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

At what temperature will the RMS speed of a gas be 1000 m/s if its molar mass is 0.044 kg/mol?

At what temperature will the RMS speed of a gas be 1000 m/s if its molar mass is 0.044 kg/mol?

Step 1: Write down the formula for RMS speed: v_rms = sqrt(3RT/M).
Step 2: Rearrange the formula to solve for temperature (T): T = (v_rms^2 * M) / (3R).
Step 3: Identify the values needed: 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) = 1765.5 K.
Step 8: The final temperature is approximately 500 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 is related to temperature and molar mass.
Ideal Gas Law – The relationship between temperature, pressure, volume, and the number of moles of a gas, which can be applied to derive the RMS speed formula.
Gas Constant (R) – A constant used in the ideal gas law, typically expressed in units of J/(mol·K), which is essential for calculations involving gas properties.