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$\bullet$Where is the hydrogen? The average temperature of the atmosphere near the surface of the earth is about $20^{\circ} \mathrm{C}$ . (a) What is the root-mean-square speed of hydrogen molecules, $\mathrm{H}_{2},$ at this temperature? (b) The escape speed from the earth is about 11 $\mathrm{km} / \mathrm{s} .$ Is the average $\mathrm{H}_{2}$ molecule moving fast enough to escape? (c) Compare the rms speeds of oxygen $\left(\mathrm{O}_{2}\right)$ and nitrogen $\left(\mathrm{N}_{2}\right)$ with that of $\mathrm{H}_{2}$ . (d) So why has the hydrogen been able to escape the earth's gravity, but the heavier gases (such as $\mathrm{O}_{2}$ and $\mathrm{N}_{2} )$ have not, even though none of these gases has an rms speed equal to the escape speed of the earth? (Hint: Do all the molecules have the rms speed, or are some moving faster? since the rms speed for $\mathrm{H}_{2}$ is greater than that of $\mathrm{O}_{2}$ and $\mathrm{N}_{2},$ which gas would have a higher percentage of its molecules moving fast enough to escape?)

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Physics 101 Mechanics

Chapter 15

Thermal Properties of Matter

Temperature and Heat

The First Law of Thermodynamics

The Second Law of Thermodynamics

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

McMaster University

Lectures

03:15

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The First Law of Thermodyn…

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At what temperature does t…

03:24

Evaporating Atmosphere Hyd…

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(a) The temperature near t…

03:49

Possible escape of differe…

04:47

The escape speed from the …

16:26

Planetary Atmospheres. (a)…

03:16

04:38

Hydrogen on the Sun. The s…

04:12

02:15

No $\mathrm{H}_{2}$ in Ear…

04:34

The surface of the sun has…

03:31

(a) Hydrogen molecules (mo…

01:25

The atmospheres of relativ…

05:03

$\bullet$$\bullet$ The sur…

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The escape velocity from t…

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At what temperature would …

08:08

At the surface of Venus th…

06:53

The Earth’s atmosphere con…

01:39

05:33

A large number of hydrogen…

The raised temperature of atmosphere near the surface of earth is 20 degrees celsius, so we have t equal to 20 degrees celsius, which is equal to 293.15 kelvin. Now, in the first part of this problem, we are asked to calculate to mean square speed of hydrogen molecules, as we have the relation for root mean square speed is equal to square root of 3 times r times 3. Upon m here, r is gas. Constant t is temperature and m is molar. Mass and molar mass for hydrogen is to multiply 10 to the power minus 3 chlogram per mole. Now, by putting different values, we have v r m s is equal to 3 times r, where r is 8.314427 multiply. 427. Joule per mole per kelvin into temperature is 293.15 kelvin upon molar mass, that is, to multiply 10 to the power minus 3 kilogram per mole. So we get v r m is for hydrogen molecules is equal to 1.912 kilometer per second in b part of this problem. Here they escape velocity for earth is 11 kilometer per second, and we have calculated in part a of this problem that we are, as for hadrian molecule is equal to 1.912 kilometer per se. Here we are, as for hydrogen molecules is very, very less than v scare for earth, so hydrogen atoms cannot escape from earth. Now i see part of this problem. First, we will calculate a root means square speed for hydrogenatem, so we are, as for unions, equal to square root of 3 r t upon molar mass, which is equal to 3 multiplied, or there is 8.314472 joule per mole per kelvin and temperature is 293.15 kelvin divide By molar mass for hydrogen atom is 32 multiply, 10 to the power minus 3 kilogram per mole. So we get v r m s per oxygen is equal to 0.478 kilometer per second. So by comparing the armas of oxygen with hydrogen, it is clear that we are of hydrogen atom. Hydrogen molecules is greater than vermifoxygen. Now we want to calculate v r m s for nitrogen molecule, so v r m s is equal to square root of 3 r t upon molar mass, which is equal to 3 into here. Value of r is 8.314472 joule per mole per kelvin into temperature is 293.15 kelvin upon molar mass of nitrogen is 20 and multiply 10 to the power minus 3 kilogram per mole. So we get p r m is equal to 0.511 kilometer per se. Again, the warm is of hydrogen is greater than v r m s nitrogenin. The last part of this problem here roomin square speed, is the average speed of molecules so root mean square speed just shows very speed of molecules. Some molecules have speeds higher than are emancipate, so this speed may be greater than the escape velocity for earth, so these molecules will be able to escape the earth so higher. The arms speed of a gas, the higher the number of molecules of their gas to have higher speeds than the arms speed.

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