An Iron Nucleus's Binding Energy

solved MassEnergy

Q. What is the binding energy of the nucleus of iron (A = 56 $A = 56$ , Z = 26 $Z = 26$ ), given its mass 55.934939 u $m = 55.934939$ u, with mH = 1.007825 u $m_{H} = 1.007825$ u and mn = 1.008665 u $m_{n} = 1.008665$ u, in MeV?

Inputs

nucleons

A

protons

Z

mass

m

55.93 u

hydrogen

m_{H}

1.008 u

neutron

m_{n}

1.009 u

Outputs

energy

E_{b}

How it's solved

Solved with MassEnergy

Given

$A = 56$
$Z = 26$
$m = 55.934939$ u
$m_{H} = 1.007825$ u
$m_{n} = 1.008665$ u

Find $E_{b}$ .

Working

$m = 55.934939$ u $= 9.28821514 \times 10^{−26}$ kg
$m_{H} = 1.007825$ u $= 1.67353278 \times 10^{−27}$ kg
$m_{n} = 1.008665$ u $= 1.67492764 \times 10^{−27}$ kg
mass lost in binding(iron) $dm = Z \cdot m_{H} + (A - Z) \cdot m_{n} - m$
$dm = 26 \cdot 1.67353278 \times 10^{−27} + (56 - 26) \cdot 1.67492764 \times 10^{−27} - 9.28821514 \times 10^{−26}$
$dm = 8.77530136 \times 10^{−28}$ kg
energy of the lost mass(iron) $E_{b} = dm \cdot c^{2}$
$E_{b} = 8.77530136 \times 10^{−28} \cdot {(2.99792458 \times 10^{8})}^{2}$
$E_{b} = 492.3$ MeV
$E_{b} = 7.88684754 \times 10^{−11}$ J $= 492.258305$ MeV
$E_{b} \approx 492.3$ MeVto 4 significant figures

Conclusion

The iron nucleus weighs less than its protons and neutrons apart; that lost mass, as energy, is what holds it together.

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 :an_iron_nucleus_s_binding_energy do4  kind :numerical5  describes "%{energy} of the nucleus of iron (A = %{nucleons}, Z = %{protons}), given its mass %{mass}, with " \6            "mH = %{hydrogen} and mn = %{neutron}, in MeV?"7  given nucleons: "56"8  given protons: "26"9  given mass: "55.934939 u"10  given hydrogen: "1.007825 u"11  given neutron: "1.008665 u"12  asks energy: "What is the binding energy"13end

solution.rb

1require_relative "../questions/an_iron_nucleus_s_binding_energy"2require_relative "../science/nuclei/mass_energy"34solve :an_iron_nucleus_s_binding_energy, using: MassEnergy do5  objects :iron67  take nucleons: iron.mass_number8  take protons: iron.atomic_number9  take mass: iron.mass10  take hydrogen: m_H11  take neutron: m_n1213  energy = iron.binding_energy14  answer energy:, in: "MeV", figures: 41516  conclusion "The iron nucleus weighs less than its protons and neutrons apart; that lost mass, as energy, is " \17             "what holds it together."18end

playground.rb

1require_relative "../solutions/an_iron_nucleus_s_binding_energy"23play_with :an_iron_nucleus_s_binding_energy do4  title "An Iron Nucleus's Binding Energy"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 iron	$E$	J	—
mass of iron	$m$	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 iron	$dm$	kg	—
atomic number of iron	$Z$	—	—
mass number of iron	$A$	—	0 … ∞
binding energy of ground	$E_{b,g}$	J	—
binding energy of iron	$E_{b}$	J	—
q value of ground	$Q_{g}$	J	—
q value of iron	$Q$	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 = m \cdot c^{2}

einsteins mass energy(iron)

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

mass lost in binding(ground)

dm = Z \cdot m_{H} + (A - Z) \cdot m_{n} - m

mass lost in binding(iron)

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

energy of the lost mass(ground)

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

energy of the lost mass(iron)

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

energy a reaction gives out(ground)

Q = dm \cdot c^{2}

energy a reaction gives out(iron)