Question

The speeds of a group of ten molecules are \(2.0,3.0,4.0, \ldots\), \(11 \mathrm{~km} / \mathrm{s}\). ( \(a\) ) Find the average speed of the group. (b) Calculate the root-mean-square speed of the group.

Short answer

(a) The average speed of the group is 6.0 km/s (b) The root-mean-square speed of the group is approximately 7.10 km/s.

Step-by-step solution

Calculate Average Speed

The average speed can be calculated by adding up all the molecule speeds, and then dividing by the count of molecules. Here are the speeds: \(2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0\) km/s, totaling ten molecules. Summing these up equals 60 km/s. Thus, the average speed equals \(\frac{60 \text{ km/s}}{10} = 6.0 \text{ km/s}\).

Calculate Square of Each Speed

In order to calculate the root-mean-square speed, we first need to find the square of each molecule's speed. The squared speeds are: \(4.0, 9.0, 16.0, 25.0, 36.0, 49.0, 64.0, 81.0, 100.0, 121.0\) (km/s)\(^2\).

Calculate Root-Mean-Square Speed

The root-mean-square speed is found by taking the square root of the average of these squared speeds. Add up the squared speeds to get 505 (km/s)\(^2\), and divide by the number of molecules (10) to find the average, which equals 50.5 (km/s)\(^2\). Taking the square root of this gives approximately 7.10 km/s.

Key concepts

Average Speed

To understand average speed in the context of molecular speeds, think of it as the typical speed at which a molecule travels within a group. Calculating the average speed is straightforward. Follow these steps:

• Sum up all of the individual speeds.
• Divide the total by the number of molecules.

In the exercise, we have ten molecules traveling at different speeds ranging from 2.0 km/s to 11.0 km/s. The total comes to 60.0 km/s when all these speeds are added together. Dividing by ten molecules, the result is 6.0 km/s as the average speed. This provides a simple snapshot of how fast the molecules are moving overall.

Root-Mean-Square Speed

Root-mean-square (RMS) speed offers a different perspective. It reflects the average speed of molecules on a more mathematically intricate level. The key here is the emphasis on the 'square'. It's particularly useful in assessing kinetic energy in gases. Here's how you calculate it:

• Square each molecule's speed individually.
• Take the average of these squared speeds.
• Finally, take the square root of this average.

For the given speeds, we calculate each squared speed and sum them up to 505 (km/s) extsuperscript{2}. Averaging over ten molecules gives 50.5 (km/s) extsuperscript{2}. The square root of this value yields approximately 7.10 km/s as the RMS speed.

Kinetic Theory of Gases

The kinetic theory of gases provides a framework for understanding gaseous behavior. It ties together molecular movements and gas properties like pressure and temperature. The theory is based on several assumptions:

• Gases consist of numerous small particles (molecules).
• These molecules move in random directions with various speeds.
• Collisions between molecules or with container walls are perfectly elastic.

These assumptions lead to the relationship between temperatures and molecular speeds. Higher temperatures correspond to higher average molecular speeds. This fundamental notion links to both average and RMS speeds, showing how molecular motion represents macroscopic gas properties.

Statistical Mechanics

Statistical mechanics takes the concepts of classical mechanics into the micro-world of particles, such as gas molecules. By applying statistics to vast numbers of particles, it bridges the gap between macroscopic and microscopic phenomena. Two key aspects:

• It considers probabilities of different molecular states and speeds.
• Derives macroscopic properties from microscopic behaviors.

In the context of our exercise, statistical mechanics helps explain why average and RMS speeds differ. Each reflects different statistical measures of the molecular motion within a gas. Insights from statistical mechanics allow us to predict properties of complex systems, reinforcing reliable models of real-world gases.