Derivation of the average translational kinetic energy of a molecule

In summary, the conversation discusses the derivation of the average translational kinetic energy of a molecule and how velocities are added. The textbook states that the square of the magnitude of the 3-d velocity vector is always equal to the sum of the squared components. However, when using the average values for each component, the equation simplifies to v^2 = 3(vx^2)av. The importance of subscripts is emphasized in understanding this concept.
  • #1
NeuronalMan
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Homework Statement



Hello, this is not actually a homework problem. I just can't seem to understand the derivation of the average translational kinetic energy of a molecule. I am startled by the way the velocities are added.


Homework Equations




My undergraduate level textbook says that (vx^2)av = (vy^2)av = (vz^2)av, but then it says (v^2)av = (vx^2)av + (vy^2)av + (vz^2)av = 3(vx^2)av

How can velocities being vectors be added this way? I am surely missing something here.


The Attempt at a Solution

 
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  • #2
I thinking you are missing the importance of subscripts. It is certainly true that the square of the magnitude of the 3-d velocity vector is always

v2=vx2+vy2+vz2

what does this become when you replace the components with their averages which are all equal to each other?
 

1. What is the formula for calculating the average translational kinetic energy of a molecule?

The formula for calculating the average translational kinetic energy of a molecule is E = 1/2 mv^2, where m is the mass of the molecule and v is its velocity.

2. How is translational kinetic energy related to temperature?

Translational kinetic energy is directly related to temperature. As temperature increases, the average velocity of molecules also increases, resulting in an increase in translational kinetic energy.

3. Does the size or mass of a molecule affect its translational kinetic energy?

Yes, the mass of a molecule directly affects its translational kinetic energy. The larger the mass of a molecule, the slower its velocity will be at a given temperature, resulting in a lower translational kinetic energy.

4. Is there a limit to the translational kinetic energy of a molecule?

According to the kinetic theory of gases, there is no upper limit to the translational kinetic energy of a molecule. However, as temperature approaches absolute zero, the average velocity of molecules approaches zero, resulting in a decrease in translational kinetic energy.

5. How is the average translational kinetic energy of a molecule related to pressure?

The average translational kinetic energy of a molecule is directly proportional to pressure. As the average kinetic energy of molecules increases, there is an increase in the frequency and force of collisions, resulting in an increase in pressure.

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