Video: Identifying the Plot That Represents the Average Molecular Kinetic Energy against the Temperature in Degrees Celsius of One Mole of an Ideal Gas in a Set of Plots

Which of the following is the plot that represents the average molecular kinetic energy against the temperature in degrees Celsius of one mole of an ideal gas? [A] Plot A [B] Plot B [C] Plot C [D] Plot D [E] Plot E

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Video Transcript

Which of the following is the plot that represents the average molecular kinetic energy against the temperature in degrees Celsius of one mole of an ideal gas?

We are presented with a number of possible plots or graphs that could show the relationship between the average molecular kinetic energy and the temperature for particles of an ideal gas. In these graphs, the average molecular kinetic energy, which we can designate k.e., would be placed on the 𝑦-axis. The temperature, which we can designate capital 𝑇, would be placed on the π‘₯-axis. In this question, an ideal gas is defined as one whose particles occupy negligible volume and do not interact with each other. In other words, the particles are negligibly small compared to their distance of separation.

First, let’s quickly look at our graphs to see what type of relationships they portray. The first graph shows a linear relationship between the average kinetic energy and the temperature. As the temperature increases, the kinetic energy is decreasing. This graph therefore has a negative slope.

The second graph is also a linear graph. As the temperature increases, the kinetic energy is increasing too. This graph has a positive slope and a positive value for the 𝑦-intercept. It does not go through the origin, so it’s not a proportional graph.

The third graph is also linear. As the temperature increases, so does the kinetic energy. This graph has a positive slope. And upon extrapolation, it appears to start from the origin. This graph shows that kinetic energy and temperature are proportional to each other. We could say that the kinetic energy is equal to a constant of proportionality multiplied by the temperature.

The fourth graph is a horizontal flat line. This line has zero slope. This graph would take the form kinetic energy equals 𝑛, where 𝑛 is a positive number.

The fifth graph is nonlinear. It shows that kinetic energy is inversely proportional to temperature. In other words, by whatever factor the kinetic energy changes by, the temperature will change by the inverse of that factor. We could express this mathematically by saying that the kinetic energy is equal to a constant divided by the temperature.

To explore the relationship between the average molecular kinetic energy of gas particles against their temperature, we need to examine the kinetic molecular theory of gases in more detail. The theory assumes that gas particles are small compared to their distance of separation. This is an important part of the understanding that these gas particles do not interact with each other. And this is part of the definition of an ideal gas that we saw earlier.

The theory also assumes that the gas particles are in constant random motion, although they’re not all moving at the same speed. As the particles are in constant random motion, they can of course collide with each other. All the collisions are assumed to be elastic. This means that kinetic energy is conserved in the collision. No heat is evolved.

If we were to take a fixed number of particles of gas at a low temperature, that is, a sample of cold gas, we would find that the average speed of the gas particles is rather low. If the gas particles were to absorb heat, the particles would move faster and their temperature would increase. Temperature is a measure of the average kinetic energy of these molecules.

At any given temperature, the individual gas particles are moving at different speeds with constant collisions and constantly changing directions. As the temperature increases, the particles acquire more kinetic energy. At higher temperatures, we see the gas particles have higher average speeds. They therefore have higher average velocities. And since the kinetic energy is related to the velocity by the equation a half times the mass times the velocity squared, these particles also have higher average kinetic energy values.

In fact, the average kinetic energy of any kind of molecule of an ideal gas per mole is equal to three divided by two multiplied by the ideal gas constant, 𝑅, multiplied by the kelvin temperature, 𝑇. This equation shows us that the average molecular kinetic energy per mole of gas is directly proportional to the kelvin temperature.

The third graph shows this relationship exactly. And it would in fact be the correct answer if the question were referring to the kelvin scale. At zero kelvin or absolute zero, the particles would have zero kinetic energy and the graph should start from the origin. However, this is not the correct answer as the question is referring to the Celsius scale.

A temperature of zero Celsius as indicated on these graphs would in fact be 273 kelvin. At this temperature, our particles will have some kinetic energy. Their kinetic energies will have a positive value. The correct graph will therefore show a positive 𝑦-intercept and a linear relationship between the kinetic energy and the temperature in Celsius. The second graph is therefore the correct answer.

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