Problem 81 - Chemistry The Central Science

Question

Explain the difference between average speed and root-meansquare speed. Which is larger for a given gas sample at a fixed temperature?

Step-by-Step Solution

Answer

The difference between average speed and root-mean-square (RMS) speed lies in the way individual speeds are treated in their calculations. Average speed (\(\overline{v}\)) involves summing the absolute speeds of individual molecules and dividing by the total number of molecules, while RMS speed (\(v_{rms}\)) is determined by finding the square root of the average of the squares of individual speeds. The RMS speed is generally larger than the average speed for a gas sample at a fixed temperature, as it emphasizes larger speed values more, resulting in a higher overall speed compared to the average speed.

Step 1: Definition of Average Speed

The average speed of gas molecules in a sample is calculated by finding the sum of the absolute speeds of all individual molecules and then dividing by the total number of molecules. Mathematically, this can be represented as: \( \overline{v} = \frac{1}{N}\sum_{i=1}^{N}\left |v_i \right | \) where \ \( \overline{v} \) = average speed, \ N = total number of molecules, and \ \( |v_i |\) = absolute speed of the i-th molecule.

Step 2: Definition of Root-Mean-Square Speed

The root-mean-square (RMS) speed of gas molecules in a sample is calculated by finding the square root of the average of the square of the speeds of all individual molecules. Mathematically, this can be represented as: \( v_{rms} = \sqrt{ \frac{1}{N}\sum_{i=1}^{N}v_i^2 } \) where \ \(v_{rms}\) = root-mean-square speed, \ N = total number of molecules, and \ \(v_i\) = speed of the i-th molecule.

Step 3: Difference Between Average Speed and Root-Mean-Square Speed

The main difference between average speed and RMS speed is in the way the speeds of the individual gas molecules are treated when calculating the respective values. In the average speed calculation, the absolute values of the individual speeds are added together without emphasis on the magnitude of the speeds, while in the RMS speed calculation, the square of the individual speeds emphasizes higher speed molecules, magnifying their contribution to the overall value.

Step 4: Comparison of Average Speed and Root-Mean-Square Speed

Generally, the value of root-mean-square speed (\(v_{rms}\)) is larger than the average speed (\(\overline{v}\)) for a given gas sample at a fixed temperature. This is due to the higher weight given to the larger speed values in the RMS speed calculation, which results in a higher overall speed for the RMS speed compared to the average speed. In conclusion, the main distinction between average speed and root-mean-square speed lies in the way the individual speeds are treated in their respective calculations. The root-mean-square speed is generally larger than the average speed for a gas sample at a fixed temperature, as it puts more emphasis on larger speed values when computing the overall value.

Key Concepts

Kinetic Molecular Theory

The kinetic molecular theory (KMT) provides a conceptual model for understanding the behavior of gases at the molecular level. It's based upon several key postulates:

• Gases consist of many tiny particles (molecules) in constant, random motion.
• These particles are so small compared to the distances between them that the volume of the individual molecules can be assumed to be negligible.
• The collisions between gas particles and between particles and the container walls are perfectly elastic, meaning there is no net loss of kinetic energy during collisions.
• There are no forces of attraction or repulsion between the particles.
• The average kinetic energy of gas particles is directly proportional to the gas temperature in kelvins.

This last point relates directly to the concepts of average speed and root-mean-square speed. Temperature serves as a measure of the average kinetic energy of the particles, so as temperature increases, so does the kinetic energy and consequently the speed of the gas molecules. Understanding KMT is crucial for interpreting why certain properties of gases behave the way they do and for making sense of how average speed and RMS speed are derived and why they differ.

Gas Molecule Speeds

When we talk about the speed of gas molecules, we refer to how fast they travel in their random motion. The speeds of these molecules in a sample vary widely, as collisions can quickly change their direction and kinetic energy. There's not just one speed but a distribution of speeds, often depicted as a graph known as the Maxwell-Boltzmann distribution.

The average speed, as calculated in the exercise, provides a single value representative of the entire sample, despite the large range of individual speeds. It gives us a sense of the median rate at which particles move. However, because the distribution of speeds is skewed towards higher speeds, the RMS speed, which accounts for the 'weight' of these faster speeds by squaring them before averaging, offers a different perspective that more accurately reflects the energy of the particles. Better understanding individual molecule speeds can help us predict behaviors of gases, like diffusion rates or reaction kinetics, which are influenced by the molecular motion.

Temperature and Molecular Speed

According to the kinetic molecular theory, there's a direct correlation between the temperature of a gas and the speeds of its molecules. Higher temperatures indicate higher average kinetic energies, leading to an increase in the speed of gas molecules.

Temperature is typically measured on the Kelvin scale in scientific contexts because it starts at absolute zero, the point where molecular motion ceases. Understanding this relationship can help explain the behavior of gases under different thermal conditions. For example, heating a gas will increase its temperature, making its molecules move faster, as reflected in both the average and RMS speeds. This increase in speed with temperature is also the reason why hot air balloons rise and why substances change their state of matter.

The exercise's comparison of average speed and RMS speed is a vivid demonstration of how temperature is fundamentally linked to molecular motion. The fact that RMS speed is consistently higher than the average speed at any given temperature reinforces the idea that temperature acts as a 'baseline' kinetic energy level for the particles in a gas.