Suppose A_n represents the area enclosed by the n th orbit of a...

The radius of the $n$th Bohr orbit of a hydrogen atom is given by $r_n = r_1 n^2$, where $r_1 is the radius of the first orbit.<br><br>The area enclosed by the$n$th orbit is$A_n = \pi r_n^2$.<br><br>Substituting the expression for$r_n$, we get$A_n = \pi (r_1 n^2)^2 = \pi r_1^2 n^4$.<br><br>Since the area of the first orbit is$A_1 = \pi r_1^2$, we can write$A_n = A_1 n^4$.<br><br>Rearranging the terms, we get$\dfrac{A_n}{A_1} = n^4$.<br><br>Taking the natural logarithm on both sides, we obtain$\ln\left(\dfrac{A_n}{A_1}\right) = \ln(n^4) = 4\ln(n)$.<br><br>Comparing this with the equation of a straight line$y = mx + c$, where$y = \ln \left(\dfrac{A_n}{A_1}\right)$and x = \ln(n)$, we get the slope $m = 4$ and the $y$-intercept $c = 0$.

Therefore, the plot is a straight line with a slope of $4$ that passes through the origin.