Carbon-14 has a half-life of 5770 years. If a fossil is 23,080 years old and it has 3 Kg of Carbon-14, how much carbon-14 did it originally have?

1 Answer
Jun 21, 2016

Answer:

#"48 kg"#

Explanation:

The nuclear half-life is simply the time needed for half of the atoms present in a sample of a radioactive substance to undergo radioactive decay.

http://www.pnausa.org/harmony-todd/nuclear-101-radioactive-half-life

In your case, carbon-14 is said to have a half-life of #5770# years, which means that the mass of any sample of carbon-14 will be halved with the passing of #5770# years.

Now, if #1# half-life from now the mass of the sample will be halved, it follows that #1# half-life ago it was twice its current value.

An interesting technique to use here is to backtrack from current time, when the sample is down to #"3 kg"#, to #"23,080"# years ago.

You can thus say that #5770# years before this point, the mass of the sample was twice the current mass

#2 xx "3 kg" = "6 kg"#

How about #2# half-lives ago?

#2 xx "5770 years" = "11,540 years ago"#

The mass of the sample was

#2 xx "6 kg" = "12 kg"#

How about #3# half-lives ago?

#3 xx "5770 years" = "17,310 years ago"#

The mass of the sample was

#2 xx "12 kg" = "24 kg"#

Finally, how about #4# half-lives ago?

#4 xx "5770 years" = "23,080 years ago"#

The mass of the sample was

#2 xx "24 kg" = "48 kg"#

Therefore, #"23,080# years ago, the sample contained #"48 kg"# of carbon-14.

When the values don't allow for such simple calculations, you can use the equation

#color(blue)(|bar(ul(color(white)(a/a)"A"_t = "A"_0 * 1/2^ncolor(white)(a/a)|)))#

Here

#"A"_t# - the mass of the sample that remains after a period of time #t#
#"A"_0# - the initial mass of the sample
#n# - the number of half-lives that pass in the period of time #t#

In your case, you have

#n = ("23,080" color(red)(cancel(color(black)("years"))))/(5770color(red)(cancel(color(black)("years")))) = 4#

You will thus have

#"A"_0 = "A"_t * 2^4#

#"A"_0 = "3 kg" * 2^4 = color(green)(|bar(ul(color(white)(a/a)color(black)("48 kg")color(white)(a/a)|)))#