Isaac Newton mentioned air resistance in "Principia," and Galileo spoke of dropped objects hitting the ground simultaneously if there was no air resistance. And anyone who has felt the wind has experience the inertia of air. Nevertheless, there are experiments that can demonstrate that air has mass in ways that appeal more directly to our understanding that mass is something that can be weighed--using little more than a bicycle pump or a balloon.
Balloons compress air to a higher density than the air around it. The elasticity of the balloon works like a rubber band to return it to its original shape. Therefore, balloons can hold enough compressed air to be measurable on a school lab scale.
Weigh a balloon before and after blowing it up with air. Make sure to weigh both times any string used to tie it off. Otherwise, students may think the extra weight came from the added string.
A variation on the above experiment is to blow up two balloons to roughly equal size, tie them off, then tie them to opposite ends of a ruler. Hang the ruler from a third string, from an overhang, for example. The third string should be centered on the ruler so that the two balloons are balanced and the ruler is level. Then prick one of the balloons with a pin. The ruler will drop to the side with the balloon that is still full of air. It is preferable to prick the balloon on its side, so students don't blame vertically expelled air for the reason the balance was thrown off. Also, make sure the balloon doesn't tear into pieces when popped.
Weigh a deflated basketball. Then pump it up to high pressure and weigh it again. The same thing can be done with a bicycle tube.
Put your hand out the window when riding in a car. The pressure on your hand indicates something is pressing on it, namely the inertia from the air. This doesn't demonstrate air has weight, but it does show air has inertia, and therefore mass.
A clear demonstration that air has weight (and therefore mass) is that it adheres to the Earth. A hundred kilometers above the Earth is a near vacuum.
Another approach is to create a chemical reaction that produces gas. The reactants after the reaction will weigh less than before, if the gas is allowed to escape.
Mixing Mg with HCl produces such a reaction. So does mixing vinegar with baking soda, which produces carbon dioxide.
The same argument can be made when evaporating water or dry ice, without the need for a chemical reaction, only a phase change.
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