If the RMS speed of a gas is 400 m/s and its molar mass is 16 g/mol, what is the

If the RMS speed of a gas is 400 m/s and its molar mass is 16 g/mol, what is the temperature of the gas?

If the RMS speed of a gas is 400 m/s and its molar mass is 16 g/mol, what is the temperature of the gas?

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 = (M * v_rms^2) / (3R).
Step 3: Convert the molar mass from grams to kilograms: 16 g/mol = 0.016 kg/mol.
Step 4: Substitute the values into the formula: M = 0.016 kg/mol and v_rms = 400 m/s.
Step 5: Calculate v_rms^2: 400 m/s * 400 m/s = 160000 m^2/s^2.
Step 6: Substitute the values into the rearranged formula: T = (0.016 kg/mol * 160000 m^2/s^2) / (3R).
Step 7: Use the value of R (the gas constant): R = 8.314 J/(mol·K).
Step 8: Calculate 3R: 3 * 8.314 J/(mol·K) = 24.942 J/(mol·K).
Step 9: Now calculate T: T = (0.016 kg/mol * 160000 m^2/s^2) / 24.942 J/(mol·K).
Step 10: Perform the multiplication: 0.016 * 160000 = 2560.
Step 11: Finally, divide: T = 2560 / 24.942 ≈ 102.5 K.

RMS Speed and Temperature Relationship – The relationship between the root mean square speed of gas molecules, their molar mass, and temperature, as described by the equation v_rms = sqrt((3RT)/M).
Unit Conversion – Understanding the need to convert molar mass from grams per mole to kilograms per mole for consistency in SI units.
Gas Constant – Knowledge of the gas constant R and its value in appropriate units (8.314 J/(mol·K)).