Question

Imagine a gas diffusing through air in an arrangement like that in Figure $14-16 a$. The diffusion rate is the number of gas atoms per second diffusing from the left to the right end of the channel. The faster the atoms move, the greater is the diffusion rate, so the diffusion rate is proportional to the rms speed of the atoms. A deuterium atom is very similar to a hydrogen atom (they both have a single electron that orbits the nucleus), except that deuterium has twice the mass. (a) If hydrogen and deuterium atoms are at the same temperature, which, if either, has the greater average kinetic energy? Give your reasoning. (b) For a given temperature, which atom, if either, would you expect to have the greater diffusion rate? Why? Problem The atomic mass of a hydrogen atom is $1.0 \mathrm{u}$ and that of a deuterium atom is $2.0 \mathrm{u}$. Find the ratio of the diffusion rate of hydrogen to that of deuterium. Be sure your answer is consistent with part (b) of the Concept Questions.Concept Questions Imagine a gas diffusing through air in an arrangement like that in Figure $14-16 a$. The diffusion rate is the number of gas atoms per second diffusing from the left to the right end of the channel. The faster the atoms move, the greater is the diffusion rate, so the diffusion rate is proportional to the rms speed of the atoms. A deuterium atom is very similar to a hydrogen atom (they both have a single electron that orbits the nucleus), except that deuterium has twice the mass. (a) If hydrogen and deuterium atoms are at the same temperature, which, if either, has the greater average kinetic energy? Give your reasoning. (b) For a given temperature, which atom, if either, would you expect to have the greater diffusion rate? Why? Problem The atomic mass of a hydrogen atom is $1.0 \mathrm{u}$ and that of a deuterium atom is $2.0 \mathrm{u}$. Find the ratio of the diffusion rate of hydrogen to that of deuterium. Be sure your answer is consistent with part (b) of the Concept Questions.

Imagine a gas diffusing through air in an arrangement like that in Figure $14-16 a$. The diffusion rate is the number of gas atoms per second diffusing from the left to the right end of the channel. The faster the atoms move, the greater is the diffusion rate, so the diffusion rate is proportional to the rms speed of the atoms. A deuterium atom is very similar to a hydrogen atom (they both have a single electron that orbits the nucleus), except that deuterium has twice the mass. (a) If hydrogen and deuterium atoms are at the same temperature, which, if either, has the greater average kinetic energy? Give your reasoning. (b) For a given temperature, which atom, if either, would you expect to have the greater diffusion rate? Why? Problem The atomic mass of a hydrogen atom is $1.0 \mathrm{u}$ and that of a deuterium atom is $2.0 \mathrm{u}$. Find the ratio of the diffusion rate of hydrogen to that of deuterium. Be sure your answer is consistent with part (b) of the Concept Questions.Concept Questions Imagine a gas diffusing through air in an arrangement like that in Figure $14-16 a$. The diffusion rate is the number of gas atoms per second diffusing from the left to the right end of the channel. The faster the atoms move, the greater is the diffusion rate, so the diffusion rate is proportional to the rms speed of the atoms. A deuterium atom is very similar to a hydrogen atom (they both have a single electron that orbits the nucleus), except that deuterium has twice the mass. (a) If hydrogen and deuterium atoms are at the same temperature, which, if either, has the greater average kinetic energy? Give your reasoning. (b) For a given temperature, which atom, if either, would you expect to have the greater diffusion rate? Why? Problem The atomic mass of a hydrogen atom is $1.0 \mathrm{u}$ and that of a deuterium atom is $2.0 \mathrm{u}$. Find the ratio of the diffusion rate of hydrogen to that of deuterium. Be sure your answer is consistent with part (b) of the Concept Questions.

