The Second Orbit of Hydrogen

Q. The radius of the innermost electron orbit of a hydrogen atom is 5.3 × 10⁻¹¹ m. What is the radius of the n = 2 orbit, in metres? expects 2.12 × 10⁻¹⁰ m

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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?

The playground

Inputs

innermost

r_{i}

0.00 m

level

n_{s}

Outputs

radius

r_{s}

The solution

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 Ruby

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

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

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

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