Question

The Classical Hydrogen Atom. The electron in a hydrogen atom can be considered to be in a circular orbit with a radius of 0.0529 nm and a kinetic energy of 13.6 eV. If the electron behaved classically, how much energy would it radiate per second (see Challenge Problem 32.51)? What does this tell you about the use of classical physics in describing the atom?

Short answer

The energy required for the hydrogen atom in radiate per second is$2.125\times {10}^{10}\mathrm{per}\sec$

Step-by-step solution

Concept of the rate at which energy is emitted from an acceleration charge and the acceleration of the circular orbit. 

The rate at which energy is emitted from an acceleration charge q and the acceleration a is $\frac{\mathbf{dE}}{\mathbf{dt}}\mathbf{=}\frac{{\mathbf{q}}^{\mathbf{2}}{\mathbf{a}}^{\mathbf{2}}}{\mathbf{6}{\mathbf{\tau \tau \epsilon }}_{\mathbf{o}}{\mathbf{c}}^{\mathbf{3}}}$q is the acceleration charge, a is the acceleration of the hydrogen atom, ${\mathit{\epsilon }}_{\mathbf{0}}$is the electric constant, c is the speed of the light. The acceleration of the circular orbit is given as $\mathbf{a}\mathbf{=}\frac{{\mathbf{v}}^{\mathbf{2}}}{\mathbf{R}}$v is the velocity of the circular orbit, R is the radius of the circular orbit

Calculate the acceleration of the circular orbit

The acceleration of the circular orbit is $\mathrm{a}=\frac{{\mathrm{v}}^2}{\mathrm{R}}$Multiply numerator and denominator with $\frac{1}{2}m$we get,$\mathrm{a}=\frac{{\displaystyle \frac{1}{2}}{\mathrm{mv}}^2}{{\displaystyle \frac{1}{2}}\mathrm{mR}}$Kinetic energy of the electron is ${\mathrm{K}}_{\mathrm{E}}=13.6\mathrm{eV}$. Substitute the values

$$\begin{gathered} {\mathrm{K}}_{\mathrm{E}}=13.6\mathrm{eV}\times 1.6\times {10}^{-19}\mathrm{J}/\mathrm{ev} \\ =2.176\times {10}^{-18}\mathrm{J} \end{gathered}$$

Substitute ${\mathrm{K}}_{\mathrm{E}}$as $\frac{1}{2}\mathrm{m}$in$\mathrm{a}=\frac{{\mathrm{v}}^2}{\mathrm{R}}$ we get,

$$\begin{gathered} \mathrm{a}=\frac{{\mathrm{K}}_{\mathrm{E}}}{{\displaystyle \frac{1}{2}}\mathrm{mR}} \\ =\frac{2{\mathrm{K}}_{\mathrm{E}}}{\mathrm{mR}} \end{gathered}$$

Substitute the values we have,

$$\begin{gathered} \mathrm{a}=\frac{2\left(2.176\times {10}^{-18}\mathrm{J}\right)}{\left(9.109\times {10}^{-31}\mathrm{kg}\right)\left(0.059\times {10}^{-9}\mathrm{m}\right)} \\ =9.031\times {10}^{22}\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$

Thus, the acceleration of the circular orbit is$9.031\times {10}^{22}\mathrm{m}/{\mathrm{s}}^2$

Calculate the energy required for the hydrogen atom

The rate at which energy is emitted from an acceleration charge q and the acceleration a is$\frac{\mathrm{dE}}{\mathrm{dt}}=\frac{{\mathrm{q}}^2{\mathrm{a}}^2}{6{\mathrm{\tau \tau \epsilon }}_{\mathrm{o}}{\mathrm{c}}^3}$Substitute the values we get,

$$\begin{gathered} \frac{\mathrm{dE}}{\mathrm{dt}}=\frac{{\left(1\times {10}^{-19}\right)}^{2}9.031\times {10}^{22}}{6\mathrm{\pi }\left(8.854\times {10}^{-12}\right)\left(2.998\times {10}^8\right)} \\ =4.64\times {10}^{-8}\mathrm{Nm}/\mathrm{s} \\ =4.64\times {10}^{-8}\mathrm{J}/\mathrm{s} \end{gathered}$$

To convert energy in electron volt into energy in joules is${\mathrm{E}}_{\mathrm{ev}}={\mathrm{E}}_{\mathrm{s}}=\left(6.241\times {10}^{18}\right)$Substitute the values we get,

$$\begin{gathered} {\mathrm{E}}_{\mathrm{ev}}=4.64\times {10}^{-8}\mathrm{J}/\mathrm{s}\left(6.241\times {10}^{18}\right) \\ =2.89\times {10}^{11}\mathrm{ev}/\mathrm{s} \end{gathered}$$

Thus, the energy required for the hydrogen atom is$2.89\times {10}^{11}\mathrm{ev}/\mathrm{s}$

Calculate the energy required for the hydrogen atom

The fraction of its energy radiates per second is

$\frac{{\displaystyle \frac{\mathrm{dE}}{\mathrm{dt}}}\left(1\mathrm{s}\right)}{\mathrm{E}}=\frac{{\mathrm{E}}_{\mathrm{hydrogen}}}{{\mathrm{K}}_{\mathrm{v}}}$

Substitute the values we get,

$$\begin{gathered} \frac{{\displaystyle \frac{\mathrm{dE}}{\mathrm{dt}}}\left(1\mathrm{s}\right)}{\mathrm{E}}=\frac{2.89\times {10}^{11}}{13.6\mathrm{ev}} \\ =2.125\times {10}^{10}\mathrm{per}\sec \end{gathered}$$

The value of rate of energy emission $\frac{dE}{dt}$is larger value, it means that the electrons in hydrogen atom almost losses it all energy instantly.

Thus, the energy required for the hydrogen atom in radiate per second is $2.125\times {10}^{10}\mathrm{per}\sec$