Average kinetic energy of molecules

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Discussion Overview

The discussion revolves around the average kinetic energy of molecules, specifically its derivation and the underlying principles. Participants explore the relationship between kinetic energy, temperature, and the ideal gas law, with a focus on conceptual understanding without calculus.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Shounak requests a derivation of the average kinetic energy formula, k = 3/2 KT, without calculus.
  • Some participants explain that each degree of freedom contributes 1/2 kT of energy, noting that point objects have 3 degrees of freedom while diatomic molecules have 5.
  • A detailed derivation is provided that connects molecular collisions with pressure and kinetic energy, leading to the relationship PV = 2N(KE)/3.
  • It is noted that the N/3 factor accounts for the distribution of molecular motion in three dimensions.
  • Participants clarify that the kinetic energy discussed is translational kinetic energy, which is related to temperature, while acknowledging other forms of kinetic energy exist.
  • Another participant seeks clarification on the connection between the ideal gas law and the kinetic energy expression, indicating a desire for further explanation.

Areas of Agreement / Disagreement

Participants generally agree on the derivation process and the role of degrees of freedom in kinetic energy, but there are unresolved questions regarding the connections between different equations and concepts, indicating that multiple views remain on the topic.

Contextual Notes

Some assumptions about the ideal gas behavior and the definitions of kinetic energy types are not fully explored, leaving certain aspects of the discussion open to interpretation.

Who May Find This Useful

This discussion may be useful for students and enthusiasts of physics seeking to understand the kinetic theory of gases, the derivation of related formulas, and the conceptual underpinnings of temperature and energy in molecular systems.

shounakbhatta
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Hello,

The formula for average kinetic energy of molecule is:

k=3/2KT.

Can anyone please explain the derivation without using calculus?

Thanks,

-- Shounak
 
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Each degree of freedom contains 1/2 kT of energy; for point objects there are only 3 degrees of freedom. Diatomic molecules have 5 degrees of freedom
 
To start right at the beginning the problem is to find an expression for the pressure (force) due to molecules colliding with a wall.
Consider a molecule of mass m moving with speed (velocity) c inside a container of dimensions l x a x b (volume = lab)
The molecule strikes face ab and rebounds perfectly elastically so change in
momentum = 2mc
Force on face ab = change in momentum x Number of collisions per second
so force due to 1 molecule = 2mc/(2l/c) = mc2/l
So pressure on ab due to 1 molecule = mc2/lab = mc2/V
If there are N molecules in the container then, on average, N/3 move in the x, y and z directions.
Therefore pressure on ab due to N molecules = Nmc2/3V
or PV = Nmc2/3
Now KE of molecules = mc2/2 so PV = 2N(KE)/3
If there is 1 mole of molecules then N = Na
Therefore PV = 2Na x KE/3 for 1 mole
You also know that PV = RT for 1mole
so 2Na x KE/3 = RT
This give KE = 3/2 RT/Na
R/Na is a ratio of 2 constants and is known as Boltzmann's constant, k.

I hope that this is MORE than what you needed.
 
Thank you very much for the derivation. The N/3 factor comes into play due to the x,y,z three directions, right?
 
correct.
In addition the equation should have an 'average' velocity known as the root mean square velocity.
I did not include that to keep the typing as simple as possible.
Also the KE is called TRANSLATIONAL KINETIC ENERGY. There are other kinetic energies (rotational and vibrational) but it is translational KE that 'shows' as Temperature
 
Hello Technician,

Thank you very much for all the reply.
 
Hello jtbell,

I have just one small thing to ask you. The link that you have provided tells, the ideal gas law PV=nRT which is again:PV=2/3N(1/2mv^2).

Can you please show me how it came?

Thanks.
 

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