The Second Orbit of Hydrogen

solved BohrModel

Q. The radius of the innermost $n_{i} = 1$ electron orbit $m_{s} = m_{i}$ of a hydrogen atom $Z_{i} = 1$ , $Z_{s} = 1$ is 5.3 × 10⁻¹¹ m $r_{i} = 5.3 \times 10^{−11}$ m. What is the radius of the n = 2 $n_{s} = 2$ orbit, in metres?

Inputs

innermost

r_{i}

0.00 m

level

n_{s}

Outputs

radius

r_{s}

How it's solved

Solved with BohrModel

Given

$r_{i} = 5.3 \times 10^{−11}$ m
$n_{s} = 2$
$n_{i} = 1$ from “innermost”
$Z_{i} = 1$ from “hydrogen atom”
$Z_{s} = 1$ from “hydrogen atom”
$m_{s} = m_{i}$ from “electron orbit”

Find $r_{s}$ .

Working

speed set by the level(innermost) $v_{i} = \frac{2 \cdot π \cdot k \cdot Z_{i} \cdot e^{2}}{n_{i} \cdot h}$
$v_{i} = \frac{2 \cdot 3.14159265 \cdot 8.98755179 \times 10^{9} \cdot 1 \cdot {(1.60217663 \times 10^{−19})}^{2}}{1 \cdot 6.62607015 \times 10^{−34}}$
$v_{i} = 2.18769126 \times 10^{6}$ m/s
held in orbit by the proton(innermost) $\frac{m_{i} \cdot {v_{i}}^{2}}{r_{i}} = \frac{k \cdot e^{2}}{{r_{i}}^{2}}$
$\frac{m_{i} \cdot {(2.18769126 \times 10^{6})}^{2}}{5.3 \times 10^{−11}} = \frac{8.98755179 \times 10^{9} \cdot {(1.60217663 \times 10^{−19})}^{2}}{{(5.3 \times 10^{−11})}^{2}}$
$m_{i} = 9.095242 \times 10^{−31}$ kg
electron orbit $m_{s} = m_{i}$
$m_{s} = 9.095242 \times 10^{−31}$
$m_{s} = 9.095242 \times 10^{−31}$ kg
speed set by the level(second) $v_{s} = \frac{2 \cdot π \cdot k \cdot Z_{s} \cdot e^{2}}{n_{s} \cdot h}$
$v_{s} = \frac{2 \cdot 3.14159265 \cdot 8.98755179 \times 10^{9} \cdot 1 \cdot {(1.60217663 \times 10^{−19})}^{2}}{2 \cdot 6.62607015 \times 10^{−34}}$
$v_{s} = 1.09384563 \times 10^{6}$ m/s
angular momentum comes in steps(second) $m_{s} \cdot v_{s} \cdot r_{s} = \frac{n_{s} \cdot h}{2 \cdot π}$
$9.095242 \times 10^{−31} \cdot 1.09384563 \times 10^{6} \cdot r_{s} = \frac{2 \cdot 6.62607015 \times 10^{−34}}{2 \cdot 3.14159265}$
$r_{s} = 2.12 \times 10^{−10}$ m
$r_{s} \approx 2.12 \times 10^{−10}$ mto 3 significant figures

Conclusion

An orbit's radius grows as the square of its quantum number, so the second orbit is four times as wide as the innermost.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :the_second_orbit_of_hydrogen do4  kind :numerical5  describes "The radius of the innermost electron orbit of a hydrogen atom is %{innermost}. %{radius} of the n = " \6            "%{level} orbit, in metres?"7  given innermost: "5.3e-11 m"8  given level: "2"9  asks radius: "What is the radius"10end

solution.rb

1require_relative "../questions/the_second_orbit_of_hydrogen"2require_relative "../science/atoms/bohr_model"34solve :the_second_orbit_of_hydrogen, using: BohrModel do5  objects :innermost, :second67  take innermost: innermost.orbit_radius8  take level: second.quantum_number910  deduce_from "innermost", innermost.quantum_number => "1"11  deduce_from "hydrogen atom", innermost.atomic_number => "1", second.atomic_number => "1"12  deduce_from "electron orbit", second.mass => innermost.mass1314  radius = second.orbit_radius15  answer radius:, in: "m", figures: 31617  conclusion "An orbit's radius grows as the square of its quantum number, so the second orbit is four times " \18             "as wide as the innermost."19end

