Two Carbon Nuclei Fusing

solved MassEnergy

Q. The Q value of a nuclear reaction A + b → C + d is defined by Q = [mA + mb – mC – md]c², where the masses refer to the respective nuclei. Given m(¹²₆C) = 12.000000 u $m_{c} = 12.000000$ u, m(²⁰₁₀Ne) = 19.992439 u $m_{n} = 19.992439$ u and m(⁴₂He) = 4.002603 u $m_{h} = 4.002603$ u, what is the Q-value of the reaction ¹²₆C + ¹²₆C → ²⁰₁₀Ne + ⁴₂He ${dm}_{r} = 2 \cdot m_{c} - m_{n} - m_{h}$ , in MeV?

Inputs

carbon

m_{c}

12 u

neon

m_{n}

19.99 u

helium

m_{h}

4.003 u

Outputs

Q_{r}

How it's solved

Solved with MassEnergy

Given

$m_{c} = 12.000000$ u
$m_{n} = 19.992439$ u
$m_{h} = 4.002603$ u
${dm}_{r} = 2 \cdot m_{c} - m_{n} - m_{h}$ from “¹²₆C + ¹²₆C → ²⁰₁₀Ne + ⁴₂He”

Find $Q_{r}$ .

Working

$m_{c} = 12.000000$ u $= 1.99264688 \times 10^{−26}$ kg
$m_{n} = 19.992439$ u $= 3.3198226 \times 10^{−26}$ kg
$m_{h} = 4.002603$ u $= 6.64647865 \times 10^{−27}$ kg
¹²₆C + ¹²₆C → ²⁰₁₀Ne + ⁴₂He ${dm}_{r} = 2 \cdot m_{c} - m_{n} - m_{h}$
${dm}_{r} = 2 \cdot 1.99264688 \times 10^{−26} - 3.3198226 \times 10^{−26} - 6.64647865 \times 10^{−27}$
${dm}_{r} = 8.23295269 \times 10^{−30}$ kg
energy a reaction gives out(reaction) $Q_{r} = {dm}_{r} \cdot c^{2}$
$Q_{r} = 8.23295269 \times 10^{−30} \cdot {(2.99792458 \times 10^{8})}^{2}$
$Q_{r} = 4.618$ MeV
$Q_{r} = 7.39940887 \times 10^{−13}$ J $= 4.61834776$ MeV
$Q_{r} \approx 4.62$ MeVto 3 significant figures

Conclusion

The neon and helium nuclei weigh less than the two carbon nuclei, and the lost mass comes out as energy: the reaction is exothermic.

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_carbon_nuclei_fusing do4  kind :numerical5  describes "The Q value of a nuclear reaction A + b → C + d is defined by Q = [mA + mb – mC – md]c², where the " \6            "masses refer to the respective nuclei. Given m(¹²₆C) = %{carbon}, m(²⁰₁₀Ne) = %{neon} and " \7            "m(⁴₂He) = %{helium}, %{q} of the reaction ¹²₆C + ¹²₆C → ²⁰₁₀Ne + ⁴₂He, in MeV?"8  given carbon: "12.000000 u"9  given neon: "19.992439 u"10  given helium: "4.002603 u"11  asks q: "what is the Q-value"12end

solution.rb

1require_relative "../questions/two_carbon_nuclei_fusing"2require_relative "../science/nuclei/mass_energy"34solve :two_carbon_nuclei_fusing, using: MassEnergy do5  objects :reaction, :carbon, :neon, :helium67  take carbon: carbon.mass8  take neon: neon.mass9  take helium: helium.mass1011  deduce_from "¹²₆C + ¹²₆C → ²⁰₁₀Ne + ⁴₂He",12              reaction.mass_defect => 2 * carbon.mass - neon.mass - helium.mass1314  q = reaction.q_value15  answer q:, in: "MeV", figures: 31617  conclusion "The neon and helium nuclei weigh less than the two carbon nuclei, and the lost mass comes out as " \18             "energy: the reaction is exothermic."19end

playground.rb

1require_relative "../solutions/two_carbon_nuclei_fusing"23play_with :two_carbon_nuclei_fusing do4  title "Two Carbon Nuclei Fusing"5end

