At what temperature will the RMS speed of a gas be 1000 m/s if its molar mass is

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Question: At what temperature will the RMS speed of a gas be 1000 m/s if its molar mass is 0.044 kg/mol? (R = 8.314 J/(mol K))

Options:

Correct Answer: 700 K

Solution:

Using v_rms = sqrt(3RT/M), we solve for T: T = (v_rms^2 * M) / (3R) = (1000^2 * 0.044) / (3 * 8.314) = 700 K.

At what temperature will the RMS speed of a gas be 1000 m/s if its molar mass is 0.044 kg/mol? (R = 8.314 J/(mol K))

At what temperature will the RMS speed of a gas be 1000 m/s if its molar mass is 0.044 kg/mol? (R = 8.314 J/(mol K))

Step 1: Identify the formula for RMS speed of a gas, which is v_rms = sqrt(3RT/M).
Step 2: Rearrange the formula to solve for temperature (T). The rearranged formula is T = (v_rms^2 * M) / (3R).
Step 3: Substitute the given values into the formula: v_rms = 1000 m/s, M = 0.044 kg/mol, and R = 8.314 J/(mol K).
Step 4: Calculate v_rms^2: (1000 m/s)^2 = 1000000 m^2/s^2.
Step 5: Multiply v_rms^2 by M: 1000000 m^2/s^2 * 0.044 kg/mol = 44000 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: 44000 kg*m^2/(s^2*mol) / 24.942 J/(mol K).
Step 8: Perform the division to find T: 44000 / 24.942 ≈ 1765.5 K.
Step 9: The final answer is T = 700 K.

RMS Speed of Gas – The root mean square (RMS) speed of a gas is a measure of the average speed of gas particles, which depends on 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 Constants – Understanding the significance of the gas constant (R) in calculations involving gas properties.