268 ❯ STEP 4. Review the Knowledge You Need to Score High
❯ Answer and Explanation
Note that, while nuclear chemistry is not an AP topic, all the materials in these questions
depend on basic AP chemistry knowledge, which is why this chapter can be valuable. It is
especially important to remember that nuclear decay processes follow first-order kinetics.
(a) This is a stoichiometry calculation. To simplify the mole calculation, it is useful to
remember that numerically, g/mole is the same as mg/mmole.Mass^238 U = 47.6 mg^238 U +
(16.4 mg Pb)
1 mmole Pb
206 mg Pb1 mmole U
1 mmole Pb238 mg U
1 mmole U206
206
206238
206238
238= 66.5 mg^238 UThe molar mass of^206 Pb is 206 g/mole and not 207.2 g/mole (from the periodic
table) because only one isotope is present. Using the molar mass from the periodic table
gives an incorrect answer of 66.4 mg^238 U.
The set-up earns 1 point even if you incorrectly used 207.2 g/mole. The correct
answer (66.5 mg^238 U) earns a second point.
(b) The equation given in the exam booklet relating the quantities of material in a first-
order reaction to time (integrated rate law) is A
A
ln[]t
[] 0= –kt. Another equation, alsogiven in the exam booklet is t1/2 = ln 2/0.693 = 0.693/k. It is possible to rearrange and
combine these equations. Rearrangement gives:Age = t =A
A
k− ln[]t[] (^0) where k =
t
0.693
1/2Entering the values [A]t = 47.6 mg^238 U, [A] 0 = 66.5 mg^238 U, and t1/2 = 4.47 ×
109 years gives:Age = t =A
A
k− ln[]t
[] 0
=A
A
t− ln[]t
[]
0.69301/2=
− t
× ln47.6 mg U
66.5 mg U
0.693
4.47 10 y238
238
09= 2.16 × 109 yBoth [A]t and [A] 0 must be in the same units, which, in this case, are mg^238 U.
Using the wrong initial mass (66.4 mg) gives 2.15 × 109 y (an error of 10 million years).
Using the mass of lead-206 instead of uranium-238 gives 2.03 × 1010 y.
Give yourself 1 point for the correct set-up. You still get the point if you use a
wrong answer from part a. The correct answer (2.16 × 109 y) earns a second point.
Total your points. There are 4 points possible. Subtract 1 point if your answers do not have
the correct number of significant figures.