Question

At a certain temperature the speeds of six gaseous molecules in a container are \(2.0,2.2,2.6,2.7,3.3,\) and \(3.5 \mathrm{~m} / \mathrm{s}\). Calculate the root-mean-square speed and the average speed of the molecules. These two average values are close to each other, but the root-mean-square value is always the larger of the two. Why?

Short answer

RMS speed is 2.77 m/s; average speed is 2.7167 m/s; RMS is larger due to squaring speeds.

Step-by-step solution

Understand the Problem

We need to calculate two different types of average speeds for gaseous molecules: the root-mean-square (RMS) speed and the average speed. Both involve the given speeds of the molecules: 2.0, 2.2, 2.6, 2.7, 3.3, and 3.5 m/s.

Calculate the Average Speed

The average speed is calculated by summing all the speeds and dividing by the number of speeds. Formula: \[ \text{Average Speed} = \frac{\sum\text{Speeds}}{n} \]Calculation:\[ \text{Average Speed} = \frac{2.0 + 2.2 + 2.6 + 2.7 + 3.3 + 3.5}{6} = \frac{16.3}{6} = 2.7167 \; m/s \]

Calculate Each Speed Squared

Square each of the given speeds. This is necessary for calculating the RMS speed. Speeds Squared:- \(2.0^2 = 4.0\)- \(2.2^2 = 4.84\)- \(2.6^2 = 6.76\)- \(2.7^2 = 7.29\)- \(3.3^2 = 10.89\)- \(3.5^2 = 12.25\)

Calculate Root-Mean-Square Speed

First, sum all the squared speeds calculated in the previous step. Divide that sum by the number of speeds, then take the square root of the result. Formula: \[ \text{RMS Speed} = \sqrt{\frac{\sum\text{Speeds Squared}}{n}} \]Calculation:\[ \text{RMS Speed} = \sqrt{\frac{4.0 + 4.84 + 6.76 + 7.29 + 10.89 + 12.25}{6}} = \sqrt{\frac{46.03}{6}} = \sqrt{7.6717} = 2.77 \; m/s \]

Compare RMS Speed with Average Speed

The calculated RMS speed is 2.77 m/s, while the average speed is 2.7167 m/s. The RMS speed is indeed slightly larger. This happens because squaring each speed before averaging gives more weight to the higher speeds, causing the RMS to be larger.

Key concepts

Gaseous Molecules

Gaseous molecules are in a constant state of motion, which is why their speeds are a fascinating topic in physics and chemistry. These molecules move randomly and collide with one another as well as with the walls of their container. Each molecule in a gas has a different speed at any given moment, leading to a diverse range of velocity measurements. In the context of studying gasses, understanding their motion is crucial for predicting the behavior of a gas as a whole. Researchers often use statistical methods to measure and interpret the average speeds of these molecules. This randomness at the molecular level translates into macroscopic properties like temperature and pressure that we can measure easily.

Average Speed

The average speed is a simple arithmetic calculation that gives us insight into how fast, on average, the molecules are moving. To calculate it, you take the sum of all the speeds and divide by the number of molecules. This provides a straightforward way to understand overall molecular motion.

• Calculate the sum of all individual speeds.
• Divide this sum by the total number of speeds.

The result gives a summary of the molecular motion without any bias towards lower or higher speeds. It is a basic measure and works well for understanding general trends in molecular kinetics.

Speed Calculation

Calculating the speed of gaseous molecules involves understanding both average speed and root-mean-square (RMS) speed. These calculations help us to get a more comprehensive view of the molecular motion. For average speed, it's a simple division of the total speed sum by the number of molecules.
For RMS speed, the process is slightly more complex.

• First, you square each individual speed to emphasize larger speeds.
• Then, you take the mean of these squared speeds by dividing by the number of molecules.
• Finally, take the square root of this mean to arrive at the RMS speed.

This method tends to give more prominence to higher speeds, reflecting more energetic molecular movements in the gas.

RMS and Average Speed Comparison

The comparison between RMS speed and average speed can be insightful when analyzing gaseous molecules.
RMS speed often appears slightly larger than the average speed because the squaring process during calculation gives extra emphasis to faster molecules. This squaring step essentially magnifies the effect of higher outlier speeds, causing the RMS speed to weigh more on these larger values.
Thus, while the average speed provides a simple mean of all speeds, RMS speed offers a weighted perspective, skewed towards higher speed contributions. This characteristic of RMS speed makes it useful in contexts where greater insight into energetic dynamics is required, such as in thermal physics and kinetic theory of gases.