The Electron's Speed in the First Level

solved BohrModel

Q. Using the Bohr's model, calculate the speed of the electron in a hydrogen atom $Z = 1$ in the n = 1 $n = 1$ level, in m/s?

Inputs

level

n

Outputs

speed

v

How it's solved

Solved with BohrModel

Given

$n = 1$
$Z = 1$ from “hydrogen atom”

Find $v$ .

Working

speed set by the level(electron) $v = \frac{2 \cdot π \cdot k \cdot Z \cdot e^{2}}{n \cdot h}$
$v = \frac{2 \cdot 3.14159265 \cdot 8.98755179 \times 10^{9} \cdot 1 \cdot {(1.60217663 \times 10^{−19})}^{2}}{1 \cdot 6.62607015 \times 10^{−34}}$
$v = 2.188 \times 10^{6}$ m/s
$v \approx 2.19 \times 10^{6}$ m/sto 3 significant figures

Conclusion

The proton's pull and the rule that angular momentum comes in whole steps together fix the speed, without needing the electron's mass; higher levels are slower.

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_electron_s_speed_in_the_first_level do4  kind :numerical5  describes "Using the Bohr's model, %{speed} in a hydrogen atom in the n = %{level} level, in m/s?"6  given level: "1"7  asks speed: "calculate the speed of the electron"8end

solution.rb

1require_relative "../questions/the_electron_s_speed_in_the_first_level"2require_relative "../science/atoms/bohr_model"34solve :the_electron_s_speed_in_the_first_level, using: BohrModel do5  objects :electron67  take level: electron.quantum_number89  deduce_from "hydrogen atom", electron.atomic_number => "1"1011  speed = electron.orbital_speed12  answer speed:, in: "m/s", figures: 31314  conclusion "The proton's pull and the rule that angular momentum comes in whole steps together fix the speed, " \15             "without needing the electron's mass; higher levels are slower."16end

playground.rb

1require_relative "../solutions/the_electron_s_speed_in_the_first_level"23play_with :the_electron_s_speed_in_the_first_level do4  title "The Electron's Speed in the First Level"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)