# 2.2 Pressure, temperature, and rms speed  (Page 9/18)

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Statistical mechanics says that in a gas maintained at a constant temperature through thermal contact with a bigger system (a “reservoir”) at that temperature, the fluctuations in internal energy are typically a fraction $1\text{/}\sqrt{N}$ of the internal energy. As a fraction of the total internal energy of a mole of gas, how big are the fluctuations in the internal energy? Are we justified in ignoring them?

Which is more dangerous, a closet where tanks of nitrogen are stored, or one where tanks of carbon dioxide are stored?

One where nitrogen is stored, as excess ${\text{CO}}_{2}$ will cause a feeling of suffocating, but excess nitrogen and insufficient oxygen will not.

## Problems

In the problems in this section, assume all gases are ideal.

A person hits a tennis ball with a mass of 0.058 kg against a wall. The average component of the ball’s velocity perpendicular to the wall is 11 m/s, and the ball hits the wall every 2.1 s on average, rebounding with the opposite perpendicular velocity component. (a) What is the average force exerted on the wall? (b) If the part of the wall the person hits has an area of $3.0\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2},$ what is the average pressure on that area?

a. 0.61 N; b. 0.20 Pa

A person is in a closed room (a racquetball court) with $V=453\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}$ hitting a ball $\left(m=42.0\phantom{\rule{0.2em}{0ex}}\text{g}\right)$ around at random without any pauses. The average kinetic energy of the ball is 2.30 J. (a) What is the average value of ${v}_{x}^{2}?$ Does it matter which direction you take to be x ? (b) Applying the methods of this chapter, find the average pressure on the walls? (c) Aside from the presence of only one “molecule” in this problem, what is the main assumption in Pressure, Temperature, and RMS Speed that does not apply here?

Five bicyclists are riding at the following speeds: 5.4 m/s, 5.7 m/s, 5.8 m/s, 6.0 m/s, and 6.5 m/s. (a) What is their average speed? (b) What is their rms speed?

a. 5.88 m/s; b. 5.89 m/s

Some incandescent light bulbs are filled with argon gas. What is ${v}_{\text{rms}}$ for argon atoms near the filament, assuming their temperature is 2500 K?

Typical molecular speeds $\left({v}_{\text{rms}}\right)$ are large, even at low temperatures. What is ${v}_{\text{rms}}$ for helium atoms at 5.00 K, less than one degree above helium’s liquefaction temperature?

177 m/s

What is the average kinetic energy in joules of hydrogen atoms on the $5500\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ surface of the Sun? (b) What is the average kinetic energy of helium atoms in a region of the solar corona where the temperature is $6.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{K}$ ?

What is the ratio of the average translational kinetic energy of a nitrogen molecule at a temperature of 300 K to the gravitational potential energy of a nitrogen-molecule−Earth system at the ceiling of a 3-m-tall room with respect to the same system with the molecule at the floor?

$4.54\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}$

What is the total translational kinetic energy of the air molecules in a room of volume $23\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}$ if the pressure is $9.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}\text{Pa}$ (the room is at fairly high elevation) and the temperature is $21\phantom{\rule{0.2em}{0ex}}\text{°C}$ ? Is any item of data unnecessary for the solution?

The product of the pressure and volume of a sample of hydrogen gas at $0.00\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ is 80.0 J. (a) How many moles of hydrogen are present? (b) What is the average translational kinetic energy of the hydrogen molecules? (c) What is the value of the product of pressure and volume at $200\phantom{\rule{0.2em}{0ex}}\text{°}\text{C?}$

a. 0.0352 mol; b. $5.65\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-21}\phantom{\rule{0.2em}{0ex}}\text{J;}$ c. 139 J

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