Gold and Silver Nuclei

solved NuclearSize

Q. What is approximately the ratio of the nuclear radii of the gold isotope (A = 197 $A_{g} = 197$ ) and the silver isotope (A = 107 $A_{s} = 107$ )?

Inputs

gold

A_{g}

197

silver

A_{s}

107

Outputs

ratio?

How it's solved

Solved with NuclearSize

Given

$A_{g} = 197$
$A_{s} = 107$

Find $\frac{R_{g}}{R_{s}}$ .

Working

radius grows as the cube root(silver) $R_{s} = R_{0} \cdot {A_{s}}^{0.333333333}$
$R_{s} = R_{0} \cdot 4.7474594$
radius grows as the cube root(gold) $R_{g} = R_{0} \cdot {A_{g}}^{0.333333333}$
$R_{g} = R_{0} \cdot 5.81864787$
$\frac{R_{g}}{R_{s}} = 1.22563$

Conclusion

A nucleus's radius grows only as the cube root of its nucleons, so gold, with nearly twice silver's nucleons, is only about a quarter wider.

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 :gold_and_silver_nuclei do4  kind :numerical5  describes "%{ratio} of the gold isotope (A = %{gold}) and the silver isotope (A = %{silver})?"6  given gold: "197"7  given silver: "107"8  asks ratio: "What is approximately the ratio of the nuclear radii"9end

solution.rb

1require_relative "../questions/gold_and_silver_nuclei"2require_relative "../science/nuclei/nuclear_size"34solve :gold_and_silver_nuclei, using: NuclearSize do5  objects :gold, :silver67  take gold: gold.mass_number8  take silver: silver.mass_number910  answer find_relation(gold.nuclear_radius / silver.nuclear_radius), figures: 31112  conclusion "A nucleus's radius grows only as the cube root of its nucleons, so gold, with nearly twice " \13             "silver's nucleons, is only about a quarter wider."14end

playground.rb

1require_relative "../solutions/gold_and_silver_nuclei"23play_with :gold_and_silver_nuclei do4  title "Gold and Silver Nuclei"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 gold	$R_{g}$	m	—
mass number of gold	$A_{g}$	—	0 … ∞
nuclear radius of silver	$R_{s}$	m	—
mass number of silver	$A_{s}$	—	0 … ∞
nuclear density of ground	$ρ_{g}$	kg/m³	—
mass of ground	$m_{g}$	kg	0 … ∞
nuclear density of gold	$ρ_{g}$	kg/m³	—
mass of gold	$m_{g}$	kg	0 … ∞
nuclear density of silver	$ρ_{s}$	kg/m³	—
mass of silver	$m_{s}$	kg	0 … ∞
coulomb barrier of ground	$U_{b,g}$	J	—
atomic number of ground	$Z_{g}$	—	0 … ∞
coulomb barrier of gold	$U_{b,g}$	J	—
atomic number of gold	$Z_{g}$	—	0 … ∞
coulomb barrier of silver	$U_{b,s}$	J	—
atomic number of silver	$Z_{s}$	—	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_{g} = R_{0} \cdot {A_{g}}^{0.333333333}

radius grows as the cube root(gold)

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

radius grows as the cube root(silver)

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

density of a nucleus(ground)

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

density of a nucleus(gold)

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

density of a nucleus(silver)

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

two alike nuclei just touching(ground)

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

two alike nuclei just touching(gold)

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

two alike nuclei just touching(silver)