Kinetic Theory of Gases Average question

In summary, the book discusses how to calculate the root mean square velocity of a gas using kinetic energy. This velocity is useful because it is the speed of a molecule that has an energy equal to the average energy of all the molecules in the gas. The average energy of the gas molecules is related (proportional to) the temperature of the gas.
  • #1
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Hey I'm studying the Kinetic Theory of gases and although it's in a chemistry
book it employs physics concepts ergo I have come here to ask you nice people :biggrin:

The 5th point is where I have a problem, the rest is just information for you.

I am assuming a particle is in a cube (ideal situations here) & it traverses a
distance l with a momentum p=mv.

1. As it travels forward it will hit the cube wall and the resultant change in
momentum will be;
[tex]change \ in \ momentum \ = \ (mv) \ - \ (-mv) \ = \ 2mv[/tex]

2. The number of collisions per second will be;
[tex]No.\ of \ collisions \ per \ second \ = \ \frac{velocity}{distance} \ = \ \frac{v}{l}[/tex]

3. The Change in momentum per second will be;
[tex]Change \ in \ momentum \ per \ second = 2mv \frac{v}{l} \ = \ \frac{2mv^2}{l}[/tex]

4. I've assumed one particle in one direction. Assuming n particles with
a total change in momentum per second of;
[tex]total \ change \ in \ momentum \ per \ second \ = \ \frac{2m}{l} \ \displaystyle\sum_{i=1}^n v_i^2 [/tex]

5. Here is where I have the problem. The text says to do this next;

Define the root square mean velocity, [tex]\overline{v} [/tex] such that;
[tex]\overline{v}^2 \ = \ \left( \ \frac{1}{n} \ \sum_{i=1}^n v_i^2 \right) [/tex]

What does this mean? Is this some mathematical concept from probabilities? I haven't done probabilities in math in like 5 years & even then that was in school :yuck: . Where did the [(2m)/l] fraction go?

Using this concept the book goes on to show that;
[tex] The \ total \ change \ in \ momentum \ = \ [2mn \overline{v}^2 ] / l [/tex]

(that is a fraction with l in the denominator, all the rest in the numerator [latex issues] )
I can't understand this (or any of the further concepts on pressure etc...) without getting this concept.
 
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  • #2
The root mean square velocity is useful because it is the speed of a molecule that has an energy equal to the average energy of all the molecules in the gas. This is because the kinetic energy of a molecule is [itex]mv^2/2[/itex]. If you add up all the squares of all the velocities of the molecules. multiply by m/2 and divide by n, you will get the average energy of the gas molecules. This is useful because the temperature of the gas is directly related (proportional to) the average energy of the gas molecules.

AM
 
  • #3
I don't see any logic nor reason for doing that mathematically.

I understand that you are converting to using kinetic energy but I mean, it seems as though you're just arbitrarily multiplying one side by m/2 then dividing by n.

What about multiplying both sides of the equation?
Or multiplying top and bottom by a clever choice of 1?

http://www.betz.lu/media/users/charel/math07.gif
 
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  • #4
2mv^2/l gives the force for just a single molecule of velocity v.If there are n such molecules all moving with the same velocity the total force would be n*2mv^2/l.The chance of the molecules all moving with the same velocity is very remote so instead of using v we use v bar(the mean squared velocity)which leads to the final equation you wrote.The 2m/l fraction hasn't gone anywhere.
 
  • #5
OH!

God it's so easy lol...

I must be so tired that it's obviously affecting my study @ this stage.

So the average velocity is just the sum of the velocities of all of the particles divided by the number of particles.

Great stuff, it makes even more sense conceptually to use this seeing as they won't all be hitting the walls at the same time. yes, thanks a lot. :)

------have to stop being scared away by sigma's :p-----
 

1. What is the Kinetic Theory of Gases?

The Kinetic Theory of Gases is a scientific theory that describes the behavior of gases in terms of the movement and interaction of their individual particles. It states that gases are made up of tiny particles that are in constant, random motion and that the pressure of a gas is caused by the collisions of these particles with the walls of their container.

2. What are the key assumptions of the Kinetic Theory of Gases?

The Kinetic Theory of Gases is based on the following key assumptions: 1) Gas particles are in constant, random motion; 2) Gas particles are point masses with no volume; 3) Gas particles do not interact with each other except during collisions; 4) Collisions between gas particles and with the walls of their container are perfectly elastic; and 5) The average kinetic energy of gas particles is directly proportional to the temperature of the gas.

3. How does the Kinetic Theory of Gases explain the relationship between temperature and pressure?

According to the Kinetic Theory of Gases, an increase in temperature leads to an increase in the average kinetic energy of gas particles. This increase in kinetic energy results in more frequent and energetic collisions between gas particles and the walls of their container, leading to a higher pressure.

4. Can the Kinetic Theory of Gases be applied to all gases?

The Kinetic Theory of Gases is a universal theory that can be applied to all gases, regardless of their chemical composition or properties. However, it is most accurate for gases at low pressures and high temperatures, where the behavior of gas particles is closer to the assumptions of the theory.

5. What are some real-life applications of the Kinetic Theory of Gases?

The Kinetic Theory of Gases has many practical applications, such as in the design and operation of gas-based technologies like refrigerators, air conditioners, and engines. It is also used in weather forecasting models and in the study of atmospheric gases. Additionally, the Kinetic Theory of Gases is essential for understanding the behavior of gases in chemical reactions and industrial processes.

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