Chapter 26, Problem 78QAP

To determine

The missing values in the table for n =1, 2, 3, 4 and 5.

Answer to Problem 78QAP

The final table is,

n	Ln	rn	Kn	En	vn
	$L_n=\frac{nh}{2\pi }$   $\text{Kg}\cdot {\text{meter}}^{\text{2}}\cdot {\text{s}}^{\text{-1}}$	$r_n=\frac{n^2{h}^2}{4{\pi }^{2}mk{e}^2}$ meter	$K_n=\frac{m{k}^2{e}^4}{2n^2{\hslash }^2}$ Joule	${\text{E}}_{\text{n}}\text{=}-\frac{m{k}^2{e}^4}{2n^2{\hslash }^2}$ Joule	$v_n=\frac{2\pi k{e}^2}{nh}$   $\text{meter}\cdot {\text{s}}^{\text{-1}}$
n= 1	$1.06\times {10}^{-34}$	$0.536\times {10}^{-10}$	$21.44\times {10}^{-19}$	$-21.44\times {10}^{-19}$	$21.47\times {10}^5$
n= 2	$2.12\times {10}^{-34}$	$\begin{array}{c}2.144\times {10}^{-10}\end{array}$	$5.36\times {10}^{-19}$	$-5.36\times {10}^{-19}$	$10.73\times {10}^5$
n= 3	$3.18\times {10}^{-34}$	$4.824\times {10}^{-10}$	$2.38\times {10}^{-19}$	$-2.38\times {10}^{-19}$	$7.16\times {10}^5$
n= 4	$4.24\times {10}^{-34}$	$8.576\times {10}^{-10}$	$1.34\times {10}^{-19}$	$-1.34\times {10}^{-19}$	$5.36\times {10}^5$
n= 5	$5.3\times {10}^{-34}$	$13.4\times {10}^{-10}$	$0.85\times {10}^{-19}$	$-0.85\times {10}^{-19}$	$4.29\times {10}^5$

Explanation of Solution

Given info:

The given table is,

n	Ln	rn	Kn	En	vn
n= 1
n= 2
n= 3
n= 4
n= 5

Formula used:

The expressions for angular momentum, speed, orbital radius, kinetic energy and total energy are:

  $\begin{array}{l}{L}_n=\frac{nh}{2\pi }\\ r_n=\frac{n^2{h}^2}{4{\pi }^{2}mk{e}^2}\\ K_n=\frac{m{k}^2{e}^4}{2n^2{\hslash }^2}\\ E_n\text{=}-\frac{m{k}^2{e}^4}{2n^2{\hslash }^2}\\ v_n=\frac{2\pi k{e}^2}{nh}\end{array}$

Calculation:

All values are given in SI unit. The values of different constants as considered for calculation are
  $\begin{array}{l}k=8.99\times {10}^9\text{ N}\cdot {\text{m}}^{\text{2}}\cdot {\text{Coulomb}}^{\text{-2}}\\ m=9.11\times {10}^{-31}\text{ kg}\\ h\text{=1}\text{.055}\times {\text{10}}^{\text{-34}}\text{ Joule-sec}\\ e\text{=1}\text{.6}\times {\text{10}}^{\text{-19}}\text{ Coulomb}\end{array}$

n	Ln	rn	Kn	En	vn
	$L_n=\frac{nh}{2\pi }$   $\text{Kg}\cdot {\text{meter}}^{\text{2}}\cdot {\text{s}}^{\text{-1}}$	$r_n=\frac{n^2{h}^2}{4{\pi }^{2}mk{e}^2}$ meter	$K_n=\frac{m{k}^2{e}^4}{2n^2{\hslash }^2}$ Joule	${\text{E}}_{\text{n}}\text{=}-\frac{m{k}^2{e}^4}{2n^2{\hslash }^2}$ Joule	$v_n=\frac{2\pi k{e}^2}{nh}$   $\text{meter}\cdot {\text{s}}^{\text{-1}}$
n= 1	$1.06\times {10}^{-34}$	$0.536\times {10}^{-10}$	$21.44\times {10}^{-19}$	$-21.44\times {10}^{-19}$	$21.47\times {10}^5$
n= 2	$2.12\times {10}^{-34}$	$\begin{array}{c}2.144\times {10}^{-10}\end{array}$	$5.36\times {10}^{-19}$	$-5.36\times {10}^{-19}$	$10.73\times {10}^5$
n= 3	$3.18\times {10}^{-34}$	$4.824\times {10}^{-10}$	$2.38\times {10}^{-19}$	$-2.38\times {10}^{-19}$	$7.16\times {10}^5$
n= 4	$4.24\times {10}^{-34}$	$8.576\times {10}^{-10}$	$1.34\times {10}^{-19}$	$-1.34\times {10}^{-19}$	$5.36\times {10}^5$
n= 5	$5.3\times {10}^{-34}$	$13.4\times {10}^{-10}$	$0.85\times {10}^{-19}$	$-0.85\times {10}^{-19}$	$4.29\times {10}^5$