playground.rb

1require_relative "../solutions/the_second_orbit_of_hydrogen"23play_with :the_second_orbit_of_hydrogen do4  title "The Second Orbit of Hydrogen"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	BohrModel
coulomb constant	$k$	N m² C⁻²	—	BohrModel
elementary charge	$e$	C	—	BohrModel
plancks constant	$h$	J s	—	BohrModel
mass	$m$	kg	0 … ∞	BohrModel
orbit radius	$r$	m	—	BohrModel
orbital speed	$v$	m/s	0 … ∞	BohrModel
binding energy	$E_{b}$	J	—	BohrModel
quantum number	$n$	—	—	BohrModel
atomic number	$Z$	—	—	BohrModel
potential energy	$U$	J	—	BohrModel
total energy	$E$	J	—	BohrModel
photon energy	$E_{p}$	J	—	BohrModel
frequency	$ν$	Hz	0 … ∞	BohrModel
binding energy of ground	$E_{b,g}$	J	—
orbit radius of ground	$r_{g}$	m	—
binding energy of innermost	$E_{b,i}$	J	—
orbit radius of innermost	$r_{i}$	m	—
binding energy of second	$E_{b,s}$	J	—
orbit radius of second	$r_{s}$	m	—
mass of ground	$m_{g}$	kg	0 … ∞
orbital speed of ground	$v_{g}$	m/s	0 … ∞
mass of innermost	$m_{i}$	kg	0 … ∞
orbital speed of innermost	$v_{i}$	m/s	0 … ∞
mass of second	$m_{s}$	kg	0 … ∞
orbital speed of second	$v_{s}$	m/s	0 … ∞
atomic number of ground	$Z_{g}$	—	—
quantum number of ground	$n_{g}$	—	—
atomic number of innermost	$Z_{i}$	—	—
quantum number of innermost	$n_{i}$	—	—
atomic number of second	$Z_{s}$	—	—
quantum number of second	$n_{s}$	—	—
potential energy of ground	$U_{g}$	J	—
potential energy of innermost	$U_{i}$	J	—
potential energy of second	$U_{s}$	J	—
total energy of ground	$E_{g}$	J	—
total energy of innermost	$E_{i}$	J	—
total energy of second	$E_{s}$	J	—
photon energy of ground	$E_{p,g}$	J	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
photon energy of innermost	$E_{p,i}$	J	—
frequency of innermost	$ν_{i}$	Hz	0 … ∞
photon energy of second	$E_{p,s}$	J	—
frequency of second	$ν_{s}$	Hz	0 … ∞

E_{b,e} = \frac{k \cdot e^{2}}{2 \cdot r_{e}}

energy to free the electron

\frac{m_{e} \cdot {v_{e}}^{2}}{r_{e}} = \frac{k \cdot e^{2}}{{r_{e}}^{2}}

held in orbit by the proton

v_{e} = \frac{2 \cdot π \cdot k \cdot Z_{e} \cdot e^{2}}{n_{e} \cdot h}

speed set by the level

m_{e} \cdot v_{e} \cdot r_{e} = \frac{n_{e} \cdot h}{2 \cdot π}

angular momentum comes in steps

U_{e} = \frac{- k \cdot e^{2}}{r_{e}}

energy of its place

E_{e} = \frac{- k \cdot e^{2}}{2 \cdot r_{e}}

energy of the atom

E_{p,l} = h \cdot ν_{l}

one photon for one jump

E_{b,g} = \frac{k \cdot e^{2}}{2 \cdot r_{g}}

energy to free the electron(ground)

E_{b,i} = \frac{k \cdot e^{2}}{2 \cdot r_{i}}

energy to free the electron(innermost)

E_{b,s} = \frac{k \cdot e^{2}}{2 \cdot r_{s}}

energy to free the electron(second)

\frac{m_{g} \cdot {v_{g}}^{2}}{r_{g}} = \frac{k \cdot e^{2}}{{r_{g}}^{2}}

held in orbit by the proton(ground)

\frac{m_{i} \cdot {v_{i}}^{2}}{r_{i}} = \frac{k \cdot e^{2}}{{r_{i}}^{2}}

held in orbit by the proton(innermost)

\frac{m_{s} \cdot {v_{s}}^{2}}{r_{s}} = \frac{k \cdot e^{2}}{{r_{s}}^{2}}

held in orbit by the proton(second)

v_{g} = \frac{2 \cdot π \cdot k \cdot Z_{g} \cdot e^{2}}{n_{g} \cdot h}

speed set by the level(ground)

v_{i} = \frac{2 \cdot π \cdot k \cdot Z_{i} \cdot e^{2}}{n_{i} \cdot h}

speed set by the level(innermost)

v_{s} = \frac{2 \cdot π \cdot k \cdot Z_{s} \cdot e^{2}}{n_{s} \cdot h}

speed set by the level(second)

m_{g} \cdot v_{g} \cdot r_{g} = \frac{n_{g} \cdot h}{2 \cdot π}

angular momentum comes in steps(ground)

m_{i} \cdot v_{i} \cdot r_{i} = \frac{n_{i} \cdot h}{2 \cdot π}

angular momentum comes in steps(innermost)

m_{s} \cdot v_{s} \cdot r_{s} = \frac{n_{s} \cdot h}{2 \cdot π}

angular momentum comes in steps(second)

U_{g} = \frac{- k \cdot e^{2}}{r_{g}}

energy of its place(ground)

U_{i} = \frac{- k \cdot e^{2}}{r_{i}}

energy of its place(innermost)

U_{s} = \frac{- k \cdot e^{2}}{r_{s}}

energy of its place(second)

E_{g} = \frac{- k \cdot e^{2}}{2 \cdot r_{g}}

energy of the atom(ground)

E_{i} = \frac{- k \cdot e^{2}}{2 \cdot r_{i}}

energy of the atom(innermost)

E_{s} = \frac{- k \cdot e^{2}}{2 \cdot r_{s}}

energy of the atom(second)

E_{p,g} = h \cdot ν_{g}

one photon for one jump(ground)

E_{p,i} = h \cdot ν_{i}

one photon for one jump(innermost)

E_{p,s} = h \cdot ν_{s}

one photon for one jump(second)

m_{s} = m_{i}

electron orbit