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 𝑥-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
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
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
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