# Chemistry: How Fast Do Gas Molecules Move?

## How Fast Do Gas Molecules Move?

How fast do the molecules in a gas move? We've said several times that they go "really fast," but how fast is "really fast"?

To answer this question, we first need to look at some of the factors that determine the speed of gas molecules.

### The Mass of the Molecules

The KMT says that the kinetic energy of the molecules in a gas is proportional only to the temperature in Kelvin. As a result, heavy objects and light ones have the same kinetic energies at the same temperature.

Let's imagine that I'm a very bad driver (this isn't much of a stretch). During the last ice storm, the roads got very slick and I drove my car into the fence surrounding my workplace at approximately five miles an hour.

When I destroyed the fence, my car was like a heavy gas molecule. Here's an interesting question: How fast would a bicyclist need to be going to destroy the fence with the same amount of kinetic energy? If you guessed "really, really fast," you're right! Because bicycles are much lighter than my car, they need a lot more speed to get the same amount of kinetic energy as my car.

Likewise, if two molecules have the same amount of kinetic energy, the lighter one will move more quickly than the heavy one. In other words, the velocity of the molecules in a gas depends on their masses!

### The Temperature of the Gas

The temperature of a gas is also important in determining its molecular speed. Because the KMT states that the amount of kinetic energy is dependent on the temperature, the temperature will determine how fast the molecules go in the first place. Of course, at a given temperature lighter molecules always move more quickly than heavier ones (as we saw just a few paragraphs ago), but *all* molecules will move more quickly if we boost the temperature of the gas.

### Putting It All Together: The Root Mean Square (rms) Velocity

Taking the mass of the molecules and the temperature into consideration, we find that the average velocity of the molecules in a gas can be described by a term called the "root mean square (rms) velocity." The rms velocity of a gas is calculated using the following equation:

*u*_{rms}= √^{3RT}⁄_{M}

In this equation, R represents the ideal gas constant (which for this equation is always 8.314 J/mol K), T represents the temperature of the gas in Kelvin, and *M* represents the molar mass of the compound in kilograms (see The Mole for more on molar mass).

##### Chemistrivia

You may have noticed that the units for R are different here than when we mentioned it earlier. The units "J/mol K" are equivalent to "L atm/mol K," and make the math work out better.

For example, the rms velocity for ammonia at room temperature is found by plugging the temperature (298 K) and the molar mass (0.0170 kg/mol) into this equation with the ideal gas constant. Using this equation, the rms velocity of ammonia would be:

##### You've Got Problems

Problem 1: What is the average velocity of hydrogen molecules at STP?

*u*_{rms}= √^{3(8.314J / molK)(298K)}⁄_{0.0170kg / mol}= 661*m*/ sec

This is pretty darn fast!

### The Random Walk

Our earlier calculation found that ammonia molecules move 661 m/sec at room temperature. If ammonia moves this quickly, why don't we immediately smell it whenever our neighbor across the street mops his floor?

If ammonia molecules traveled straight from your neighbor's floor to your nose, you would smell it almost immediately. However, molecules don't travel in straight paths—rather, they bump into each other in random fashion.

To illustrate what I mean, imagine that the Olympic committee has decided to make the marathon more interesting by blindfolding all the runners. These runners will eventually finish the marathon even if they're blindfolded. Unfortunately, it will probably take them days to finish the race because they won't be traveling on a straight path—instead, they'll be bumping into trees, spectators, each other, etc. The path that these runners take is referred to as a "random walk" because they'll be traveling very quickly in random directions.

Molecules do the same thing. Though ammonia molecules travel 661 m/sec at room temperature, they take a long time to cross a room because they keep bumping into things. The length of time it takes for molecules to travel from one place to another depends not only on their rms velocity, but also on the distance between collisions, called the "mean free path."

Excerpted from The Complete Idiot's Guide to Chemistry © 2003 by Ian Guch. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with **Alpha Books**, a member of Penguin Group (USA) Inc.

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