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We can get the average kinetic energy of a molecule, 1 2 mv 2 size 12{ { { size 8{1} } over { size 8{2} } } ital "mv" rSup { size 8{2} } } {} , from the left-hand side of the equation by canceling N size 12{N} {} and multiplying by 3/2. This calculation produces the result that the average kinetic energy of a molecule is directly related to absolute temperature.

KE ¯ = 1 2 m v 2 ¯ = 3 2 kT size 12{ {overline {"KE"}} = { {1} over {2} } m {overline {v rSup { size 8{2} } }} = { {3} over {2} } ital "kT"} {}

The average translational kinetic energy of a molecule, KE ¯ size 12{ {overline {"KE"}} } {} , is called thermal energy     . The equation KE ¯ = 1 2 m v 2 ¯ = 3 2 kT size 12{ {overline { size 11{"KE"}}} = { {1} over {2} } m {overline { size 11{v rSup { size 8{2} } }}} = { {3} over {2} } ital "kT"} {} is a molecular interpretation of temperature, and it has been found to be valid for gases and reasonably accurate in liquids and solids. It is another definition of temperature based on an expression of the molecular energy.

It is sometimes useful to rearrange KE ¯ = 1 2 m v 2 ¯ = 3 2 kT size 12{ {overline { size 11{"KE"}}} = { {1} over {2} } m {overline { size 11{v rSup { size 8{2} } }}} = { {3} over {2} } ital "kT"} {} , and solve for the average speed of molecules in a gas in terms of temperature,

v 2 ¯ = v rms = 3 kT m , size 12{ sqrt { {overline {v rSup { size 8{2} } }} } =v rSub { size 8{"rms"} } = sqrt { { {3 ital "kT"} over {m} } } ,} {}

where v rms size 12{v rSub { size 8{"rms"} } } {} stands for root-mean-square (rms) speed.

Calculating kinetic energy and speed of a gas molecule

(a) What is the average kinetic energy of a gas molecule at 20 . 0 º C size 12{"20" "." 0°C} {} (room temperature)? (b) Find the rms speed of a nitrogen molecule ( N 2 ) size 12{ \( N rSub { size 8{2} } \) } {} at this temperature.

Strategy for (a)

The known in the equation for the average kinetic energy is the temperature.

KE ¯ = 1 2 m v 2 ¯ = 3 2 kT size 12{ {overline {"KE"}} = { {1} over {2} } m {overline {v rSup { size 8{2} } }} = { {3} over {2} } ital "kT"} {}

Before substituting values into this equation, we must convert the given temperature to kelvins. This conversion gives T = ( 20 . 0 + 273 ) K = 293 K . size 12{T= \( "20" "." 0+"273" \) " K=293 K" "." } {}

Solution for (a)

The temperature alone is sufficient to find the average translational kinetic energy. Substituting the temperature into the translational kinetic energy equation gives

KE ¯ = 3 2 kT = 3 2 1 . 38 × 10 23 J/K 293 K = 6 . 07 × 10 21 J . size 12{ {overline {"KE"}} = { {3} over {2} } ital "kT"= { {3} over {2} } left (1 "." "38" times "10" rSup { size 8{ - "23"} } " J/K" right ) left ("293"" K" right )=6 "." "07" times "10" rSup { size 8{ - "21"} } `J "." } {}

Strategy for (b)

Finding the rms speed of a nitrogen molecule involves a straightforward calculation using the equation

v 2 ¯ = v rms = 3 kT m , size 12{ sqrt { {overline {v rSup { size 8{2} } }} } =v rSub { size 8{"rms"} } = sqrt { { {3 ital "kT"} over {m} } } ,} {}

but we must first find the mass of a nitrogen molecule. Using the molecular mass of nitrogen N 2 size 12{N rSub { size 8{2} } } {} from the periodic table,

m = 2 14 . 0067 × 10 3 kg/mol 6 . 02 × 10 23 mol 1 = 4 . 65 × 10 26 kg . size 12{m= { {2 left ("14" "." "0067" right ) times "10" rSup { size 8{ - 3} } `"kg/mol"} over {6 "." "02" times "10" rSup { size 8{"23"} } `"mol" rSup { size 8{ - 1} } } } =4 "." "65" times "10" rSup { size 8{ - "26"} } `"kg" "." } {}

Solution for (b)

