CHEMISTRY 101 – Root mean square velocity of gas molecules

CHEMISTRY 101 – Root mean square velocity of gas molecules


According to kinetic molecular theory, the average kinetic energy of a gas is directly proportional to the temperature in Kelvin. If we have a mixture of gases with different molar masses at a given temperature, for the average kinetic energy of each gas to be the same, lighter gas particles will travel faster on average. At any given temperature, there’s a distribution of velocities for gas particles. Some are moving faster, some slower. This distribution will be important when we discuss Collision Theory and reaction rates. Although there is a distribution of velocities, we can talk about the root mean square velocity, which you can think of as an average velocity. The root mean square velocity is equal to the square root of 3 times the ideal gas law constant, R, times temperature divided by molar mass. We have to pay particular attention to units in this equation. The ideal gas law constant has units of joules per mole Kelvin instead of liter atmospheres per mole Kelvin. So the constant is 8.314 joules per mole Kelvin. The joules in the ideal gas law constant also means that our molar mass must be in kilograms per mole instead of grams per mole, because one joule is 1 kilogram meters squared per second squared. Let’s practice with our root mean square velocity equation. Our problem reads, “Calculate the root mean square velocity of gaseous xenon atoms at 215° C.” We’ll start with our root mean square velocity equation. In this equation, we need to use the value of the ideal gas law constant with units of joules per mole Kelvin. We also need to convert our temperature to Kelvin and our molar mass from grams per mole to kilograms per mole. If we substitute in 1 kilogram meters squared per second squared for joules in our ideal gas law constant, we’ll see how units cancel out. Kilograms cancel out, grams cancel out, Kelvin cancel out, moles cancel out, and we’re left with units of meters squared per second squared. These units are inside the square root, so the units for our root mean square velocity are just meters per second. Completing the calculation, we get 304 meters per second.

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