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The forward reaction in a chemical equation:
Forward reaction proceeds left→right; at equilibrium, forward rate = backward rate, not zero.
For each $t\in\mathbb{R}$, let $[t]$ denote the greatest integer $\leq t$. Evaluate: $\displaystyle\lim_{x\to0^+} x\left(\left[\dfrac{1}{x}\right]+\left[\dfrac{2}{x}\right]+\cdots+\left[\dfrac{15}{x}\right]\right)$
For large $N=1/x$, $[k/x]\approx kN$ with error at most 1.
$x\sum_{k=1}^{15}\left[\dfrac{k}{x}\right] \approx x\sum_{k=1}^{15}\dfrac{k}{x} = \sum_{k=1}^{15}k = \dfrac{15\cdot16}{2} = 120$
More rigorously: $[k/x] = k/x - \{k/x\}$ where $0\leq\{k/x\}<1$. Then $x\cdot\{k/x\}
A body moves eastward at $30\text{ m/s}$. After 10 s its velocity becomes $40\text{ m/s}$ northward. The magnitude of average acceleration is:
Change in velocity: $\Delta\vec{v} = 40\hat{j} - 30\hat{i}$
$|\Delta v| = \sqrt{30^2 + 40^2} = \sqrt{900+1600} = \sqrt{2500} = 50\text{ m/s}$
Average acceleration $= \dfrac{50}{10} = \mathbf{5\text{ m/s}^2}$
- Explains the line spectrum of hydrogen (and hydrogen-like ions)
- Correctly predicts wavelengths of Lyman, Balmer, Paschen series
- Cannot explain multi-electron atoms
- Cannot explain Zeeman effect (splitting in magnetic field)
- Cannot explain Stark effect (splitting in electric field)
- Cannot explain bond formation
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