Two Deuterons Meeting

solved NuclearSize

Q. What is the height of the potential barrier for a head on collision of two deuterons $Z = 1$ , in keV? (Hint: The height of the potential barrier is given by the Coulomb repulsion between the two deuterons when they just touch each other. Assume that they can be taken as hard spheres of radius 2.0 fm $R = 2.0$ fm.)

Inputs

radius

R

2 fm

Outputs

barrier

U_{b}

How it's solved

Solved with NuclearSize

Given

$R = 2.0$ fm
$Z = 1$ from “two deuterons”

Find $U_{b}$ .

Working

$R = 2.0$ fm $= 2 \times 10^{−15}$ m
two alike nuclei just touching(deuteron) $U_{b} = \frac{k \cdot {(Z \cdot e)}^{2}}{2 \cdot R}$
$U_{b} = \frac{8.98755179 \times 10^{9} \cdot {(1 \cdot 1.60217663 \times 10^{−19})}^{2}}{2 \cdot 2 \times 10^{−15}}$
$U_{b} = 360$ keV
$U_{b} = 5.76769388 \times 10^{−14}$ J $= 359.991137$ keV
$U_{b} \approx 3.6 \times 10^{2}$ keVto 2 significant figures

Conclusion

Two deuterons must be brought to touching against their electric repulsion before they can fuse, and the energy that takes is the barrier.

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 :two_deuterons_meeting do4  kind :numerical5  describes "%{barrier} for a head on collision of two deuterons, in keV? (Hint: The height of the potential " \6            "barrier is given by the Coulomb repulsion between the two deuterons when they just touch each other. " \7            "Assume that they can be taken as hard spheres of radius %{radius}.)"8  given radius: "2.0 fm"9  asks barrier: "What is the height of the potential barrier"10end

solution.rb

1require_relative "../questions/two_deuterons_meeting"2require_relative "../science/nuclei/nuclear_size"34solve :two_deuterons_meeting, using: NuclearSize do5  objects :deuteron67  take radius: deuteron.nuclear_radius89  deduce_from "two deuterons", deuteron.atomic_number => "1"1011  barrier = deuteron.coulomb_barrier12  answer barrier:, in: "keV", figures: 21314  conclusion "Two deuterons must be brought to touching against their electric repulsion before they can fuse, " \15             "and the energy that takes is the barrier."16end

playground.rb

1require_relative "../solutions/two_deuterons_meeting"23play_with :two_deuterons_meeting do4  title "Two Deuterons Meeting"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	NuclearSize
coulomb constant	$k$	N m² C⁻²	—	NuclearSize
elementary charge	$e$	C	—	NuclearSize
size of a nucleon s share	$R_{0}$	m	—	NuclearSize
mass number	$A$	—	0 … ∞	NuclearSize
atomic number	$Z$	—	0 … ∞	NuclearSize
mass	$m$	kg	0 … ∞	NuclearSize
nuclear radius	$R$	m	—	NuclearSize
nuclear density	$ρ$	kg/m³	—	NuclearSize
coulomb barrier	$U_{b}$	J	—	NuclearSize
nuclear radius of ground	$R_{g}$	m	—
mass number of ground	$A_{g}$	—	0 … ∞
nuclear radius of deuteron	$R$	m	—
mass number of deuteron	$A$	—	0 … ∞
nuclear density of ground	$ρ_{g}$	kg/m³	—
mass of ground	$m_{g}$	kg	0 … ∞
nuclear density of deuteron	$ρ$	kg/m³	—
mass of deuteron	$m$	kg	0 … ∞
coulomb barrier of ground	$U_{b,g}$	J	—
atomic number of ground	$Z_{g}$	—	0 … ∞
coulomb barrier of deuteron	$U_{b}$	J	—
atomic number of deuteron	$Z$	—	0 … ∞

R_{n} = R_{0} \cdot {A_{n}}^{0.333333333}

radius grows as the cube root

ρ_{n} = \frac{m_{n}}{\frac{4 \cdot π \cdot {R_{n}}^{3}}{3}}

density of a nucleus

U_{b,n} = \frac{k \cdot {(Z_{n} \cdot e)}^{2}}{2 \cdot R_{n}}

two alike nuclei just touching

R_{g} = R_{0} \cdot {A_{g}}^{0.333333333}

radius grows as the cube root(ground)

R = R_{0} \cdot A^{0.333333333}

radius grows as the cube root(deuteron)

ρ_{g} = \frac{m_{g}}{\frac{4 \cdot π \cdot {R_{g}}^{3}}{3}}

density of a nucleus(ground)

ρ = \frac{m}{\frac{4 \cdot π \cdot R^{3}}{3}}

density of a nucleus(deuteron)

U_{b,g} = \frac{k \cdot {(Z_{g} \cdot e)}^{2}}{2 \cdot R_{g}}

two alike nuclei just touching(ground)

U_{b} = \frac{k \cdot {(Z \cdot e)}^{2}}{2 \cdot R}

two alike nuclei just touching(deuteron)