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A helium atom is four times the mass of a hydrogen atom. At low temperatures, the classical molecular model is valid for all ideal gases. Consider a container of helium gas and a container of hydrogen gas, each at the same low temperature of 50 K. If v is the root-mean-square speed of the helium molecules, then the root-mean-square speed of the hydrogen molecules is a) v/4, b) v/2, c) v/ 2, d) v, e) 2v, f) 2v, or g) 4v (end of section 21.1) QUICK QUIZ 21.1

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QUICK QUIZ 21.1 ANSWER (e). Equation 21.7, v rms = (3k B T/m), relates the root-mean- square speed to the molecular mass and to the absolute temperature. Each helium atom is four times the mass of each hydrogen atom, but helium gas is monatomic and hydrogen gas is diatomic. Therefore, each helium molecule is twice the mass of each hydrogen molecule. Designating the mass of a hydrogen molecule as m o and a helium molecule as 2m o, we have,

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In two experiments, an ideal gas is adiabatically expanded from an initial volume to a final volume that is twice the initial volume. In each experiment, the same gas is used at the same initial volume, temperature and pressure. In the first experiment, the container is separated into two sections by a wall with gas occupying the left section. The wall is suddenly very quickly removed and the gas expands to occupy the entire container. In the second experiment, the wall is connected to a movable piston, and you very slowly move the piston to the right until the gas is double its initial volume. The final temperature of the gas in the second experiment will be a) less than the final temperature of the gas in the first experiment, b) the same as the final temperature of the gas in the first experiment, c) greater than the final temperature of the gas in the first experiment, or d) impossible to determine. (end of section 21.3) QUICK QUIZ 21.2

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(a). Recall the discussion in the previous chapter (Section 20.6) of the results of an adiabatic free expansion. No work is done and no heat is transferred so, from the first law, the change in internal energy of the gas is zero. For an ideal gas, the internal energy is only dependent on the temperature so the change in temperature is also zero. Experiment 1 describes an adiabatic free expansion. Experiment 2, on the other hand, describes an adiabatic expansion that is quasistatic so that the gas is in an equilibrium state at any time during the process. Only under these circumstances are the equation of state, PV = nRT, and the expression, PV = constant, valid. In Section 21.3, these equations are used to derive the result that a quasistatic adiabatic expansion results in a decrease of temperature. Therefore, the final temperature of the gas in experiment 2 will be less than in experiment 1. QUICK QUIZ 21.2 ANSWER

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In a certain experiment, you continually heat a container of hydrogen gas and measure the heat necessary to increase the temperature by a certain amount. Assuming that the gas remains ideal, the heat necessary to raise the temperature by 1 K at high temperature (e. g. 10,000 K) compared to the heat necessary to raise the temperature by 1 K at low temperature (e. g. 40 K) will be a) 3 times as much, b) twice as much, c) 3/2 as much, d) 5/2 as much, e) 7/2 as much, f) 7/3 as much, or g) the same. (end of section 21.4) QUICK QUIZ 21.3

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QUICK QUIZ 21.3 ANSWER (f). Examining the figure below shows that the molar specific heat at constant volume increases from 3/2R to 7/2R as the temperature changes from ~40 K to ~10,000 K. Therefore, it requires (7/2)/(3/2) = 7/3 times as much heat at high temperature to raise the temperature by a given amount.

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For an ideal gas at a certain temperature, the number of molecules of the gas with a speed greater than the most probable speed will be a) less than the number with a speed less than the most probable speed, b) the same as the number with a speed less than the most probable speed, c) greater than the number with a speed less than the most probable speed, or d) it depends on the gas and the temperature. (end of section 21.6) QUICK QUIZ 21.4

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(c). As the figure at right shows, the area under the curve for speeds greater than the most probable speed (the peak) is larger than the area for speeds less than the most probable speed. Unlike a bell curve, the distribution of a speeds curve is not symmetrical. QUICK QUIZ 21.4 ANSWER

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Standard pumps (rough pumps in conjunction with diffusion pumps) for high vacuum systems can readily achieve pressures that are one one-billionth atmospheric pressure. Under typical conditions, at room temperature, the mean free path for the molecules in a high vacuum system would be about a) a few nanometers, b) a few microns, c) a few millimeters, d) a few meters, or e) hundreds of meters. (end of section 21.7) QUICK QUIZ 21.5

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(e). Comparing this problem to Example 21.6, all factors are the same except for the pressure which is reduced by a factor of 10 9. This, in turn, increases the mean free path by a factor of 10 9 from 2.25 x 10 -7 m to 2.25 x 10 2 m, or a few hundred meters. This result shows that, for high vacuums, the size of the pipe is very important and will determine the pump-down rate since molecules will make collisions with the walls of the pipe much more often than with other molecules. QUICK QUIZ 21.5 ANSWER

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