If the RMS speed of a gas is 250 m/s, what is the temperature if the molar mass

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Question: If the RMS speed of a gas is 250 m/s, what is the temperature if the molar mass is 0.028 kg/mol?

Options:

Correct Answer: 300 K

Solution:

Using v_rms = sqrt(3RT/M), we can rearrange to find T = (v_rms^2 * M) / (3R) = 300 K.

If the RMS speed of a gas is 250 m/s, what is the temperature if the molar mass is 0.028 kg/mol?

If the RMS speed of a gas is 250 m/s, what is the temperature if the molar mass is 0.028 kg/mol?

Step 1: Identify the given values. We have the RMS speed (v_rms) = 250 m/s and the molar mass (M) = 0.028 kg/mol.
Step 2: Recall the formula for RMS speed: v_rms = sqrt(3RT/M), where R is the ideal gas constant.
Step 3: Rearrange the formula to solve for temperature (T). The rearranged formula is T = (v_rms^2 * M) / (3R).
Step 4: Substitute the known values into the rearranged formula. First, calculate v_rms^2: (250 m/s)^2 = 62500 m^2/s^2.
Step 5: Use the value of R, which is approximately 8.314 J/(mol·K).
Step 6: Substitute the values into the formula: T = (62500 m^2/s^2 * 0.028 kg/mol) / (3 * 8.314 J/(mol·K)).
Step 7: Calculate the numerator: 62500 * 0.028 = 1750.
Step 8: Calculate the denominator: 3 * 8.314 = 24.942.
Step 9: Divide the numerator by the denominator: T = 1750 / 24.942 ≈ 70.1 K.
Step 10: Verify the calculations to ensure accuracy.

RMS Speed – The root mean square speed of gas molecules, which relates to temperature and molar mass.
Ideal Gas Law – The relationship between temperature, pressure, volume, and the number of moles of a gas.
Gas Constant (R) – A constant used in the ideal gas law, typically 8.314 J/(mol·K).