Molecular Kinetic Theory Model 2

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Average molecular kinetic energy

The average molecular energy can be used to find the pressure of radiation.

Illustrative background for Kinetic energyIllustrative background for Kinetic energy ?? "content

Kinetic energy

  • We assume that a gas is homogeneous (the same everywhere) and isotropic (has the same value when measured in different directions).
  • This means that the average speed of a molecule in the x-direction must be the same as the y and the z-directions. So we can write:
    • vx=vy=vZ{v_x}={v_y}={v_Z}
  • This means that the average kinetic energy in each direction is the same.
Illustrative background for Momentum Illustrative background for Momentum  ?? "content


  • We consider a cube of length l which contains a gas.
  • If we only look at the x-direction, we can say that the time taken for a particle to travel from one end to the other is:
    • t=lvxt=\frac{l}{{v_x}}
  • If a particle collides with the wall of the box and is absorbed, we can write the change in momentum as:
    • Change in momentum = mass x velocity
    • P=mvxP=m{v_x}
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  • We can write the force exerted on the wall when it absorbs a particle as:
    • Force = change in momentum ÷ change in time
    • F=PtF=\frac{P}{t}
  • We can substitute in the equations for momentum and time to get:
    • F=mvx2lF=\frac{m{v_x}^2}{l}
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  • The pressure on the wall of the box when it absorbs one particle is given by the equation:
    • Pressure = force ÷ area
    • p=FAp=\frac{F}{A}
  • We can use the value for force and that the area is length squared to get:
    • p=mvx2l3p=\frac{m{v_x}^2}{l^3}
  • Since volume is length cubed we get:
    • p=mvx2Vp=\frac{m{v_x}^2}{V}
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Pressure 2

  • We have found the pressure for one particle if N particles hit the wall the pressure would be:
    • Total pressure = number of particles x pressure from one particle.
    • pt=Np{p_t}=Np
  • We can substitute the pressure in to give:
    • pt=Nmvx2V{p_t}=N\frac{m{v_x}^2}{V}
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Kinetic equation

  • Recalling that the velocity is the same in all directions, we can write an equation for the total velocity squared as:
    • vt2=vx2+vy2+vz2{v_t}^2= {v_x}^2+{v_y}^2+{v_z}^2
  • Because the velocities are equal, we can simplify to give:
    • vx2=13vt2{v_x}^2=\frac{1}{3}{v_t}^2
  • So the equation for pressure can be written as:
    • pV=13Nmv2pV=\frac{1}{3}Nmv^2

Molecular Kinetic Theory - Developments Over Time

Ideas about the underlying structure of materials have changed considerably over time.

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Early ideas

  • The idea that the atom, or atoms that are too small to directly view, move around is ancient. This dates from Lucretius in approx 50 BCE.
  • This idea was largely ignored because of the predominance of Aristotelian ideas about elements such as fire, earth, air and water.
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Particle nature of matter

  • The modern theory of the particle nature of matter was attributed to Bernoulli.
  • This was prior to the ideas of conservation of energy and how collisions between particles could be elastic.
  • The model did not predict anything in itself.
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More complex ideas

  • Clausius developed a much more complex set of ideas.
  • The observation of Brownian motion is the first direct evidence of the existence of particles.
  • The kinetic theory of gases became universally accepted because of Einstein and Smoluchowski’s theoretical model, which made specific predictions about Brownian motion and diffusion.

Jump to other topics

1Measurements & Errors

2Particles & Radiation


4Mechanics & Materials


6Further Mechanics & Thermal Physics (A2 only)

7Fields & Their Consequences (A2 only)

8Nuclear Physics (A2 only)

9Option: Astrophysics (A2 only)

10Option: Medical Physics (A2 only)

11Option: Engineering Physics (A2 only)

12Option: Turning Points in Physics (A2 only)

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