Problem Statement
What fraction of ideal gas molecules have speeds below the most probable speed $v_p$?
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: In terms of $u = v/v_p$, the Maxwell distribution becomes:
Step 2 — Apply the relevant physical law or equation: $$f(u) = \frac{4}{\sqrt{\pi}}u^2 e^{-u^2}$$
Step 3 — Solve algebraically for the unknown: Fraction with $u < 1$:
Step 4 — Substitute numerical values with units: $$F = \frac{4}{\sqrt{\pi}}\int_0^1 u^2 e^{-u^2}\,du$$
Step 5 — Compute and check the result: Numerical integration: $\int_0^1 u^2 e^{-u^2}\,du \approx 0.1895$.
Step 6: $$F = \frac{4}{\sqrt{\pi}}\times0.1895 = \frac{4\times0.1895}{1.7725} \approx 0.428$$
Worked Calculation
$$f(u) = \frac{4}{\sqrt{\pi}}u^2 e^{-u^2}$$
$$F = \frac{4}{\sqrt{\pi}}\int_0^1 u^2 e^{-u^2}\,du$$
$$F = \frac{4}{\sqrt{\pi}}\times0.1895 = \frac{4\times0.1895}{1.7725} \approx 0.428$$
In terms of $u = v/v_p$, the Maxwell distribution becomes:
$$f(u) = \frac{4}{\sqrt{\pi}}u^2 e^{-u^2}$$
Fraction with $u < 1$:
$$F = \frac{4}{\sqrt{\pi}}\int_0^1 u^2 e^{-u^2}\,du$$
Numerical integration: $\int_0^1 u^2 e^{-u^2}\,du \approx 0.1895$.
$$F = \frac{4}{\sqrt{\pi}}\times0.1895 = \frac{4\times0.1895}{1.7725} \approx 0.428$$
Result: About 43% of molecules have speeds below $v_p$.
Answer
$$\boxed{F = \frac{4}{\sqrt{\pi}}\times0.1895 = \frac{4\times0.1895}{1.7725} \approx 0.428}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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