A Bismuth Nucleus's Binding Energy

Q. What is the binding energy of the nucleus of bismuth (A = 209, Z = 83), given its mass 208.980388 u, with mH = 1.007825 u and mn = 1.008665 u, in MeV? expects 1640 MeV

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Q. What is the binding energy of the nucleus of bismuth (A = 209 $A = 209$ , Z = 83 $Z = 83$ ), given its mass 208.980388 u $m = 208.980388$ u, with mH = 1.007825 u $m_{H} = 1.007825$ u and mn = 1.008665 u $m_{n} = 1.008665$ u, in MeV?

The playground

Inputs

nucleons

A

209

protons

Z

mass

m

209 u

hydrogen

m_{H}

1.008 u

neutron

m_{n}

1.009 u

Outputs

energy

E_{b}

The solution

Solved with MassEnergy

Given

$A = 209$
$Z = 83$
$m = 208.980388$ u
$m_{H} = 1.007825$ u
$m_{n} = 1.008665$ u

Find $E_{b}$ .

Working

$m = 208.980388$ u $= 3.47020098 \times 10^{−25}$ 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(bismuth) $dm = Z \cdot m_{H} + (A - Z) \cdot m_{n} - m$
$dm = 83 \cdot 1.67353278 \times 10^{−27} + (209 - 83) \cdot 1.67492764 \times 10^{−27} - 3.47020098 \times 10^{−25}$
$dm = 2.92400505 \times 10^{−27}$ kg
energy of the lost mass(bismuth) $E_{b} = dm \cdot c^{2}$
$E_{b} = 2.92400505 \times 10^{−27} \cdot {(2.99792458 \times 10^{8})}^{2}$
$E_{b} = 1640$ MeV
$E_{b} = 2.62796468 \times 10^{−10}$ J $= 1640.24654$ MeV
$E_{b} \approx 1640$ MeVto 4 significant figures

Conclusion

A heavy nucleus holds many more nucleons, so its total binding energy is larger, though each nucleon in it is held a little less tightly than in iron.

The Ruby

1require_relative "../physics"23question :a_bismuth_nucleus_s_binding_energy do4  kind :numerical5  describes "%{energy} of the nucleus of bismuth (A = %{nucleons}, Z = %{protons}), given its mass %{mass}, with " \6            "mH = %{hydrogen} and mn = %{neutron}, in MeV?"7  given nucleons: "209"8  given protons: "83"9  given mass: "208.980388 u"10  given hydrogen: "1.007825 u"11  given neutron: "1.008665 u"12  asks energy: "What is the binding energy"13end

1require_relative "../questions/a_bismuth_nucleus_s_binding_energy"2require_relative "../science/nuclei/mass_energy"34solve :a_bismuth_nucleus_s_binding_energy, using: MassEnergy do5  objects :bismuth67  take nucleons: bismuth.mass_number8  take protons: bismuth.atomic_number9  take mass: bismuth.mass10  take hydrogen: m_H11  take neutron: m_n1213  energy = bismuth.binding_energy14  answer energy:, in: "MeV", figures: 41516  conclusion "A heavy nucleus holds many more nucleons, so its total binding energy is larger, though each " \17             "nucleon in it is held a little less tightly than in iron."18end

1require_relative "../solutions/a_bismuth_nucleus_s_binding_energy"23play_with :a_bismuth_nucleus_s_binding_energy do4  title "A Bismuth Nucleus's Binding Energy"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 bismuth	$E$	J	—
mass of bismuth	$m$	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 bismuth	$dm$	kg	—
atomic number of bismuth	$Z$	—	—
mass number of bismuth	$A$	—	0 … ∞
binding energy of ground	$E_{b,g}$	J	—
binding energy of bismuth	$E_{b}$	J	—
q value of ground	$Q_{g}$	J	—
q value of bismuth	$Q$	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(bismuth)

{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(bismuth)

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(bismuth)

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

energy a reaction gives out(ground)

Q = dm \cdot c^{2}

energy a reaction gives out(bismuth)