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Let Aₙ be the area enclosed by the nth orbit of a hydrogen atom. The graph of ln(Aₙ / A₁) plotted against ln(n) is:

1. A curve that does not pass through the origin
2. A straight line with slope 4
3. A monotonically increasing non-linear curve
4. A circle

Correct answer: A straight line with slope 4

Solution

In a hydrogen atom, the radius of the nth orbit is rₙ = a0 n². The area enclosed is Aₙ = pi rₙ² = pi a0² n⁴. Hence Aₙ/A₁ = n⁴. Taking natural log: ln(Aₙ/A₁) = 4 ln n. So the graph of ln(Aₙ/A₁) vs ln(n) is a straight line through the origin with slope 4.

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