The half-life of^14C is approximately 5700 years. The ratio of^14C/^12C in the a
ID: 498564 • Letter: T
Question
The half-life of^14C is approximately 5700 years. The ratio of^14C/^12C in the atmosphere is approximately 1.2 times 10^-12. If a tree was seeded and started to grow 17, 100 (= 3 times 5700) years ago and lived for 5700 years (ok, I know that is a long life for a tree, but nobody was carving their initials in it): a) Draw and concisely explain a plot of the^14C/^12C ratio in the tree for the last 17, 100 years. b) For the first 5700 years since the tree was seeded and started to grow and for the time since, give the functional form of the plot. c) What is the reaction order in [^14C] for the 11, 400 years since the tree died? Explain concisely. What is the^14C/^12C ratio in the (dead) tree now? Show your calculation.Explanation / Answer
The C14 is is found in atmosphera, animals and plants get C14 through photosintesis and from their eating.
In this case the tree will have the same ratio of C14/C12 of the atmosphera until it dies. After it dies the ammount of C14 will start to decrease.
First we have to calculate de decay constant "k"
the equation that will help us is:
t hl = (1 / k) * ln 2
k =( ln 2 ) / t
k = 0.0001216 years -1
To draw a plot we have to get the years depending the remaining ammount of C14 using the next equation.
t = 1/k * ln (1 / C ) ; C = % of C14 remaining
the ratio of C14/C12 change just by multiplying the ratio by the % of C14 remaining.
If we do it like that we will get the next numbers:
6E-14
Remember that for first 5700 years the ratio is the same, the ammount of C14 in the tree is the same.
Just put these data on excel and draw a graph.
b) For the first 5700 years the functional form of the plot is just an horizontal line
if we only consider the %C14 remaining the formula is y = 1
if we consider the ratio the plot gets the next formula y = 1.2 * 10 -12
For the last 11400 years the functional form of the plot is
y = e-.00012x
X = the %C14 remaining in the tree after it dies.
d) The C14 / C12 ratio now is 3 x 10-3
t %C14 C14/C12 5700 1 1.2E-12 6121.80331 0.95 1.14E-12 6566.41763 0.9 1.08E-12 7036.45195 0.85 1.02E-12 7534.99014 0.8 9.6E-13 8065.71375 0.75 9E-13 8633.06709 0.7 8.4E-13 9242.48375 0.65 7.8E-13 10616.2299 0.55 6.6E-13 11400 0.5 6E-13 12266.4176 0.45 5.4E-13 13234.9901 0.4 4.8E-13 14333.0671 0.35 4.2E-13 15600.7039 0.3 3.6E-13 17100 0.25 3E-13 18934.9901 0.2 2.4E-13 21300.7039 0.15 1.8E-13 24634.9901 0.1 1.2E-13 30334.9901 0.056E-14
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