Two Carbon Nuclei Fusing

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(²⁰₁₀Ne) = 19.992439 u and m(⁴₂He) = 4.002603 u, what is the Q-value of the reaction ¹²₆C + ¹²₆C → ²⁰₁₀Ne + ⁴₂He, in MeV? expects 4.62 MeV

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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?

The playground

Inputs

carbon

m_{c}

12 u

neon

m_{n}

19.99 u

helium

m_{h}

4.003 u

Outputs

Q_{r}

The solution

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 Ruby

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

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

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

The laws

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