Question

If three molecules have velocities \(0.5,1\) and 2 the ratio of rms speed and average speed is (The velocities are in \(\mathrm{km} / \mathrm{s}\) ) (A) \(0.134\) (B) \(1.34\) (C) \(1.134\) (D) \(13.4\)

Short answer

The ratio of rms speed to average speed for the given velocities is approximately 1.134.

Step-by-step solution

Calculate the average speed

To calculate the average speed, we simply add the velocities of all three molecules, and then divide the sum by the number of molecules (3). Average speed = (0.5 + 1 + 2) / 3

Solve for the average speed

Now, we can substitute the values and calculate the average speed: Average speed = (0.5 + 1 + 2) / 3 = 3.5 / 3 = 1.16667 km/s

Calculate the rms speed

To calculate the root-mean-square speed, we have to first square each molecule's velocity, add those squares together, divide the sum by the number of molecules (3), and then take the square root of the result. Rms speed = \(\sqrt{\frac{(0.5^2 + 1^2 + 2^2)}{3}}\)

Solve for the rms speed

Now, we can substitute the values and calculate the rms speed: Rms speed = \(\sqrt{\frac{(0.5^2 + 1^2 + 2^2)}{3}}\) = \(\sqrt{\frac{(0.25 + 1 + 4)}{3}}\) = \(\sqrt{\frac{5.25}{3}}\) = 1.31529 km/s

Calculate the ratio of rms speed to average speed

Finally, we will find the ratio of the rms speed and the average speed: Ratio = Rms speed / Average speed = 1.31529 / 1.16667

Solve for the ratio

Now, we can substitute the values we found in the previous steps to get the ratio: Ratio = 1.31529 / 1.16667 ≈ 1.127 Looking at the answer choices, the closest answer is (C) 1.134, hence the ratio of rms speed to average speed for the given velocities is approximately 1.134.

Key concepts

Root-Mean-Square Speed

The concept of root-mean-square (rms) speed is crucial in understanding how fast molecules in a gas move. It provides an "average" speed that is weighted by the square of the velocities, giving more importance to higher speeds. This is because velocity squared gives more weight to higher numbers, emphasizing faster molecules.
To calculate the rms speed, follow these steps:

• Square each velocity of the molecules.
• Add all these squared velocities together.
• Divide the sum by the number of molecules.
• Take the square root of the result. This is the rms speed.

The formula is given by: \[\text{RMS Speed} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + \, ...\, }{N}} \]where \(v_1, v_2, v_3\) are the velocities and \(N\) is the total number of molecules. In our example, this rms speed calculation takes the velocities of 0.5, 1, and 2 km/s, calculates their squares, finds their average, and takes the square root to find approximately 1.315 km/s.
Root-mean-square speed is useful in kinetic theory to predict the energy and dynamics of gaseous particles, providing insight beyond simple arithmetic averages by accounting for the distribution of molecular speeds.

Average Speed

Average speed is a straightforward and intuitive way of understanding the typical speed of particles. It gives a simple mean of all the velocities, balancing both slower and faster particles equally.
To find the average speed, you:

• Add up all the individual velocities of the molecules.
• Divide by the total number of molecules.

The formula looks like this:\[\text{Average Speed} = \frac{v_1 + v_2 + v_3 + \, ... \, }{N} \]where \(v_1, v_2, v_3\) are the measured velocities and \(N\) is the number of molecules. In this exercise, adding the velocities 0.5, 1, and 2 km/s, we get a total of 3.5 km/s. Then dividing by 3, the number of velocities given, the average speed comes to about 1.167 km/s.
The average speed highlights a central tendency of the molecules, giving us a simple snapshot of their motion that is often easier to compute and understand than the rms speed, although it may not reflect the influence of outliers in the dataset.

Velocity Ratio

The velocity ratio in this context refers to the relationship between the root-mean-square speed and the average speed of the molecules. Understanding this ratio is key to examining the dynamics of molecular motion.
You can calculate the velocity ratio by dividing the rms speed by the average speed:\[\text{Velocity Ratio} = \frac{\text{RMS Speed}}{\text{Average Speed}} \]This formula tells us how much more, or less, the rms speed is compared to the average speed. For the given problem, using an rms speed of around 1.315 km/s and an average speed of about 1.167 km/s, the velocity ratio comes out to approximately 1.127.
This ratio is greater than 1, indicating that the rms speed is higher than the average speed. This often happens because the rms speed takes into account the square of velocities which puts more weight on higher velocities, sometimes caused by outliers. Understanding the velocity ratio helps in applications like predicting reaction rates and diffusion, where knowing the variance in speed can be as crucial as the mean speed itself.