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In part a of the figure, circles represent molecules distributed in a gas. An arrow at each circle represents the molecule’s velocity vector. The locations of the molecules and their velocity magnitudes and directions are all randomly distributed. In part b of the figure, an arc represents the wave front of a sound wave in the gas. The velocities of molecules near the arc are oriented roughly perpendicular to the arc, and therefore parallel to the propagation direction of the wave.
(a) In an ordinary gas, so many molecules move so fast that they collide billions of times every second. (b) Individual molecules do not move very far in a small amount of time, but disturbances like sound waves are transmitted at speeds related to the molecular speeds.

Calculating temperature: escape velocity of helium atoms

To escape Earth’s gravity, an object near the top of the atmosphere (at an altitude of 100 km) must travel away from Earth at 11.1 km/s. This speed is called the escape velocity . At what temperature would helium atoms have an rms speed equal to the escape velocity?

Strategy

Identify the knowns and unknowns and determine which equations to use to solve the problem.

Solution

  1. Identify the knowns: v is the escape velocity, 11.1 km/s.
  2. Identify the unknowns: We need to solve for temperature, T . We also need to solve for the mass m of the helium atom.
  3. Determine which equations are needed.
    • To get the mass m of the helium atom, we can use information from the periodic table:
      m = M N A .
    • To solve for temperature T , we can rearrange
      1 2 m v 2 = 3 2 k B T

      to yield
      T = m v 2 3 k B .
  4. Substitute the known values into the equations and solve for the unknowns,
    m = M N A = 4.0026 × 10 −3 kg/mol 6.02 × 10 23 mol = 6.65 × 10 −27 kg

    and
    T = ( 6.65 × 10 −27 kg ) ( 11.1 × 10 3 m/s ) 2 3 ( 1.38 × 10 −23 J/K ) = 1.98 × 10 4 K .

Significance

This temperature is much higher than atmospheric temperature, which is approximately 250 K ( −25 ° C or 10 ° F ) at high elevation. Very few helium atoms are left in the atmosphere, but many were present when the atmosphere was formed, and more are always being created by radioactive decay (see the chapter on nuclear physics). The reason for the loss of helium atoms is that a small number of helium atoms have speeds higher than Earth’s escape velocity even at normal temperatures. The speed of a helium atom changes from one collision to the next, so that at any instant, there is a small but nonzero chance that the atom’s speed is greater than the escape velocity. The chance is high enough that over the lifetime of Earth, almost all the helium atoms that have been in the atmosphere have reached escape velocity at high altitudes and escaped from Earth’s gravitational pull. Heavier molecules, such as oxygen, nitrogen, and water, have smaller rms speeds, and so it is much less likely that any of them will have speeds greater than the escape velocity. In fact, the likelihood is so small that billions of years are required to lose significant amounts of heavier molecules from the atmosphere. [link] shows the effect of a lack of an atmosphere on the Moon. Because the gravitational pull of the Moon is much weaker, it has lost almost its entire atmosphere. The atmospheres of Earth and other bodies are compared in this chapter’s exercises.

A photograph of an astronaut driving the lunar rover on the moon. The image and the shadow of the rover are very sharp. The sky is dark.
This photograph of Apollo 17 Commander Eugene Cernan driving the lunar rover on the Moon in 1972 looks as though it was taken at night with a large spotlight. In fact, the light is coming from the Sun. Because the acceleration due to gravity on the Moon is so low (about 1/6 that of Earth), the Moon’s escape velocity is much smaller. As a result, gas molecules escape very easily from the Moon, leaving it with virtually no atmosphere. Even during the daytime, the sky is black because there is no gas to scatter sunlight. (credit: Harrison H. Schmitt/NASA)

Questions & Answers

Newton's second laws is call with
Dyutee Reply
Really
Arzoodan
what is mean by thermodynamics
Prasad Reply
it is study about temperature and it's equilibrium
thiru
Its the study of heat and its relation with others kind of energy
Antonio
state caulombs law clearly
constand Reply
show mathematically that an electron has the greater speed than the proton when they attract each other
ezra Reply
show mathematically that an electron has the greater speed than the proton when they attract each other
srikanta
@ezra & srikanta; for electrons: a=ke^2/(mr^2) and for protons: a=kp^2/(mr^2)
Sikandar
what is electrostatics
Hero Reply
the study of charge at rest
Gulzar
@Hero; the study of charges at rest is the electrostatics
Sikandar
okay what is electrostatic?
Abd
charge at rest
Nawal
set of character...
Arzoodan
oky
Abd
Gauss law, electric fields, dipoles,...
Antonio
good
Abd
A proton initially at rest falls through a p.d of 25000V. what speed does it gain?
Minister Reply
@Minister; use equation v= sq root(2×eV/m)
Sikandar
what is the reaction of heat on magnet
ORIZINO Reply
Magnetization decreases with increase in temperature. But in case of diamagnetic substance heat has no role on magnetization.
srikanta
what is a physical significant of electric dipole moment .
PRANAB Reply
A dipole moment it's a mechanical electrical effect used in nature
Antonio
what is the uses of carbon brushes in generator
Malik Reply
to minimize heat
constand
at what temperature is the degree Fahrenheit equal to degree Celsius
Grace Reply
Celsius and Faharaneith are different, never equal
Antonio
find their liners express of n=a+b/T² ( plot graph n against T)
Donsmart Reply
Radio Stations often advertis "instant news,,if that meens you can hear the news the instant the radio announcer speaks it is the claim true? what approximate time interval is required for a message to travel from Cairo to Aswan by radio waves (500km) (Assume the waves Casbe detected at this range )
mahmod Reply
what is growth and decay
Pawan Reply
Can someone please predict the trajectory of a point charge in a uniform electric field????
erlinda Reply
what is deference between strong force and coulomb force
zahid Reply
how do you convert temperature in degree Celsius to Fahrenheit
kwame
Celsius x 9/5 +32
Cyclone
Practice Key Terms 8

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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