_______________________ __________________

(Group Name) (Date)

Exercise on Fractions and Isotopic Composition

Manager
______________________ Recorder_____________________

Spokesperson___________________ Analyst______________________

In
this problem set you will be given questions pertaining to the isotopic
composition of various elements. Use the
following symbols for your algebraic equations

**m = mass; m _{1} = mass of species 1, m_{2} = mass of species 2..... m_{ave} = Average Mass**

**X = fraction; X _{1}
= fraction of species 1, X_{2}
= Fraction of species **2...

**Problem 1: ****Consider an element which has 3
naturally occurring isotopes. The
isotopic masses and percent abundances are:
53.9396 amu (5.845%), 55.9349 amu (91.754%) and 56.9354amu . What is the
Average Mass?**

a. Define algebraic mass variables (m_{1} = , m_{2} =.....) and write values for
those given in the problem statement, use question marks for unknowns (there
should be 4)

**m _{1}
=**

b.
Define fractional abundance variables (X_{1} =, X_{2} =.....)
and write values for those given in the problem statement, use question marks
for unknowns (there should be 3)

**X _{1} =0.05845,
X_{2} = 0.91754 , X_{3} =?**

c.
Calculate value of remaining fractional abundance variable (think about the
concept of a fraction and the fact that there are only three types of
isotopes).

**(X _{3} =1 - 0.05845 - 0.91754 = 0.02401)**

d.
Set up and algebraic equation and identify variable you are solving for. Solve
equation for unknown.

**m _{1}X_{1} + m_{2}X_{2} + m_{3}X_{3}
= m_{ave} **

e.
Substitute numerical values into algebraic equation and solve

**m _{ave} = 0.05845(53.9396amu) + 0.91754(55.9349amu)
+ 0.02401(56.9354amu) = 55.84 amu**

f.
What element is this? **Iron **

**Problem 2: Copper has two
naturally occurring isotopes. One of the
isotopes has an atomic weight of 62.9 amu and an abundance of 69.1%. What is the atomic weight (amu) of the other
isotope?**

a. Define algebraic mass variables, write down givens and unknowns
(there should be 3):

**m _{1}
=**

b.
Define fractional abundance variables, write down givens and unknowns (there
should be 2):

**X _{1} =0.691 ,
X_{2} = ?**

c.
Set up (write down) algebraic equation based on the concept of fractional
abundance of isotopes.

**m _{1}X_{1} + m_{2}X_{2} = m_{ave}
**

d.
Identify unknown you are solving for and rearrange algebraic equation to show
solution for your desired unknown.

_{}

e. You can not arithmetically solve one equation
if you have more than one unknown. You
have three unknowns, and so you need to look somewhere else for information.
What are the 2 additional unknowns? ** m _{ave} and X_{2}.**

f. What is the average atomic weight? Remember,
this is a chemistry class and you should be able to find that for any
element. Write it down in terms of your
algebraic variables.** ****m _{ave} = 63.546 amu**

g.
There are only two isotopes. How can you
determine the fractional abundance of the unknown isotope? Write it down in terms of you algebraic
variables.

**X _{2} = 1 - 0.691 = 0.309**

h.
Substitute numerical values into the algebraic equation and arithmetically
solve for the atomic weight of the other unknown.

_{}

**Problem 3: Natural Chlorine consists of **^{ 35}**Cl**** (34.97 amu) and**** ^{37}Cl**

a. Define algebraic mass variables, write down givens and unknowns
(there should be 3):

**m _{1}
=34.97 amu , m_{2} = 36.97 amu
m_{ave} = ?**

b.
Define fractional abundance variables, write down givens and unknowns (there
should be 2):

**X _{1} =? , X_{2} = ?**

c.
Set up (write down) algebraic equation based on the concept of fractional
abundance of isotopes.

**m _{1}X_{1} + m_{2}X_{2} = m_{ave}
**

d.
Identify unknown you are solving for and rearrange algebraic equation to show
solution for your desired unknown.

_{}

e. You can not arithmetically solve one equation
if you have more than one unknown. You
have three unknowns, and so you need to look somewhere else for information.
What are the 2 additional unknowns? ** m _{ave} and X_{2}.**

f. What is the average atomic weight? Remember,
this is a chemistry class and you should be able to find that for any
element. Write it down in terms of your
algebraic variables. **m _{ave}
= **

g.
There are only two isotopes. How can you
determine the fractional abundance of the additional unknown isotope (not the
one are solving for)? Write it down in
terms of you algebraic variables.

**1=X _{1} + X_{2} so X_{2}= 1 - X_{1}**

h.
Substitute this expression into the equation solving for the unknown you are
after and algebraically solve for the unknown

_{}

i.
Substitute numerical values into the algebraic equation and arithmetically
solve for the atomic weight of the other unknown.

_{}

note,
X_{2} = 1-0.76 = 0.24, so it is 24.0% of the higher isotope