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Physics

John D. Cutnell,… 7th Edition

Chapter 14, Problem 62

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Solved by Expert Alex Plastow

05/24/2021

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Since the temperature is the same for both hydrogen and deuterium, their average kinetic energy will also be the same. Therefore, neither hydrogen nor deuterium has a greater average kinetic energy. Show more…

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Imagine a gas diffusing through air in an arrangement like that in Figure $14-16 a$. The diffusion rate is the number of gas atoms per second diffusing from the left to the right end of the channel. The faster the atoms move, the greater is the diffusion rate, so the diffusion rate is proportional to the rms speed of the atoms. A deuterium atom is very similar to a hydrogen atom (they both have a single electron that orbits the nucleus), except that deuterium has twice the mass. (a) If hydrogen and deuterium atoms are at the same temperature, which, if either, has the greater average kinetic energy? Give your reasoning. (b) For a given temperature, which atom, if either, would you expect to have the greater diffusion rate? Why? Problem The atomic mass of a hydrogen atom is $1.0 \mathrm{u}$ and that of a deuterium atom is $2.0 \mathrm{u}$. Find the ratio of the diffusion rate of hydrogen to that of deuterium. Be sure your answer is consistent with part (b) of the Concept Questions.Concept Questions Imagine a gas diffusing through air in an arrangement like that in Figure $14-16 a$. The diffusion rate is the number of gas atoms per second diffusing from the left to the right end of the channel. The faster the atoms move, the greater is the diffusion rate, so the diffusion rate is proportional to the rms speed of the atoms. A deuterium atom is very similar to a hydrogen atom (they both have a single electron that orbits the nucleus), except that deuterium has twice the mass. (a) If hydrogen and deuterium atoms are at the same temperature, which, if either, has the greater average kinetic energy? Give your reasoning. (b) For a given temperature, which atom, if either, would you expect to have the greater diffusion rate? Why? Problem The atomic mass of a hydrogen atom is $1.0 \mathrm{u}$ and that of a deuterium atom is $2.0 \mathrm{u}$. Find the ratio of the diffusion rate of hydrogen to that of deuterium. Be sure your answer is consistent with part (b) of the Concept Questions.

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Key Concepts

Kinetic Energy at a Given Temperature

In kinetic theory, the average kinetic energy of gas particles is determined solely by the temperature of the system and is given by an expression such as (3/2)kT. This indicates that at the same temperature, all gases have the same average kinetic energy regardless of their masses.

Root Mean Square (RMS) Speed

The rms speed of gas particles is a measure of the speed at which particles are moving on average and is calculated using the formula v_rms = sqrt(3kT/m). This shows that, for a given temperature, lighter particles will have a higher rms speed than heavier ones.

Diffusion Rate and Molecular Speed

The diffusion rate in gases is directly related to the average speed of the gas particles. A higher rms speed means that gas particles are more likely to traverse a given distance in a shorter time, which results in an increased rate of diffusion.

Mass Dependency in Kinetic Theory

Mass plays a crucial role in determining the speed of gas particles at a given temperature. Even though different gases at the same temperature have the same average kinetic energy, lighter particles have higher speeds because the kinetic energy is distributed over a smaller mass. This difference in speed affects physical processes such as diffusion, where lighter particles tend to diffuse more rapidly than heavier ones.

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Transcript

00:02 All right, let's consider a scenario in which we have two variations, isotopes, two flavors, hydrogen.

00:15 The first one we'll just call hydrogen.

00:18 The other one called, i might be butchering this pronunciation here, but deuterium.

00:33 Okay, so if both of the temperatures are equal, which can we expect to have a greater kinetic energy? so you might recall a kinetic energy can be evaluated using 3 over 2, ultimate constant, two times temperature, equal to the relationship can be determined experimentally because temperature is directly proportional to the speed of the particles in the gas.

01:13 So i do not expect that either of them have greater kinetic energy.

01:28 And moving on, which diffuses fast.

01:34 At a given temperature.

01:38 So we can also solve for kinetic energy using this expression right here.

01:52 We know that the kinetic energy is equal because the temperature is equal.

01:58 So we would expect this expression to be true.

02:12 Say we, okay, say for hydrogen, we have a decrease in mass or it's decreased relative to deuterium.

02:25 However, if this were to happen and for this relationship to hold for any given set of atoms, particles, excuse me, i'd expect the velocity of hydrogen also need to increase to break even with the kinetic energy of deuterium.

02:44 So, i'd expect hydrogen uses faster.

02:56 It's a strange looking at.

03:07 My sloppy cursive, much better.

03:11 All right.

03:14 So we're given a problem in which...

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