The figure shows a graph between ln left| fra | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The area enclosed by the $n^{th}$ orbit is given by $A_n = \pi r_n^2$.Since the radius of the $n^{th}$ orbit in a hydrogen-like atom is proportion

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(A) The area enclosed by the $n^{th}$ orbit is given by $A_n = \pi r_n^2$.Since the radius of the $n^{th}$ orbit in a hydrogen-like atom is proportional to $n^2$ (i.e., $r_n \propto n^2$), we have:$\frac{A_n}{A_1} = \left( \frac{r_n}{r_1} \right)^2 = \left( \frac{n^2}{1^2} \right)^2 = n^4$.Taking the natural logarithm $(\ln)$ on both sides:$\ln \left( \frac{A_n}{A_1} \right) = \ln (n^4) = 4 \ln (n)$.Comparing this with the equation of a straight line $y = mx + c$, where $y = \ln \left| \frac{A_n}{A_1} \right|$, $x = \ln |n|$, $m = 4$, and $c = 0$, we see that the graph is a straight line passing through the origin with a slope of $4$.Looking at the provided graph, curve $4$ represents a straight line with a slope of $4$ (since at $\ln |n| = 1$, $\ln \left| \frac{A_n}{A_1} \right| = 4$).Therefore, the correct curve is $4$.

The figure shows a graph between $\ln \left| \frac{A_n}{A_1} \right|$ and $\ln |n|$, where $A_n$ is the area enclosed by the $n^{th}$ orbit in a hydrogen-like atom. The correct curve is

Similar Questions

An electron in an excited state of $Li^{2+}$ ion has angular momentum $\frac{3 h}{2 \pi}$. The de-Broglie wavelength of the electron in this state is $p \pi a_{0}$ (where, $a_{0} = \text{Bohr radius}$). The value of $p$ is

According to de-Broglie,the de-Broglie wavelength for an electron in an orbit of radius $5.3 \times 10^{-11} \ m$ of a hydrogen atom is $10^{-10} \ m$. The principal quantum number for this electron is:

In Bohr's model,the atomic radius of the first orbit is $r_0$,then the radius of the third orbit is

If an electron in a hydrogen atom jumps from the third orbit to the second orbit,the frequency of the emitted radiation is given by (where $c$ is the speed of light and $R$ is the Rydberg constant):

The magnitude of angular momentum,orbit radius,and frequency of revolution of an electron in a hydrogen atom corresponding to quantum number $n$ are $L, r$,and $f$ respectively. According to Bohr's theory of the hydrogen atom,which of the following is constant for all orbits?

Vedclass Test Series

Exam Paper Generator

Online Exam Module