- #1
Lissajoux
- 82
- 0
Homework Statement
The binding energy of a nucleus is given by:
[tex]E_{b}=a_{1}A-a_{2}A^{\frac{2}{3}}-a_{3}Z^{2}A^{-\frac{1}{3}}-a_{4}\left(Z-\frac{A}{2}\right)^{2}A^{-1}\pm a_{5}A^{-\frac{1}{2}}[/tex]
For a given set of isobars, [itex]A[/itex] constant, [itex]E_{b}[/itex] will be maximised at the value of [itex]Z[/itex] that satisfies:
[tex]\frac{dE_{b}}{dZ}=0[/tex]
a) Find this derivative and solve for [itex]Z[/itex] in terms of the [itex]a_{i}[/itex] and [itex]A[/itex]
b) Using the expression derived in part a) and given values of [itex]a_{i}[/itex], find the value of [itex]Z[/itex] which maximises [itex]E_{b}[/itex] when [itex]A=25[/itex]. Round [itex]Z[/itex] to the nearest integer value.
c) Use the periodic table and the derived value of [itex]Z[/itex] from part b) to determine which element of mass number 25 is a stable isotope.
Homework Equations
Within the problem statement and subsequent solution attempt.
The Attempt at a Solution
a) This is what I have for the differentiated equation:
[tex]\frac{dE_{b}}{dZ}=-2Za_{3}A^{-\frac{1}{3}}-a_{4}\left(2ZA^{-1}-1\right)=0[/tex]
.. this is correct? I think that's all I have to do for this part.
b) I know what the values of [itex]A, a_{3}, a_{4}[/itex] are, so I can put these into the equation, but then how to I 'solve it for Z' from that? I guess that need to find a value of [itex]Z[/itex] such that [itex]\frac{dE_{b}}{dZ}=0[/itex] right? Not sure how to do this.
c) Once I know what [itex]Z[/itex] is, how do I determine which element of mass number 25 is a stable isotope? I don't really understand this question part.
Last edited: