Humidity and the Ideal Gas Law

Before reading, please check out our previous blog post about how humidity affects the density of air.

In our previous post, the question was posed: Is it easier to hit a home run on a humid night or a dry night at the same ball park? The following explanation will help us figure this out.

So what is an ideal gas anyway? The ideal gas law is used to assume that certain properties about a gas will remain constant. The law states that the product of the pressure and the volume of a gas is equal to the product of the amount of substance (expressed as moles), the universal gas constant, and the temperature of the gas. As an equation:

Let’s say we have a cubic foot of dry air. It is about 78% nitrogen (N2=28 grams/mole), 21% oxygen (O2=32 grams/mole) and a small amount of other gases. For details on how these masses were calculated, see our post about the molar masses of gases. These molecules will move in and out of our cube of air as they please. Now, let us add some humidity into the equation. Water vapor (H2O=18 grams/mole) now takes up some of the space that the nitrogen and oxygen molecules used to occupy.

Now that we know which particles have a greater molar mass, let us prove that density is proportional to molar mass. To do this, we will break down the ideal gas equation to solve for density. The variable M represents the molar mass of the gas molecule. The rest of the variables will remain constant.

In the final step, m/V was changed to density. This derivation proves that the density of an air molecule is directly proportional to its molar mass. Thus, the density of water vapor (18 grams/mole) is less than dry air (28.5 grams/mole).

With the water added to the mixture, the portion of heavy molecules (air) has decreased and the portion of light molecules (water vapor) has increased in our cube. Now our cube has the same volume but a smaller mass. Therefore, the density of the cube of gases has decreased. How does this relate to the home run you may ask?

The instant the ball cracks off the bat, it is running into countless air and water vapor molecules as it flies towards the outfield fence. If the air less humid or if the pressure is greater, the baseball will feel greater effects of resistance due to drag. This will cause it to slow at a faster rate and not travel as far. We can conclude that the best place and conditions for the home run derby would be at the Colorado Rockies stadium in Denver (alt>5000 ft) on a humid night.

The next time you step outside on a hot humid summer night, you may want to say “wow the air density is quite low tonight”.

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