You have two identical containers, one containing gas A and...

You have two identical containers, one containing gas $A$ and the other gas $B .$ The masses of these molecules are $m_{A}=3.34 \times 10^{-27} \mathrm{kg}$ and $m_{B}=5.34 \times 10^{-26} \mathrm{kg} .$ Both gases are under the same pressure and are at $10.0^{\circ} \mathrm{C}$ . (a) Which molecules $(A$ or $B)$ have greater translational kinetic energy per molecule and rms speeds? (b) Now you want to raise the temperature of only one of these containers so that both gases will have the same only one of these containers so that both gases will have the same rms speed. For which gas should you raise the temperature? (c) At what temperature will you accomplish your goal? (d) Once you have accomplished your goal, which molecules $(A$ or $B)$ now have greater average translational kinetic energy per molecule?

You have two identical containers, one containing gas $A$ and the other gas $B .$ The masses of these molecules are $m_{A}=3.34 \times 10^{-27} \mathrm{kg}$ and $m_{B}=5.34 \times 10^{-26} \mathrm{kg} .$ Both gases are under the same pressure and are at $10.0^{\circ} \mathrm{C}$ . (a) Which molecules $(A$ or $B)$ have greater translational kinetic energy per molecule and rms speeds? (b) Now you want to raise the temperature of only one of these containers so that both gases will have the same only one of these containers so that both gases will have the same rms speed. For which gas should you raise the temperature? (c) At what temperature will you accomplish your goal? (d) Once
you have accomplished your goal, which molecules $(A$ or $B)$ now have greater average translational kinetic energy per molecule?

Key Concepts

Equipartition Theorem

This principle states that in thermal equilibrium each degree of freedom contributes equally to the total energy of a system. For translational motion of an ideal gas, each molecule has an average kinetic energy of (3/2)kT, showing that the average translational kinetic energy per molecule depends only on the temperature and not on the molecular mass.

Root Mean Square (rms) Speed

The rms speed is defined as the square root of the average of the squares of the speeds of the molecules and is given by the formula v_rms = ?(3kT/m) for an ideal gas. It connects temperature, molecular mass, and molecular speed, illustrating that for a given temperature, lighter molecules move faster than heavier ones.

Kinetic Energy–Temperature Relationship

In ideal gases, the average translational kinetic energy per molecule is directly proportional to the absolute temperature, as expressed by (3/2)kT. This relationship implies that altering the temperature of a gas changes its kinetic energy uniformly, regardless of individual molecular masses.

Molecular Mass and Speed

The equation for rms speed shows a dependence on molecular mass: for the same temperature, a larger molecular mass results in a lower rms speed. This inverse square root relationship explains why gases with different molecular masses, when at the same temperature, have different speeds even though each molecule has the same average kinetic energy.

Temperature Adjustment for Equal Speeds

When seeking to equalize the rms speeds of gases with different molecular masses, one must adjust the temperature of the gas with the heavier molecules. Increasing its temperature boosts the rms speed according to v_rms = ?(3kT/m), thereby allowing a setting where both lighter and heavier molecules can achieve the same speed, even though their kinetic energies will differ due to the temperature change.

Transcript

00:01 Hello, so for the first part of this problem, i'm going to write down the expressions we need to make the analysis.

00:11 So the average kinetic energy, the average kinetic energy, the gas, i'll be given by 3 over 2, kt, k is the both -man constant, t is the temperature, and the ritzman -speed is also.

00:42 So given by 3 k t over m.

01:00 So if we compare these two equations, so for the kinetic energy, the 3 over 2 is a constant, a constant, a constant.

01:12 So what it means is that the kinetic energy only depends on the temperature.

01:17 So from the question, because the temperature is the same, what it means is that a k -e, the average translational kinetic energy is the same because the temperature is the same and the kinetic energy depends only on the temperature so it's the same now when we go to the root mean square speed you realize that the it depends on the temperature and the mass so for the temperature side it's okay but for the mass so when you look at the masses of a and b they are different and when you look at the relationship between the vrms and the mass is an inverse relationship what it means is that if you have lighter molecules they're going to have a very high vrms so when you look at the mass a and b you realize that a is lighter than b so what it means is that a will have a higher rms than the b and so we are saying that because the mass of a is smaller than the mass of b then a is going to have a high vrms than b so the vrms so still on the a part the vrms of a is greater than the vrms of b.

03:28 So let's go to the b part.

03:32 For the b part, we want to know, so given this condition, we want to know which of the temperatures should we raise so that they become equal.

03:46 So like i said, the mass of a is lighter than the mass of b.

03:57 And so if we want to raise any temperature, then it's got to be b.

04:02 Because the question is asking that we want to to raise the temperature for one of them so that they have the same vrms.

04:15 So if you want to raise the temperature of any of the containers so that the vrms are the same.

04:22 So here you see in the part a, now the vrms of a is greater than the vrms of b.

04:29 So if you want the vrms to be equal, then you want to increase the temperature of b.

04:38 So for the v part, you're going to raise temperature of b so that they can have the same vrms.

04:53 Let's look at c.

04:57 So for c, let me check the question again quickly.

05:04 So for c, we want to know the temperature at which we will accomplish this goal.

05:11 All right.

05:12 So let's go back to vrms formula...

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