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Various Speeds of Gas Molecules

The motion of molecules in a gas is characterized by any of the following three speeds:
 
Root mean square speed It is defined as the square root of the mean of the squares of the speed of different molecules, i.e.,
 
57741.png
  • From the expression for pressure of ideal gas,
     
    57734.png
     
    57728.png
     
    57716.png 
  • 57710.png 57704.png
  • 57698.png
     
    [as M = NAm and R = NAk]
     
    ∴ Root mean square velocity,
     
    57692.png
Important Points
  • With rise in temperature, the rms speed of gas molecules increases as 58131.png
  • With increase in molecular weight, therms speed of gas molecules decreases as
     
    58125.png
  • The rms speed of gas molecules is 58118.png times that of the speed of sound in gas.
     
    As 58111.png and 58105.png
     
    ∴ 58099.png
  • The rms speed of gas molecules does not depend on the pressure of gas (if temperature remains constant) because P ∝ ρ (Boyle’s law.) If pressure is increased n times, then density will also increase by n times but vrms remains constant.
 
Most probable speed The particles of a gas have a range of speeds. This is defined as the speed which is possessed by maximum fraction of total number of molecules of the gas, i.e., if speeds of 10 molecules of a gas are 1, 2, 2, 3, 3, 3, 4, 5, 6, 6 km/s, then the most probable speed is 3 km/s, as maximum fraction of total molecules possess this speed.
 
As M is the molecular weight of gas PV = μRT and mass of gas = μM,
 
∴ Most probable speed,
57661.png
 
Average speed It is the arithmetic mean of the speeds of molecules in a gas at given temperature.
 
57655.png
and according to kinetic theory of gases
 
Average speed, 57649.png
 
Note:
  • vrms > vav > vmp (order remembering trick) (RAM)
  • vrms : vav : vmp = 58089.png
  • For oxygen gas molecules, vrms = 461 m/s, vav = 424.7 m/s, and vrms = 376.4 m/s.




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