The laws it drew on

quantity	written	units	within	from
speed of light	$c$	m/s	—	MassEnergy
hydrogen atom mass	$m_{H}$	kg	—	MassEnergy
neutron mass	$m_{n}$	kg	—	MassEnergy
mass	$m$	kg	0 … ∞	MassEnergy
energy	$E$	J	—	MassEnergy
atomic number	$Z$	—	—	MassEnergy
mass number	$A$	—	0 … ∞	MassEnergy
mass defect	$dm$	kg	—	MassEnergy
binding energy	$E_{b}$	J	—	MassEnergy
q value	$Q$	J	—	MassEnergy
energy of ground	$E_{g}$	J	—
mass of ground	$m_{g}$	kg	0 … ∞
energy of reaction	$E_{r}$	J	—
mass of reaction	$m_{r}$	kg	0 … ∞
energy of carbon	$E_{c}$	J	—
mass of carbon	$m_{c}$	kg	0 … ∞
energy of neon	$E_{n}$	J	—
mass of neon	$m_{n}$	kg	0 … ∞
energy of helium	$E_{h}$	J	—
mass of helium	$m_{h}$	kg	0 … ∞
mass defect of ground	${dm}_{g}$	kg	—
atomic number of ground	$Z_{g}$	—	—
mass number of ground	$A_{g}$	—	0 … ∞
mass defect of reaction	${dm}_{r}$	kg	—
atomic number of reaction	$Z_{r}$	—	—
mass number of reaction	$A_{r}$	—	0 … ∞
mass defect of carbon	${dm}_{c}$	kg	—
atomic number of carbon	$Z_{c}$	—	—
mass number of carbon	$A_{c}$	—	0 … ∞
mass defect of neon	${dm}_{n}$	kg	—
atomic number of neon	$Z_{n}$	—	—
mass number of neon	$A_{n}$	—	0 … ∞
mass defect of helium	${dm}_{h}$	kg	—
atomic number of helium	$Z_{h}$	—	—
mass number of helium	$A_{h}$	—	0 … ∞
binding energy of ground	$E_{b,g}$	J	—
binding energy of reaction	$E_{b,r}$	J	—
binding energy of carbon	$E_{b,c}$	J	—
binding energy of neon	$E_{b,n}$	J	—
binding energy of helium	$E_{b,h}$	J	—
q value of ground	$Q_{g}$	J	—
q value of reaction	$Q_{r}$	J	—
q value of carbon	$Q_{c}$	J	—
q value of neon	$Q_{n}$	J	—
q value of helium	$Q_{h}$	J	—

E_{b} = m_{b} \cdot c^{2}

einsteins mass energy

{dm}_{n} = Z_{n} \cdot m_{H} + (A_{n} - Z_{n}) \cdot m_{n} - m_{n}

mass lost in binding

E_{b,n} = {dm}_{n} \cdot c^{2}

energy of the lost mass

Q_{r} = {dm}_{r} \cdot c^{2}

energy a reaction gives out

E_{g} = m_{g} \cdot c^{2}

einsteins mass energy(ground)

E_{r} = m_{r} \cdot c^{2}

einsteins mass energy(reaction)

E_{c} = m_{c} \cdot c^{2}

einsteins mass energy(carbon)

E_{n} = m_{n} \cdot c^{2}

einsteins mass energy(neon)

E_{h} = m_{h} \cdot c^{2}

einsteins mass energy(helium)

{dm}_{g} = Z_{g} \cdot m_{H} + (A_{g} - Z_{g}) \cdot m_{n} - m_{g}

mass lost in binding(ground)

{dm}_{r} = Z_{r} \cdot m_{H} + (A_{r} - Z_{r}) \cdot m_{n} - m_{r}

mass lost in binding(reaction)

{dm}_{c} = Z_{c} \cdot m_{H} + (A_{c} - Z_{c}) \cdot m_{n} - m_{c}

mass lost in binding(carbon)

{dm}_{n} = Z_{n} \cdot m_{H} + (A_{n} - Z_{n}) \cdot m_{n} - m_{n}

mass lost in binding(neon)

{dm}_{h} = Z_{h} \cdot m_{H} + (A_{h} - Z_{h}) \cdot m_{n} - m_{h}

mass lost in binding(helium)

E_{b,g} = {dm}_{g} \cdot c^{2}

energy of the lost mass(ground)

E_{b,r} = {dm}_{r} \cdot c^{2}

energy of the lost mass(reaction)

E_{b,c} = {dm}_{c} \cdot c^{2}

energy of the lost mass(carbon)

E_{b,n} = {dm}_{n} \cdot c^{2}

energy of the lost mass(neon)

E_{b,h} = {dm}_{h} \cdot c^{2}

energy of the lost mass(helium)

Q_{g} = {dm}_{g} \cdot c^{2}

energy a reaction gives out(ground)

Q_{r} = {dm}_{r} \cdot c^{2}

energy a reaction gives out(reaction)

Q_{c} = {dm}_{c} \cdot c^{2}

energy a reaction gives out(carbon)

Q_{n} = {dm}_{n} \cdot c^{2}

energy a reaction gives out(neon)

Q_{h} = {dm}_{h} \cdot c^{2}

energy a reaction gives out(helium)

{dm}_{r} = 2 \cdot m_{c} - m_{n} - m_{h}

¹²₆C + ¹²₆C → ²⁰₁₀Ne + ⁴₂He