For a gas with a molar mass of 32 g/mol at 273 K, what is the RMS speed?

For a gas with a molar mass of 32 g/mol at 273 K, what is the RMS speed?

Step 1: Identify the formula for RMS speed, which is v_rms = sqrt(3RT/M).
Step 2: Identify the values needed for the formula: R (the gas constant) is 8.314 J/(mol·K), T (temperature) is 273 K, and M (molar mass) is 32 g/mol.
Step 3: Convert the molar mass from grams per mole to kilograms per mole for consistency in units: 32 g/mol = 0.032 kg/mol.
Step 4: Substitute the values into the formula: v_rms = sqrt(3 * 8.314 * 273 / 0.032).
Step 5: Calculate the numerator: 3 * 8.314 * 273 = 6816.342.
Step 6: Divide the result by the molar mass: 6816.342 / 0.032 = 213,011.31.
Step 7: Take the square root of the result: sqrt(213,011.31) = 461.36 m/s.
Step 8: Round the final answer to the nearest whole number: v_rms ≈ 461 m/s.

RMS Speed Calculation – The question tests the ability to calculate the root mean square (RMS) speed of a gas using the formula v_rms = sqrt(3RT/M), where R is the gas constant, T is the temperature in Kelvin, and M is the molar mass in kg/mol.
Unit Conversion – The question requires understanding of unit conversions, particularly converting molar mass from grams per mole to kilograms per mole.
Gas Laws – The question is rooted in the kinetic theory of gases, which relates temperature, molar mass, and molecular speed.