Question

You are given the following group of particles \(\left(N_{n}\right.\) represents the number of particles that have a speed \(v_{n}\) ): $$\begin{array}{lc}N_{n} & v_{n}(\mathrm{~km} / \mathrm{s}) \\\\\hline 2 & 1.0 \\\4 & 2.0 \\\6 & 3.0 \\\8 & 4.0 \\\2 & 5.0\end{array}$$ (a) Compute the average speed \(v_{\mathrm{av}} .(b)\) Compute the rootmean- square speed \(v_{\text {rms. }}\) (c) Among the five speeds shown, which is the most probable speed \(v_{\mathrm{p}}\) for the entire group?

Short answer

The average speed (\(v_{av}\)) is 2.73 km/s. The rootmean-square speed (\(v_{rms}\)) is 3.27 km/s. The most probable speed (\(v_p\)) is 4.0 km/s.

Step-by-step solution

Calculate average speed

The average speed \(v_{av}\) is given by: \[ v_{av} = \frac{\text{total sum of all speeds}}{\text{total sum of all instances}} = \frac{1*2 + 2*4 + 3*6 + 4*8 + 5*2} {2 + 4 + 6 + 8 + 2} = \frac{60}{22} = 2.73 \, \mathrm{km/s}\]

Calculate rootmean-square speed

The rootmean-square speed \(v_{rms}\) is given by:\[ v_{rms} = \sqrt{\frac{\text{sum of the squares of all speeds}}{\text{total sum of all instances}}} = \sqrt{\frac{1^2*2 + 2^2*4 + 3^2*6 + 4^2*8 + 5^2*2}{2 + 4 + 6 + 8 + 2}} = \sqrt{\frac{120}{22}} = 3.27 \, \mathrm{km/s}\]

Determine the most probable speed

The most probable speed \(v_p\) for the entire group is the one associated with the highest number of instances. Let's look directly at the data: 2 instances of 1.0 km/s. 4 instances of 2.0 km/s. 6 instances of 3.0 km/s. 8 instances of 4.0 km/s. 2 instances of 5.0 km/s. From these counts, 4.0 km/s has the highest number of instances (8), so it is the most probable speed.

Key concepts

Average Speed

The concept of average speed in the kinetic theory of gases is fundamental. It helps us understand how fast the particles in a gas are moving on average. To compute the average speed, we consider the individual speeds of all particles and the number of particles traveling at each specific speed.
In our example, particles are moving at different speeds: 1.0, 2.0, 3.0, 4.0, and 5.0 km/s, with varying numbers of particles at each speed. To find the average speed, we first multiply the number of particles (or instances) by their respective speeds. Then, we add all these products together, providing the total sum of all speeds multiplied by their occurrences.
Finally, we divide this sum by the total number of particles. The formula looks like this:

• Average Speed, \(v_{av}\), = \( \frac{\sum_{i} N_{i} \cdot v_{i}}{\sum_{i} N_{i}} \)

Applying this formula to our problem gives an average speed of approximately 2.73 km/s.

Root-Mean-Square Speed

Root-mean-square speed (or RMS speed) provides a measure of the speed of particles in a gas that accounts for all velocities and helps us understand energy distribution. It's particularly valuable in thermodynamics to relate molecular speeds to temperature. The root-mean-square speed is calculated by taking the square root of the average of the squares of all present speeds.
First, we square each individual speed and multiply by the number of particles traveling at that speed. This captures the kinetic energy contribution of each group of particles. By summing these value and dividing by the total number of particles, we find the average of these squared speeds.
Finally, taking the square root of this average gives us the root-mean-square speed. The formula for RMS speed is:

• RMS Speed, \(v_{rms}\), = \( \sqrt{\frac{\sum_{i} N_{i} \cdot v_{i}^2}{\sum_{i} N_{i}}} \)

For our particle scenario, this formula results in an RMS speed of approximately 3.27 km/s.

Most Probable Speed

Most probable speed is a concept that helps identify the speed at which the majority of particles in a gas are moving. It is the speed associated with the highest number of particles at a given instance. This is particularly useful in determining the behavior of a large collection of particles.
In the given data, particles are moving at different speeds. To find the most probable speed, we should look for the speed that is paired with the largest number of particles. Essentially, it is a straightforward count of how many particles are traveling at each speed.
From our example, 4.0 km/s is the speed at which the largest number of particles, specifically 8 particles, is moving. Hence, the most probable speed is 4.0 km/s. Understanding this concept helps us quickly assess the dynamic state of a gas just by observing the particle distributions.