Respuesta :
Solution: For calculating nuclear binding energy we need to find the binding energy and that can be calculated by mass defect. Mass defect is the excess mass present in any isotope.
[tex]_{13}^{27}\textrm{Al}[/tex] nucleus having 13 protons as it is visible from atomic number and number of neutrons is 14 which is calculated from [tex]\text{atomic mass= number of neutrons + number of protons}[/tex].
As it is given in the question that [tex]m_p= 1.0072765amu[/tex] and [tex]m_n=1.0086649amu[/tex], we can calculate the mass of the isotope.
[tex]\text{Mass of Isotope}= (13\times1.0072765)amu+(14\times1.0086649)amu[/tex]
[tex]\text{Mass of isotope}=27.2159031amu[/tex]
Mass defect or [tex]\Delta m[/tex] = mass of isotope - atomic mass.
[tex]\Delta m=27.2159031-26.9815386[/tex]
[tex]\Delta m=0.2343645amu[/tex]
As the mass is expressed in kilograms do to convert it into kilograms, we need to use the conversion [tex](1amu=1.6606\times10^-^2^7kg)[/tex]
[tex]\Delta m=(0.2343645\frac{amu}{nucleus})(1.6606\times10^-^2^7\frac{kg}{amu})[/tex]
[tex]\Delta m=3.891856\times10^-^2^6kg/nucleus[/tex]
To calculate energy binding energy we will use the formula
[tex]\Delta E=\Delta mc^2[/tex]
where c= speed of light [tex](2.9979\times10^8m/s)[/tex]
[tex]\Delta E=(3.891856\times10^-^2^6kg/nucleus)(2.9979\times10^8m/s)^2[/tex]
[tex]\Delta E=3.497768\times10^-^1^1 J/nucleus[/tex]
Nuclear binding energy is expressed in kJ/mol of nuclei, so for that we need to convert Joules to kJ by conversion [tex](1kJ=1000J)[/tex] and converting individual particles into moles, we need to multiply it by avagadro's number that is [tex]6.022\times10^2^3nuclei/mol[/tex].
[tex]\text{Nuclear Binding Energy}= (3.497768\times10^-^1^1\frac{J}{nucleus})(\frac{1kJ}{1000J})(6.022\times10^2^3\frac{nuclei}{mol})[/tex]
[tex]N.B.E.=2.106355\times10^1^1\text{kJ/mol of nuclei}[/tex]
Amount of energy to break down the energy of a single bond of [tex]_{13}^{27}\textrm{Al}[/tex] nuclei is [tex]2.106355\times10^1^1kJ/\text{mol of nuclei}[/tex]
