At what temperature will the RMS speed of a gas be 500 m/s if its molar mass is

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Question: At what temperature will the RMS speed of a gas be 500 m/s if its molar mass is 0.02 kg/mol? (2000)

Options:

Correct Answer: 500 K

Solution:

Using v_rms = sqrt(3RT/M), rearranging gives T = (v_rms^2 * M) / (3R). Substituting values gives T = 500 K.

At what temperature will the RMS speed of a gas be 500 m/s if its molar mass is 0.02 kg/mol? (2000)

At what temperature will the RMS speed of a gas be 500 m/s if its molar mass is 0.02 kg/mol? (2000)

Step 1: Write down the formula for RMS speed: v_rms = sqrt(3RT/M).
Step 2: Identify the variables: v_rms = 500 m/s, M = 0.02 kg/mol, and R = 8.314 J/(mol·K).
Step 3: Rearrange the formula to solve for temperature (T): T = (v_rms^2 * M) / (3R).
Step 4: Calculate v_rms^2: (500 m/s)^2 = 250000 m^2/s^2.
Step 5: Multiply v_rms^2 by M: 250000 m^2/s^2 * 0.02 kg/mol = 5000 kg·m^2/(s^2·mol).
Step 6: Calculate 3R: 3 * 8.314 J/(mol·K) = 24.942 J/(mol·K).
Step 7: Divide the result from Step 5 by the result from Step 6: 5000 kg·m^2/(s^2·mol) / 24.942 J/(mol·K) = 200.0 K.
Step 8: The final temperature is T = 200.0 K.

RMS Speed of Gas – The root mean square (RMS) speed of a gas is a measure of the average speed of particles in a gas, which is related to temperature and molar mass.
Ideal Gas Law – The relationship between temperature, molar mass, and RMS speed is derived from the ideal gas law and kinetic theory.
Gas Constant (R) – The universal gas constant (R) is a key value in gas equations, typically 8.314 J/(mol·K).