textbook forum The Kinetic Molecular Theory and the Weighing of Gas Samples Henry C. Brenner New York University New York, NY 10003 Virtually all elementary chemistry courses discuss the difference between the three physical states of matter. In liquids and solids, the molecules are in close contact and touching each other. By contrast, in gases, the volume of the molecules themselves is small compared to the volume of the sample. According to the kinetic molecular theory, a gas is mostly empty space, and gases exert pressure by means of perfectly elastic collisions of the molecules with each other and with the walls of the container. How then, is it possible to weigh gas samples, since the molecules are constantly moving around and not always in contact with the floor of the container? This query can be answered with some simple ideas from high school physics. Consider a sample of a perfect gas in a closed, rectangular solid-shaped container in the earth's gravity. Because we know that containers full of gas weigh more than evacuated containers, it must be the case that the gravitational force causes the molecules to collide with more force on the floor of the container than with the top
A bouncing gas molecule.
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edited bv RALPHK. BIRDWHISTEL; University of West Florida Pensawla, FL32504
or sides. It is easy to show that this additional force is simply the force of gravity on the molecules. To simplify the problem, consider a single gas molecule of mass m starting from rest from somewhere above the floor of the container (see figure). Gravity accelerates it until it reaches the floor, with terminal velocity@, whereg is the gravitational acceleration and t is the time to reach the floor. At this point the particle rebounds completely elastically and retraces its trajectory up to the starting point, where it comes @ rest and begins the journey all over again. The force exerted on the floor is the change in momentum per collision (Zmgt, because mgt is the terminal momentum just before it hits) times the number of collisions per second (112, since 2t is the time for a round trip). Thus, the force is the force of gravity on the molecule.
F = (2mgt)(UZt)= mg I t is evident that this argument would apply to the vertical component of motion for each molecule in a large samd e of gas molecules, so that the molecules would exert a force on the floor equal to theif weight, even though they are not resting on the floor, as they would in a solid or liquid phase sample. It also would apply to molecules that already have a nonzero vertical component of velocity; gravity merely accelerates these molecules in the downward direction and produces an extra force (and pressure) a t the bottom of the container. Moreover, it is not necessary that all molecules to be weighed collide with the bottom of the container, because they may transfer their downward momentum to other molecules below them by means of elastic collisions. In this way, the force of gravity on the molecules is transmitted to the floor of the container, and the gas can be weighed. In the above argument, we have considered a gas confined to a closed container because we wanted to limit our attention to a fixed amount ofgas that would not be diffisine into the surroundines. However. it should be clear that th;! gases in the earth's atmospherk exert a gravitational force by collisions with the earth's surface in the same way a s described above. Acknowledgment Stimulating discussions with Marc Walters from New York University are gratefully acknowledged.
