversions is crucial in physics and chemistry, as it allows for correct calculations and interpretations.

root mean square speed

Root Mean Square (RMS) speed gives us insight into the velocity of gas particles. It is key in understanding the thermal properties of gases and provides a measure of the speed of particles regardless of their individual velocities. The formula for RMS speed is:\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]

• Shows the relationship between kinetic energy and temperature.
• Higher RMS speeds indicate hotter gases.
• Important for calculating molecular motion in different temperature conditions.

By rearranging this formula, as shown in the exercise, you can solve for temperature, demonstrating the interconnected nature of these physical properties.