Question

Explain why the average velocity of air molecules in a closed auditorium is zero but their root-mean-square speed or average speed is not zero.

Short answer

Answer: The average velocity of air molecules in a closed auditorium is zero because their random motion results in no net displacement in any specific direction. However, their RMS speed and average speed are non-zero as they only consider the magnitude of the molecules' speeds, not the direction of their motion.

Step-by-step solution

Defining Average Velocity

Average velocity is defined as the total displacement of an object divided by the total time it takes to cover that displacement. Mathematically, the average velocity (v_avg) can be represented as: \[ v_\text{avg} = \frac{\Delta x}{\Delta t} \] Where Δx is the displacement and Δt is the time interval.

Defining RMS Speed

Root-mean-square (RMS) speed is a measure of the average speed of particles in a system, taking into account the fact that they might be moving in different directions. RMS speed is calculated by squaring individual particle speeds, averaging those squared speeds, and then taking the square root of that average. Mathematically, RMS speed (v_rms) can be represented as: \[ v_\text{rms} = \sqrt{\frac{1}{N} \sum_{i=1}^N v_i^2} \] Where N is the number of particles and vi is the speed of the ith particle.

Defining Average Speed

Average speed is defined as the total distance covered divided by the total time it takes to cover that distance. Unlike average velocity, which considers the change in position (displacement), average speed only considers the magnitude of the distance traveled. Mathematically, average speed (v_s) can be represented as: \[ v_\text{s} = \frac{\text{Total Distance}}{\Delta t} \]

Explaining why Average Velocity is Zero

In a closed auditorium, air molecules are constantly moving in random directions with random speeds. As a result, there is no overall net displacement in any specific direction. If we add up all the individual displacements of each molecule, the result will be zero due to their random motion. Therefore, the average velocity of these air molecules is zero.

Explaining why RMS Speed and Average Speed are non-zero

Even though the average velocity of air molecules is zero, their RMS speed and average speed are not zero. This is because these two quantities are measures of the magnitude of the molecules' speeds and are always positive. Since the air molecules are in constant motion, their individual speeds are non-zero. When calculating the RMS speed and average speed, the random directions of motion do not cancel out, as they do for average velocity, because RMS speed and average speed only consider the magnitude of the molecules' speeds. Thus, these quantities are non-zero. In conclusion, the average velocity of air molecules in a closed auditorium is zero because their random motion results in no net displacement in any direction. However, their RMS speed and average speed are nonzero as they only consider the magnitude of the molecules' speeds, not the direction of their motion.