Substituting this mass and the value for k size 12{k} {} into the equation for v rms size 12{v rSub { size 8{"rms"} } } {} yields

v rms = 3 kT m = 3 1 . 38 × 10 23 J/K 293 K 4 . 65 × 10 –26 kg = 511 m/s . size 12{v rSub { size 8{"rms"} } = sqrt { { {3 ital "kT"} over {m} } } = sqrt { { {3 left (1 "." "38" times "10" rSup { size 8{–"23"} } " J/K" right ) left ("293 K" right )} over {4 "." "65" times "10" rSup { size 8{"–26"} } " kg"} } } ="511"" m/s" "." } {}

Discussion

Note that the average kinetic energy of the molecule is independent of the type of molecule. The average translational kinetic energy depends only on absolute temperature. The kinetic energy is very small compared to macroscopic energies, so that we do not feel when an air molecule is hitting our skin. The rms velocity of the nitrogen molecule is surprisingly large. These large molecular velocities do not yield macroscopic movement of air, since the molecules move in all directions with equal likelihood. The mean free path (the distance a molecule can move on average between collisions) of molecules in air is very small, and so the molecules move rapidly but do not get very far in a second. The high value for rms speed is reflected in the speed of sound, however, which is about 340 m/s at room temperature. The faster the rms speed of air molecules, the faster that sound vibrations can be transferred through the air. The speed of sound increases with temperature and is greater in gases with small molecular masses, such as helium. (See [link] .)

In part a of the figure, circles represent molecules distributed in a gas. Attached to each circle is a vector representing velocity. The circles have a random arrangement, while the vector arrows have random orientations and lengths. In part b of the figure, an arc represents a sound wave as it passes through a gas. The velocity of each molecule along the peak of the wave is roughly oriented parallel to the transmission direction of the wave.
(a) There are many molecules moving so fast in an ordinary gas that they collide a billion 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.

Questions & Answers

a 50kg mass is placed on a frictionless piston fitted to a gas cylinder .If 149 kelvin of heat is supplied to the cylinder increasing the internal energy by 100 joules,determine the height through which the mass of the piston raise
Lawal Reply
what is thermodynamics
wana Reply
thermodynamic is a branch of physics that teaches on the relationship about heat and anyother form of energy
Emmanuel
could you please help solve question on thermodynamics
Lawal
if a mass of 149 of heat is supplied and there's an increase in internal energy of 100jouls,find the raise in height
Lawal
if l cary box and stop is ther any work
Tamirat Reply
no that because u have moved no distance. for work to be performed a force needs to be applied and a distance needs to be moved
Emmanuel
Different between fundamental unit and derived unit
Alimi Reply
fundamental unit are independent quantities that do not depend on any other unit while derived unit are quantities that depend on two or more units for it definition
Emmanuel
what is nuclear fission
Sadik Reply
hello
Shawty
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Shawty
miss your absence here...
Shawty
what is a vector
Temitayo Reply
vectors are quantities that have numerical value or magnitude and direction.
Muhammad
what is regelation
Oladele
vector is any quantity that has magnitude and direction
Emmanuel
Physics is a physical science that deals with the study of matter in relation to energy
Divine Reply
Hi
Jimoh
hello
Salaudeen
hello
Sadik
Yes
Maxamuud
hi everyone
Muhammad
what is physics
Rhema Reply
physics is a physical science that deals with the study of matter in relation to energy
Osayuwa
a15kg powerexerted by the foresafter 3second
Firdos Reply
what is displacement
Xolani Reply
movement in a direction
Jason
hello
Hosea
Hey
Smart
haider
Explain why magnetic damping might not be effective on an object made of several thin conducting layers separated by insulation? can someone please explain this i need it for my final exam
anas Reply
Hi
saeid
hi
Yimam
Hi
Jimoh
An object made of several thin conducting layers separated by insulation may not be affected by magnetic damping because the eddy current produced in each layer due to induction will be very small and the opposing magnetic flux produced by the eddy currents will be very small
Muhammad
What is thê principle behind movement of thê taps control
Oluwakayode Reply
while
Hosea
what is atomic mass
thomas Reply
this is the mass of an atom of an element in ratio with the mass of carbon-atom
Chukwuka
show me how to get the accuracies of the values of the resistors for the two circuits i.e for series and parallel sides
Jesuovie Reply
Explain why it is difficult to have an ideal machine in real life situations.
Isaac Reply
tell me
Promise
what's the s . i unit for couple?
Promise
its s.i unit is Nm
Covenant
Force×perpendicular distance N×m=Nm
Oluwakayode
İt iş diffucult to have idêal machine because of FRİCTİON definitely reduce thê efficiency
Oluwakayode
It is difficult to have an ideal machine in real life situation because in ideal machines all the input energy should be converted to output energy . But , some part of energy is always lost in overcoming friction and input energy is always greater than output energy . Hence , no machine is ideal.
Muhammad
Practice Key Terms 1

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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