Maxwell-Boltzmann Distribution - Online Tutor, Practice Problems & Exam Prep

The Maxwell-Boltzmann distribution provides insight into the speeds of gas molecules at various temperatures, illustrating the probability of finding molecules at certain velocities. Key velocities on the distribution curve include the probable speed, mean speed, and root mean square speed. The probable speed is the most common velocity among gas molecules, while the mean speed is the average velocity, calculated using the equation:

$\sqrt{\frac{8}{RT}/M}$

where R is the gas constant and M is the molar mass. The root mean square speed, given by:

$\sqrt{\frac{3}{RT}/M}$

represents the square root of the average of the squares of the velocities, and it is the highest on the curve, indicating the fastest molecules. These speeds are crucial for understanding gas behavior and are represented on a distribution curve, with the Y-axis indicating the probability and the X-axis representing velocity. The curve shows that at any given temperature, there is a variety of molecular velocities, with the root mean square speed being the greatest, followed by the mean speed, and the probable speed being the least.

Maxwell-Boltzmann Distribution

So the Maxwell Boltzmann distribution is a probability distribution that describes the speed of ideal gases at a given temperature. So as temperature changes, the velocities of gas molecules will vary. Now a probability distribution itself is just the region of the curve that shows the relative number of gas molecules.

So we choose a temperature and we can get the relative number of gas molecules traveling at that velocity. Now to do this, we use a distribution function to determine these varying velocities. But it's beyond the scope of this class. So this complex formula, don't worry about it too much. You're not going to see it within this course.

Instead, we're going to use our memory tool that just remember 283 for your velocity O. We'll see different types of velocities when looking at a distribution curve, and we'll use 283 to help us remember which is which.

Maxwell-Boltzmann Distribution

So like I said earlier, just remember 283 for your velocity. So here we have three different velocities or speed that involved with a distribution curve. So here we have two, then we have 8:00 and then we have three. The first speed or velocity represents the speed at the top of the curve that represents the largest number of molecules with that speed. This is called your probable speed. So if you had to guess, you would say a majority of gases within a container have this speed most likely.

Next we have the average speed of gaseous molecules. What's another word for average mean? So here mean speed and it's equal to 8RT. Now next we have the speed that is the square root of the average speed squared. So that's just our root mean square speed. If you've watched this video earlier, you know it's pretty familiar to us. Here, root mean square speed equals 3RT. Now, when we're talking about MRT, M here represents the molar mass of the gas in kilograms per mole. So all of these are kilograms per mole because we're talking about speed or velocity.

Remember RR becomes 8.314 and it'll be joules over moles times K. And then temperature as always is in Kelvin when we're dealing with calculations. Now this probable speed, mean speed and root mean square speed can be transfixed or put onto a distribution curve. So let's take a look at this distribution curve here. So we're going to say here that in our distribution curve on our Y axis we have our probability distribution. This can be seen as the likelihood of X number of gas molecules within a container existing there.

And on our X axis we have our velocity from zero all the way up to 1350 meters per second. We can see at the very top we have our probable speed. What this is telling me, it's telling me at 400 meters per second, most likely if you're going to look at a particular gas, it has the most likely chance of falling here around 400 meters per second because that's where the curve is the highest. Next, right next to it is we have our average or mean speed. Notice here that our probable speed is around 400 meters per second and our average or mean speed is a little bit higher than that and then even higher than both of them is our root mean square speed. We see that it kind of falls close to 600 meters per second.

So here this distribution curve is just showing us the varying velocities for a collection of gases and what we need to understand in terms of velocity of increasing velocity. We would say that root mean square speed we can see is the highest 'cause it's closest to 600. Then we can see next would be our average or mean speed, and then we would see that our probable speed would be the lowest in terms of velocity when it comes to this chart. So that's the way we need to think about it in terms of the different types of velocities or speeds that exist at any given temperature. That's the whole idea behind the Maxwell distribution curve.

