The Hydrogen Atom's Size

solved BohrModel

Q. It is found experimentally that 13.6 eV $E_{b} = 13.6$ eV of energy is required to separate a hydrogen atom into a proton and an electron. What is the orbital radius of the electron, in metres?

Inputs

energy

E_{b}

13.6 eV

Outputs

radius

r

How it's solved

Solved with BohrModel

Given

$E_{b} = 13.6$ eV

Find $r$ .

Working

$E_{b} = 13.6$ eV $= 2.17896022 \times 10^{−18}$ J
energy to free the electron(electron) $E_{b} = \frac{k \cdot e^{2}}{2 \cdot r}$
$2.17896022 \times 10^{−18} = \frac{8.98755179 \times 10^{9} \cdot {(1.60217663 \times 10^{−19})}^{2}}{2 \cdot r}$
$r = 5.294 \times 10^{−11}$ m
$r \approx 5.3 \times 10^{−11}$ mto 2 significant figures

Conclusion

The electron is held as far out as the energy needed to free it allows; that sets the size of the atom.

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_hydrogen_atom_s_size do4  kind :numerical5  describes "It is found experimentally that %{energy} of energy is required to separate a hydrogen atom into a " \6            "proton and an electron. %{radius}, in metres?"7  given energy: "13.6 eV"8  asks radius: "What is the orbital radius of the electron"9end

solution.rb

1require_relative "../questions/the_hydrogen_atom_s_size"2require_relative "../science/atoms/bohr_model"34solve :the_hydrogen_atom_s_size, using: BohrModel do5  objects :electron67  take energy: electron.binding_energy89  radius = electron.orbit_radius10  answer radius:, in: "m", figures: 21112  conclusion "The electron is held as far out as the energy needed to free it allows; that sets the size of the atom."13end

playground.rb

1require_relative "../solutions/the_hydrogen_atom_s_size"23play_with :the_hydrogen_atom_s_size do4  title "The Hydrogen Atom's Size"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 electron	$E_{b}$	J	—
orbit radius of electron	$r$	m	—
mass of ground	$m_{g}$	kg	0 … ∞
orbital speed of ground	$v_{g}$	m/s	0 … ∞
mass of electron	$m$	kg	0 … ∞
orbital speed of electron	$v$	m/s	0 … ∞
atomic number of ground	$Z_{g}$	—	—
quantum number of ground	$n_{g}$	—	—
atomic number of electron	$Z$	—	—
quantum number of electron	$n$	—	—
potential energy of ground	$U_{g}$	J	—
potential energy of electron	$U$	J	—
total energy of ground	$E_{g}$	J	—
total energy of electron	$E$	J	—
photon energy of ground	$E_{p,g}$	J	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
photon energy of electron	$E_{p}$	J	—
frequency of electron	$ν$	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} = \frac{k \cdot e^{2}}{2 \cdot r}

energy to free the electron(electron)

\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 \cdot v^{2}}{r} = \frac{k \cdot e^{2}}{r^{2}}

held in orbit by the proton(electron)

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 = \frac{2 \cdot π \cdot k \cdot Z \cdot e^{2}}{n \cdot h}

speed set by the level(electron)

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

angular momentum comes in steps(ground)

m \cdot v \cdot r = \frac{n \cdot h}{2 \cdot π}

angular momentum comes in steps(electron)

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

energy of its place(ground)

U = \frac{- k \cdot e^{2}}{r}

energy of its place(electron)

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

energy of the atom(ground)

E = \frac{- k \cdot e^{2}}{2 \cdot r}

energy of the atom(electron)

E_{p,g} = h \cdot ν_{g}

one photon for one jump(ground)

E_{p} = h \cdot ν

one photon for one jump(electron)