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Chapter 19: Problem 13

The speeds of 11 molecules are 2.0, 3.0, 4.0, . . . , 12 km/s.What are their (a) average speed and (b) rms speed?

Short Answer

Expert verified
The average speed is 7.0 km/s, and the rms speed is approximately 7.68 km/s.

Step by step solution

01

List All The Speeds

We have been given the speeds of molecules as: 2.0, 3.0, 4.0, ..., 12.0 km/s. There are a total of 11 speeds. We will denote them using: \(v_1 = 2.0\), \(v_2 = 3.0\), ..., \(v_{11} = 12.0\).
02

Calculate Average Speed

The average speed (\(v_{avg}\)) is calculated by finding the sum of all the speeds and dividing by the number of speeds. \[v_{avg} = \frac{v_1 + v_2 + v_3 + \, ... \, + v_{11}}{11} = \frac{2.0 + 3.0 + 4.0 + \, ... \, + 12.0}{11}\]The sum of an arithmetic series can be calculated using the formula \( S = \frac{n}{2} (a + l) \), where \(a\) is the first term, \(l\) is the last term, and \(n\) is the number of terms. Substituting the values:\[S = \frac{11}{2} (2.0 + 12.0) = \frac{11}{2} \times 14.0 = 77.0\]Now, \(v_{avg} = \frac{77.0}{11} = 7.0\, \text{km/s}\).
03

Calculate RMS Speed

The root-mean-square (rms) speed is the square root of the average of the squares of the speeds. First, we find the squares of each speed:\(v_1^2 = 4.0\), \(v_2^2 = 9.0\), ..., \(v_{11}^2 = 144.0\).Now, finding the mean of these squared speeds:\[\text{Mean of squares} = \frac{4.0 + 9.0 + 16.0 + \, ... \, + 144.0}{11}\]The sum of squares from 1 to 12 is given by the formula for the sum of squares of an arithmetic series, \[\sum_{i=2}^{12} i^2 = \, \sum_{i=1}^{12} i^2 - 1^2 = \frac{12(12 + 1)(2 \cdot 12 + 1)}{6} - 1 = 650 - 1 = 649\]Then, the mean of squares is: \[\frac{649}{11} = 59.0\]The rms speed is:\[v_{rms} = \sqrt{59.0} \approx 7.68\, \text{km/s}\]
04

Final Result Summary

The average speed of the molecules is found to be \(7.0\, \text{km/s}\), and the root-mean-square speed is approximately \(7.68\, \text{km/s}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Speed
In the world of gases, molecules are constantly in motion. When we want to know how fast these molecules are moving on average, we calculate their average speed. To do this, we take the sum of the speeds of all the individual molecules and divide that sum by the total number of molecules. This gives us the average.
The formula for calculating the average speed, denoted as \(v_{avg}\), is:
\[v_{avg} = \frac{v_1 + v_2 + v_3 + \, ... \, + v_{11}}{11}\]
  • "\(v_1\), \(v_2\), ..., \(v_{11}\)" represent the speeds of each molecule.
  • The numerator is the sum of all these speeds.
  • The denominator is the count of the molecules, which is 11 in this case.
Calculating it, we discovered that the average speed of the given set of molecules is 7.0 km/s. This value tells us how fast on average these molecules are traveling.
RMS Speed
Root-mean-square (RMS) speed is another way of describing how fast molecules in a gas are moving. Unlike average speed, RMS speed takes into account the energy content of the gas molecules.
This is because it involves taking the square of each speed first, finding their average, and then taking the square root of that average.
To calculate RMS speed, represented as \(v_{rms}\), the formula is:
\[v_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2 + \, ... \, + v_{11}^2}{11}}\]
  • First, square the speed of each molecule.
  • Add all these squared speeds together.
  • Divide by 11, the total number of molecules.
  • Finally, take the square root of the resulting average.
In our example, the RMS speed was calculated to be approximately 7.68 km/s. This gives us insight into the kinetic energy of the gas, which is directly related to temperature.
Molecular Speeds
Molecular speeds in a gas can vary significantly. While average and RMS speeds give us useful measures, they do not tell the whole story.
The distribution of molecular speeds is affected by factors such as temperature and the type of gas.
Three important measures of speed in the kinetic theory of gases are:
  • Average Speed: provides a simple mean velocity that's useful for general comparisons.
  • RMS Speed: correlates directly with kinetic energy and temperature.
  • Most Probable Speed: the speed at which the maximum number of molecules are moving; not explicitly covered here but essential for full comprehension.
Understanding these speeds helps us predict how gas molecules behave under different conditions. This knowledge is crucial in fields like thermodynamics and physical chemistry, where gas behaviors are fundamental to processes like reactions and heat transfers.

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