Concept explainers
A vessel contains 1.00 × 104 oxygen molecules at 500 K. (a) Make an accurate graph of the Maxwell speed distribution function versus speed with points at speed intervals of 100 m/s. (b) Determine the most probable speed from this graph. (c) Calculate the average and rms speeds for the molecules and label these points on your graph. (d) From the graph, estimate the fraction of molecules with speeds in the range 300 m/s to 600 m/s.
(a)
The graph of the Maxwell speed distribution function versus speed with points at speed intervals of
Answer to Problem 38AP
The of the Maxwell speed distribution function versus speed with points at speed intervals of
Explanation of Solution
The Maxwell distribution curve is the graph between the distribution of speed and the change in speed or speed interval.
The number of molecules of oxygen in vessel is
Write the expression of Maxwell’s speed distribution function.
Here,
The mass of the molecules of oxygen
Here,
The molecular mass of the oxygen molecules in
Substitute
Substitute
Substitute the values of
0 | 0 |
100 | 1.64 |
200 | 5.86 |
300 | 10.88 |
400 | 14.78 |
500 | 16.33 |
600 | 15.39 |
700 | 12.7 |
800 | 9.31 |
900 | 6.13 |
1000 | 3.64 |
1100 | 1.961 |
1200 | 0.96 |
1300 | 0.43 |
1400 | 0.18 |
1500 | 0.07 |
On the basis of the table, a graph is plotted below;
(b)
The most probable speed from the graph.
Answer to Problem 38AP
The most probable speed is
Explanation of Solution
The most probable speed occurs where
Conclusion:
Therefore, the most probable speed is
(c)
The average and rms speeds for the molecules and label these points on the graph.
Answer to Problem 38AP
The average and rms speeds for the molecules is
Explanation of Solution
Write the expression of average velocity.
The mass of the molecules of oxygen
Substitute
The molecular mass of the oxygen molecules in
Substitute
Thus, the average speed is
Write the expression of rms velocity.
Substitute
Substitute
Thus, the rms velocity of the oxygen molecules is
The graph of Maxwell’s curve is shown below;
The point
Conclusion:
Therefore, the average and rms speeds for the molecules is
(d)
The fraction of molecules with the speed in the range of
Answer to Problem 38AP
The fraction of molecules with the speed in the range of
Explanation of Solution
The figure given below shows the Maxwell’s curve,
The area under the distribution curve in the range
Conclusion:
Write the area under the curve in the range
Therefore, the fraction of molecules with the speed in the range of
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Chapter 20 Solutions
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