Maxwell-Boltzmann Distribution Example 1

So Here it says to calculate the most probable speed of F₂ molecules at 335 Kelvin. So remember most probable speed is VP=2RT over the molar mass. So here we're going to say that we have 2∗8.314. Now remember 8.314 uses the units of joules over moles times K.

But remember that one Joule equals kilograms times meter squared over second squared. So I can substitute this in for Joules. When we do that, that's going to give me my units of kilograms times meter squared divided by seconds squared, and then bring down moles times Kelvin as well, times 335 Kelvin divided by the molar mass in kilograms per mole.

For F₂ molecules we have two fluorines, and according to the periodic table, there are atomic masses are 19 grams respectively, so that gives me 38g per mole. Converting the grams to kilograms means that we'll have 0.038kg per mole cancel out units, moles cancel out, Kelvin's cancel out, kilograms cancel out.

So we'll have here meter squared over second squared, and we take the square root. When we do that, we get 383 meters per second as the most probable speed for F₂ molecules.

Calculate the molar mass of an unknown gas if its average speed is 920 m/s at 303 K.

• A. 0.0105 kg/mol
• B. 0.0136 kg/mol
• C. 0.0238 kg/mol
• D. 0.0262 kg/mol
• E. 0.0281 kg/mol

Here’s what students ask on this topic:

What does the Maxwell-Boltzmann distribution show?

The Maxwell-Boltzmann distribution is a statistical law that describes the distribution of speeds among particles in a gas at a certain temperature, assuming the gas is in thermal equilibrium. This distribution shows that in a sample of gas, there will be a few particles with very low speeds, most particles with moderate speeds, and a few particles with very high speeds. The shape of the distribution curve is determined by the temperature of the gas and the mass of the particles. As the temperature increases, the peak of the curve flattens and broadens, indicating that a greater number of particles are moving at higher speeds. The Maxwell-Boltzmann distribution is fundamental in the field of statistical mechanics and is crucial for understanding phenomena such as diffusion, viscosity, and the rate of chemical reactions.

How do you read the Maxwell-Boltzmann distribution?

The Maxwell-Boltzmann distribution is a statistical distribution that describes the energy distribution of particles in a gas that is in thermal equilibrium. To read the Maxwell-Boltzmann distribution, you'll typically look at a graph where the x-axis represents particle speeds (or sometimes energies), and the y-axis represents the number of particles or the probability of finding a particle with that speed.

The curve you see on the graph is the Maxwell-Boltzmann distribution curve. It starts at zero speed, rises to a peak, and then falls off again as speed increases. The peak of the curve corresponds to the most probable speed—not the average or the maximum speed, but the speed that most particles are likely to have.

The area under the curve represents the total number of particles in the system. As temperature increases, the peak of the curve will shift to the right, indicating that the most probable speed increases. The curve also becomes flatter and broader, showing that particles are more likely to have a wider range of speeds at higher temperatures.

How do you draw the Maxwell-Boltzmann distribution?

To draw the Maxwell-Boltzmann distribution, which describes the distribution of speeds of particles in a gas, you'll need to follow these steps:

1. Set Up Axes: On a graph, label the horizontal axis as 'Speed' (v) and the vertical axis as 'Number of Particles' or 'Probability Density' (f(v)).
2. Plot the Curve: The Maxwell-Boltzmann distribution is not a straight line but a curve that peaks and then tapers off. The peak represents the most probable speed. To plot this, you would typically use a formula for the distribution, $f\left(v\right)=A*v^2*e^{-B{v}^2}$, where A and B are constants that depend on the temperature and mass of the particles.
3. Identify the Peak: The curve starts at the origin (0,0), rises to a peak, and then gradually falls off towards the horizontal axis as speed increases. The peak occurs at a speed where the kinetic energy is most probable, not at the highest speed.
4. Asymptotic Tail: As the speed increases further, the curve approaches the speed axis asymptotically, meaning it gets closer and closer to the axis but never actually touches it.
5. Temperature Variation: If you're showing multiple distributions at different temperatures, remember that