Calculate Root Mean Square Velocity of Gas Particles

This example problem demonstrates how to calculate the root mean square velocity of particles in an ideal gas. This value is the square root of the average velocity-squared of molecules in a gas. While the value is an approximation, especially for real gases, it offers useful information when studying kinetic theory. Root Mean Square Velocity Problem What is the average velocity or root mean square velocity of a molecule in a sample of oxygen at 0  °C? Solution Gases consist of atoms or molecules that move at different speeds in random directions. The root means square velocity (RMS velocity) is a way to find a single velocity value for the particles.  The average velocity of gas particles is found using the root mean square velocity formulaÃŽ ¼rms (3RT/M) ½whereÃŽ ¼rms root mean square velocity in m/secR ideal gas constant 8.3145 (kg ·m2/sec2)/K ·molT absolute temperature in KelvinM mass of a mole of the gas in kilograms. Really, the RMS calculation gives you root mean square speed, not velocity. This is because velocity is a vector quantity, which has  magnitude and direction. The RMS calculation only gives the magnitude or speed.The temperature must be converted to Kelvin and the molar mass must be found in kg to complete this problem. Step 1 Find the absolute temperature using the Celsius to Kelvin conversion formula:T  °C 273T 0 273T 273 K Step 2 Find molar mass in kg:From the periodic table, the molar mass of oxygen 16 g/mol.Oxygen gas (O2) is comprised of two oxygen atoms bonded together. Therefore:molar mass of O2 2 x 16molar mass of O2 32 g/molConvert this to kg/mol:molar mass of O2 32 g/mol x 1 kg/1000 gmolar mass of O2 3.2 x 10-2 kg/mol Step 3 Find ÃŽ ¼rmsÃŽ ¼rms (3RT/M) ½ÃŽ ¼rms [3(8.3145 (kg ·m2/sec2)/K ·mol)(273 K)/3.2 x 10-2 kg/mol] ½ÃŽ ¼rms (2.128 x 105 m2/sec2) ½ÃŽ ¼rms 461 m/sec Answer The average velocity or root mean square velocity of a molecule in a sample of oxygen at 0  °C is 461 m/sec.