# 2.2 Pressure, temperature, and rms speed  (Page 10/18)

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What is the gauge pressure inside a tank of $4.86\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}\text{mol}$ of compressed nitrogen with a volume of $6.56\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}$ if the rms speed is 514 m/s?

If the rms speed of oxygen molecules inside a refrigerator of volume $22.0\phantom{\rule{0.2em}{0ex}}\text{ft}{\text{.}}^{3}$ is 465 m/s, what is the partial pressure of the oxygen? There are 5.71 moles of oxygen in the refrigerator, and the molar mass of oxygen is 32.0 g/mol.

21.1 kPa

The escape velocity of any object from Earth is 11.1 km/s. At what temperature would oxygen molecules (molar mass is equal to 32.0 g/mol) have root-mean-square velocity ${v}_{\text{rms}}$ equal to Earth’s escape velocity of 11.1 km/s?

The escape velocity from the Moon is much smaller than that from the Earth, only 2.38 km/s. At what temperature would hydrogen molecules (molar mass is equal to 2.016 g/mol) have a root-mean-square velocity ${v}_{\text{rms}}$ equal to the Moon’s escape velocity?

458 K

Nuclear fusion, the energy source of the Sun, hydrogen bombs, and fusion reactors, occurs much more readily when the average kinetic energy of the atoms is high—that is, at high temperatures. Suppose you want the atoms in your fusion experiment to have average kinetic energies of $6.40\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-14}\phantom{\rule{0.2em}{0ex}}\text{J}$ . What temperature is needed?

Suppose that the typical speed $\left({v}_{\text{rms}}\right)$ of carbon dioxide molecules (molar mass is 44.0 g/mol) in a flame is found to be 1350 m/s. What temperature does this indicate?

$3.22\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{K}$

(a) Hydrogen molecules (molar mass is equal to 2.016 g/mol) have ${v}_{\text{rms}}$ equal to 193 m/s. What is the temperature? (b) Much of the gas near the Sun is atomic hydrogen (H rather than ${\text{H}}_{2}\right).$ Its temperature would have to be $1.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{7}\phantom{\rule{0.2em}{0ex}}\text{K}$ for the rms speed ${v}_{\text{rms}}$ to equal the escape velocity from the Sun. What is that velocity?

There are two important isotopes of uranium, ${}^{235}\text{U}$ and ${}^{238}\text{U}$ ; these isotopes are nearly identical chemically but have different atomic masses. Only ${}^{235}\text{U}$ is very useful in nuclear reactors. Separating the isotopes is called uranium enrichment (and is often in the news as of this writing, because of concerns that some countries are enriching uranium with the goal of making nuclear weapons.) One of the techniques for enrichment, gas diffusion, is based on the different molecular speeds of uranium hexafluoride gas, ${\text{UF}}_{6}$ . (a) The molar masses of ${}^{235}\text{U}$ and ${}^{238}{\text{UF}}_{6}$ are 349.0 g/mol and 352.0 g/mol, respectively. What is the ratio of their typical speeds ${v}_{\text{rms}}$ ? (b) At what temperature would their typical speeds differ by 1.00 m/s? (c) Do your answers in this problem imply that this technique may be difficult?

a. 1.004; b. 764 K; c. This temperature is equivalent to $915\phantom{\rule{0.2em}{0ex}}\text{ºF}$ , which is high but not impossible to achieve. Thus, this process is feasible. At this temperature, however, there may be other considerations that make the process difficult. (In general, uranium enrichment by gaseous diffusion is indeed difficult and requires many passes.)

The partial pressure of carbon dioxide in the lungs is about 470 Pa when the total pressure in the lungs is 1.0 atm. What percentage of the air molecules in the lungs is carbon dioxide? Compare your result to the percentage of carbon dioxide in the atmosphere, about 0.033%.

Dry air consists of approximately $78\text{%}\phantom{\rule{0.2em}{0ex}}\text{nitrogen},21\text{%}\phantom{\rule{0.2em}{0ex}}\text{oxygen},\text{and}\phantom{\rule{0.2em}{0ex}}1\text{%}\phantom{\rule{0.2em}{0ex}}\text{argon}$ by mole, with trace amounts of other gases. A tank of compressed dry air has a volume of 1.76 cubic feet at a gauge pressure of 2200 pounds per square inch and a temperature of 293 K. How much oxygen does it contain in moles?

65 mol

(a) Using data from the previous problem, find the mass of nitrogen, oxygen, and argon in 1 mol of dry air. The molar mass of ${\text{N}}_{2}$ is 28.0 g/mol, that of ${\text{O}}_{2}$ is 32.0 g/mol, and that of argon is 39.9 g/mol. (b) Dry air is mixed with pentane $\left({\text{C}}_{5}{\text{H}}_{12},$ molar mass 72.2 g/mol), an important constituent of gasoline, in an air-fuel ratio of 15:1 by mass (roughly typical for car engines). Find the partial pressure of pentane in this mixture at an overall pressure of 1.00 atm.

(a) Given that air is $21%$ oxygen, find the minimum atmospheric pressure that gives a relatively safe partial pressure of oxygen of 0.16 atm. (b) What is the minimum pressure that gives a partial pressure of oxygen above the quickly fatal level of 0.06 atm? (c) The air pressure at the summit of Mount Everest (8848 m) is 0.334 atm. Why have a few people climbed it without oxygen, while some who have tried, even though they had trained at high elevation, had to turn back?

a. 0.76 atm; b. 0.29 atm; c. The pressure there is barely above the quickly fatal level.

(a) If the partial pressure of water vapor is 8.05 torr, what is the dew point? $\left(760\phantom{\rule{0.2em}{0ex}}\text{torr}=1\phantom{\rule{0.2em}{0ex}}\text{atm}=101,325\phantom{\rule{0.2em}{0ex}}\text{Pa}\right)$ (b) On a warm day when the air temperature is $35\phantom{\rule{0.2em}{0ex}}\text{°C}$ and the dew point is $25\phantom{\rule{0.2em}{0ex}}\text{°C}$ , what are the partial pressure of the water in the air and the relative humidity?

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