Maxwell-Boltzmann Distribution: Videos & Practice Problems

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

The Maxwell-Boltzmann distribution is a key concept in understanding the behavior of ideal gases, particularly in relation to their molecular speeds at a specific temperature. This distribution illustrates how the velocities of gas molecules change as temperature varies, highlighting that not all molecules move at the same speed. In essence, a probability distribution provides a visual representation of the relative number of gas molecules at different velocities, allowing us to identify how many molecules are traveling at a given speed. While the detailed mathematical formulation of this distribution is complex and not necessary for this course, it is important to grasp the general idea that temperature influences molecular speed.To simplify the understanding of this concept, a useful memory tool is the number 283, which can help recall the typical velocity associated with gas molecules in this context. By focusing on this number, students can better navigate the variations in molecular speeds represented in the distribution curve without needing to delve into the intricate details of the distribution function itself.

Maxwell-Boltzmann Curve

In the study of gas molecules, understanding the different types of speeds is crucial for analyzing their behavior within a container. Three key velocities are often discussed: probable speed, mean speed, and root mean square speed. The probable speed represents the velocity at which the largest number of gas molecules are found, indicating the most likely speed of the gas particles. This is typically the peak of the distribution curve.

The mean speed, also referred to as average speed, is calculated using the formula:

$$\text{Mean Speed} = \sqrt{\frac{8RT}{M}}$$

In this equation, \( R \) is the ideal gas constant (8.314 J/(mol·K)), \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas in kilograms per mole.

Another important speed is the root mean square speed, which is derived from the average speed and is given by the formula:

$$\text{Root Mean Square Speed} = \sqrt{\frac{3RT}{M}}$$

In this context, \( M \) again represents the molar mass of the gas, and the calculations are performed under the same conditions of temperature and pressure. The root mean square speed is typically higher than both the probable and mean speeds, reflecting the distribution of molecular speeds in a gas.

When visualizing these speeds on a distribution curve, the y-axis represents the probability distribution, indicating the likelihood of gas molecules having a certain speed, while the x-axis shows the velocity ranging from 0 to 1350 meters per second. The peak of the curve corresponds to the probable speed, which is often lower than the mean speed, and both are lower than the root mean square speed. This hierarchy of speeds illustrates that as the velocity increases, the root mean square speed is the highest, followed by the mean speed, and finally the probable speed.

This understanding of the Maxwell distribution curve is essential for grasping the kinetic theory of gases, as it highlights how molecular speeds vary at a given temperature and the statistical nature of gas behavior.

Most Probable Speed Example

To calculate the most probable speed of fluorine (F₂) molecules at a temperature of 335 Kelvin, we utilize the formula for most probable speed, which is given by:

\( v_p = \sqrt{\frac{2RT}{M}} \)

In this equation, \( R \) represents the ideal gas constant, which is \( 8.314 \, \text{J/(mol·K)} \). To ensure consistency in units, we note that 1 Joule can be expressed as \( \text{kg} \cdot \text{m}^2/\text{s}^2 \). Therefore, substituting this into our equation gives us:

\( v_p = \sqrt{\frac{2 \times 8.314 \, \text{kg} \cdot \text{m}^2/\text{s}^2 \cdot \text{mol}^{-1} \cdot \text{K}^{-1} \times 335 \, \text{K}}{M}} \)

Next, we need to determine the molar mass \( M \) of F₂. Each fluorine atom has an atomic mass of 19 grams, so for F₂, the molar mass is:

\( M = 2 \times 19 \, \text{g/mol} = 38 \, \text{g/mol} \)

Converting grams to kilograms, we find:

\( M = 0.038 \, \text{kg/mol} \)

Now, substituting \( M \) back into the equation, we can cancel out the units appropriately:

\( v_p = \sqrt{\frac{2 \times 8.314 \times 335}{0.038}} \)

After performing the calculations and simplifying the units, we find that the most probable speed \( v_p \) for F₂ molecules at 335 Kelvin is:

\( v_p \approx 383 \, \text{m/s} \)

This value represents the speed at which the majority of F₂ molecules are expected to travel at the given temperature.

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

Answer: (C)

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