When to use KE = 3/2 RT vs 1/2 mv^2

[OiOcha]

Homework Statement

The question asks whether the kinetic energy of a particle is increased if mass is increased. Particle fulfills criteria of an ideal gas.

Homework Equations

KE = 3/2RT = 1/2 mv^2

The Attempt at a Solution

My confusion is how can one derivation be dependent on mass and the other does not. Which do we use under which circumstances? Does KE depend on mass or does it not?

 

[QuantumQuest]

My confusion is how can one derivation be dependent on mass and the other does not. Which do we use under which circumstances? Does KE depend on mass or does it not?

As you say,

Particle fulfills criteria of an ideal gas.

What does this tell you about the two equations of KE? I'll hint you on the kinetic molecular theory. What are its postulates?

 

[haruspex]

how can one derivation be dependent on mass and the other does not.

Are you sure the other is not? Mass can affect it without necessarily appearing as m in the formula.

 

[OiOcha]

So what I'm getting is
KE = 3/2 RT for ideal gases. The kinetic molecular theory states that an ideal gas' KE is only affected by temperature. With that said, I'm still not understanding how/why mass does or does not play a role.

 

[QuantumQuest]

The kinetic molecular theory states that an ideal gas' KE is only affected by temperature

OK, why is that? Can you think of a reason for this?

 

[OiOcha]

Ideal gases are considered to be volumeless point masses. Would that mean it has zero/negligible mass, therefore we don't take it into account?

 

[QuantumQuest]

Ideal gases are considered to be volumeless point masses. Would that mean it has zero/negligible mass, therefore we don't take it into account?

I'll quote two postulates from Kinetic Molecular Theory

- Gases are composed of a large number of particles that behave like hard, spherical objects in a state of constant, random motion.
- Collisions between gas particles or collisions with the walls of the container are perfectly elastic. None of the energy of a gas particle is lost when it collides with another particle or with the walls of the container.

Add to these, that the average kinetic energy of a collection of gas particles, depends only on the temperature of the gas. Now, from these, can you see what is each equation's meaning? Can you see from the set of the postulates of Kinetic Molecular Theory, what is implied for the mass of each particle?

 

[haruspex]

So what I'm getting is
KE = 3/2 RT for ideal gases. The kinetic molecular theory states that an ideal gas' KE is only affected by temperature. With that said, I'm still not understanding how/why mass does or does not play a role.

But what is temperature, according to the theory? QuantumQuest's hints should help.

 

[OiOcha]

I'm sorry but I am still unable to see the connection or how one leads to the other.

Here's where I'm at:
As temperature is the measure of average kinetic energy of gas molecules, so if temperature is higher, that means average kinetic energy is higher. Since average kinetic energy is higher, we can also say that the molecules are moving with higher speed (which relates temperature to speed). Since collisions are elastic, that energy is never lost when molecules hit the walls of a container or each other. With that, there's a connection between temperature -->KE-->velocity, that's how we get KE= 1/2 mv^2 = 3/2 RT

EDIT: The textbook question I'm working on is regarding particles being ejected from a cathode ray tube. Electrons are being ejected. The question asks how does the kinetic energy of ejected protons compare to kinetic energy of ejected electrons. Given that both follow ideal gas behavior and temperature of the cathode is constant.

 

[haruspex]

so if temperature is higher, that means average kinetic energy is higher. Since average kinetic energy is higher, we can also say that the molecules are moving with higher speed

... or are more massive.

 

[OiOcha]

Would it be correct to say that because protons are more massive, they would have to be moving slower in order to have the same kinetic energy as electrons (less massive but greater speed). Also the fact that they have the same kinetic energy is inferred from the given that both systems are at the same temperature?

 

[haruspex]

Would it be correct to say that because protons are more massive, they would have to be moving slower in order to have the same kinetic energy as electrons (less massive but greater speed). Also the fact that they have the same kinetic energy is inferred from the given that both systems are at the same temperature?

Yes. But it's good that you clarified the set-up in post #9. From your original post, I took it that the mass was changing but the velocity remaining the same.