The root mean square (rms) speed of a particle in an ideal gas can be calculated using the formula:

$v_{rms} = \sqrt{\frac{3k_B T}{m}}$

Where:

• $k_B$ is the Boltzmann constant ($1.38 \times 10^{-23} \, \text{J/K}$),
• $T$ is the temperature ($9.02 \times 10^5 \, \text{K}$),
• $m$ is the mass of the particle ($9.11 \times 10^{-31} \, \text{kg}$).

Let's compute:

1. Calculate the numerator: $3k_B T$: $3k_B T = 3 \times (1.38 \times 10^{-23}) \times (9.02 \times 10^5)$

2. Divide by the mass $m$: $\frac{3k_B T}{m}$

3. Take the square root to find $v_{rms}$.

Let me calculate this for you. math

Constants

k_B = 1.38e-23 # Boltzmann constant in J/K T = 9.02e5 # Temperature in K m = 9.11e-31 # Mass of the electron in kg

Calculate the rms speed

v_rms = math.sqrt((3 * k_B * T) / m) v_rmsThe root mean square (rms) speed of the free electrons is approximately $6.40 \times 10^6 \, \text{m/s}$.

Let me know if you would like more details or have any questions.

Here are 5 questions to expand your understanding:

1. How is the Boltzmann constant related to thermodynamic systems?
2. What is the significance of the rms speed in ideal gas behavior?
3. How does temperature affect the rms speed of particles?
4. Can the pressure in the problem be used to cross-check results?
5. How does the mass of the particle influence its rms speed?

Tip: Always double-check your units when using equations involving physical constants to avoid errors.