Let's break down the problem step by step to clarify the concepts of most probable speed (v_mp) and root mean square speed (v_rms) in the context of the kinetic theory of gases. You have a sample of 10 molecules with different speeds, and you've already identified the most probable speed correctly. Now, let's analyze the calculations for v_rms and see where the discrepancy arises.
Understanding the Speed Distribution
You have the following distribution of speeds among 10 molecules:
- 2 molecules at 100 m/s
- 3 molecules at 200 m/s
- 4 molecules at 300 m/s
- 1 molecule at 50 m/s
Calculating Most Probable Speed (v_mp)
The most probable speed (v_mp) is the speed at which the largest number of molecules are moving. In your case, 4 molecules are moving at 300 m/s, which is indeed the highest count. Thus, you correctly identified:
v_mp = 300 m/s
Calculating Root Mean Square Speed (v_rms)
Now, let's calculate the root mean square speed using the formula:
v_rms = sqrt((v1² + v2² + v3² + ... + vn²) / N)
Where N is the total number of molecules. In your case, N = 10. Let's compute it step by step:
Step 1: Squaring the Speeds
We need to square each speed and multiply by the number of molecules at that speed:
- For 100 m/s: (100 m/s)² * 2 = 10000 * 2 = 20000
- For 200 m/s: (200 m/s)² * 3 = 40000 * 3 = 120000
- For 300 m/s: (300 m/s)² * 4 = 90000 * 4 = 360000
- For 50 m/s: (50 m/s)² * 1 = 2500 * 1 = 2500
Step 2: Summing the Values
Now, let's sum these results:
Total = 20000 + 120000 + 360000 + 2500 = 501500
Step 3: Dividing by the Total Number of Molecules
Now, we divide this sum by the total number of molecules (N = 10):
Mean of squares = 501500 / 10 = 50150
Step 4: Taking the Square Root
Finally, we take the square root to find v_rms:
v_rms = sqrt(50150) ≈ 224.1 m/s
Identifying the Error
The discrepancy in your initial calculation of v_rms = 367.4 m/s likely arose from using an incorrect method or formula. The ratio v_rms/v_mp is not a standard calculation and can lead to confusion. Instead, always use the direct calculation method outlined above for v_rms.
Summary of Results
To summarize:
- Most Probable Speed (v_mp) = 300 m/s
- Root Mean Square Speed (v_rms) ≈ 224.1 m/s
By following the correct steps for calculating v_rms, you can avoid errors and ensure accurate results. If you have any further questions or need clarification on any part of this process, feel free to ask!
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