Chapter 4: Q16P (page 158)

A hydrogenic atom consists of a single electron orbiting a nucleus with Z protons. ( $Z = 1$ would be hydrogen itself, $Z = 2$ is ionized helium , $Z = 3$ is doubly ionized lithium, and so on.) Determine the Bohr energies $E n (Z)$ , the binding energy $E_{1} (Z)$ , the Bohr radius $a (Z)$ , and the Rydberg constant R $(Z)$ for a hydrogenic atom. (Express your answers as appropriate multiples of the hydrogen values.) Where in the electromagnetic spectrum would the Lyman series fall, for $Z = 2$ and $Z = 3$ ? Hint: There’s nothing much to calculate here— in the potential (Equation 4.52) $Z e^{2}$ , so all you have to do is make the same substitution in all the final results.
$V (r) = - \frac{e^{2}}{4 π \dot{o_{0}}} \frac{1}{r}$ (4.52).

Short Answer

Expert verified

The electronic spectrum would be the Lyman seires fall, for $z = 2$ and $z = 3$ is at, $z = 2$ is $3.04 \times 10^{- 8} m .$

At $z = 3$ is $1.35 \times 10^{- 8} m .$

Step by step solution

Given

The potential is given by:

$V (r) = - \frac{e^{2}}{4 π \dot{o_{0}}} \frac{1}{r}$

Electronic spectrum

Consider a hydrogen, if irradiated with light it will be excited to another energy level. The energies of these photons are easily calculated from the Bohr energy formula:

$E_{n} = - \frac{1}{n^{2}} \frac{m e^{4}}{2 h^{2} {(4 Τ Τ \in_{0})}^{2}}$

Lyman lines range from $n_{i} = 2 t o n_{i} = \infty (w i t h n_{f} = 1)$ ;

the wavelengths range from

Finding the electronic spectrum

$\frac{1}{λ_{2}} = R (1 - \frac{1}{4}) = \frac{3}{4} R λ_{2} = \frac{4}{3 R}$

Down to

role="math" localid="1658210586427" $\frac{1}{λ_{1}} = R (1 - \frac{1}{\infty}) = R \Rightarrow R_{1} = \frac{1}{R}$

For $Z = 2$

$λ_{1} = \frac{1}{4 R} = \frac{1}{4 (1.097 \times 10^{7})} = 2.28 \times 10^{- 8} m$

$λ_{2} = \frac{1}{3 R} = 3.04 \times 10^{- 8} m,$

Ultraviolet.

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Short Answer

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Given

Electronic spectrum

Finding the electronic spectrum

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