A Jump Between Two Levels

solved BohrModel

Q. A difference of 2.3 eV $E_{p} = 2.3$ eV separates two energy levels in an atom. What is the frequency of radiation emitted when the atom makes a transition from the upper level to the lower level, in hertz?

Inputs

gap

E_{p}

2.3 eV

Outputs

frequency

ν

How it's solved

Solved with BohrModel

Given

$E_{p} = 2.3$ eV

Find $ν$ .

Working

$E_{p} = 2.3$ eV $= 3.68500626 \times 10^{−19}$ J
one photon for one jump(radiation) $E_{p} = h \cdot ν$
$3.68500626 \times 10^{−19} = 6.62607015 \times 10^{−34} \cdot ν$
$ν = 5.561 \times 10^{14}$ Hz
$ν \approx 5.6 \times 10^{14}$ Hzto 2 significant figures

Conclusion

The atom loses exactly the energy between the two levels and sends it out as one photon, whose frequency is that energy over Planck's constant.

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 :a_jump_between_two_levels do4  kind :numerical5  describes "A difference of %{gap} separates two energy levels in an atom. %{frequency} when the atom makes a " \6            "transition from the upper level to the lower level, in hertz?"7  given gap: "2.3 eV"8  asks frequency: "What is the frequency of radiation emitted"9end

solution.rb

1require_relative "../questions/a_jump_between_two_levels"2require_relative "../science/atoms/bohr_model"34solve :a_jump_between_two_levels, using: BohrModel do5  objects :radiation67  take gap: radiation.photon_energy89  frequency = radiation.frequency10  answer frequency:, in: "Hz", figures: 21112  conclusion "The atom loses exactly the energy between the two levels and sends it out as one photon, whose " \13             "frequency is that energy over Planck's constant."14end

playground.rb

1require_relative "../solutions/a_jump_between_two_levels"23play_with :a_jump_between_two_levels do4  title "A Jump Between Two Levels"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 radiation	$E_{b}$	J	—
orbit radius of radiation	$r$	m	—
mass of ground	$m_{g}$	kg	0 … ∞
orbital speed of ground	$v_{g}$	m/s	0 … ∞
mass of radiation	$m$	kg	0 … ∞
orbital speed of radiation	$v$	m/s	0 … ∞
atomic number of ground	$Z_{g}$	—	—
quantum number of ground	$n_{g}$	—	—
atomic number of radiation	$Z$	—	—
quantum number of radiation	$n$	—	—
potential energy of ground	$U_{g}$	J	—
potential energy of radiation	$U$	J	—
total energy of ground	$E_{g}$	J	—
total energy of radiation	$E$	J	—
photon energy of ground	$E_{p,g}$	J	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
photon energy of radiation	$E_{p}$	J	—
frequency of radiation	$ν$	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(radiation)

\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(radiation)

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(radiation)

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(radiation)

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(radiation)

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(radiation)

E_{p,g} = h \cdot ν_{g}

one photon for one jump(ground)

E_{p} = h \cdot ν

one photon for one jump(radiation)