Question #5e9e6

1 Answer
Jul 19, 2016

Answer:

#"2.25 years"#

Explanation:

Your tool of choice here will be the equation

#color(blue)(|bar(ul(color(white)(a/a)A_t = A_0 * 1/2^n color(white)(a/a)|)))#

Here

#A_t# - the amount undecayed after a period of time #t#
#A_0# - the initial mass of the radioactive isotope
#n# - the number of half-lives that pass in a period of time #t#

In your case, you know that it took #9# years for a sample of a given radioactive isotope to decay from #"448 g"#, the initial mass of the sample, to #"28 g"#, the mass that remains undecayed.

Your goal here will be to use the above equation to find the value of #n#, the number of half-lives that pass in #9# years. Plug in your values to find

#28 color(red)(cancel(color(black)("g"))) = 448 color(red)(cancel(color(black)("g"))) * 1/2^n#

Rearrange to find

#2^n = 448/28 = 16#

Since #16# can be written as a power of #2#

#16 = 2 * 2 * 2 * 2 = 2^4#

you will have

#2^n = 2^4#

This implies that

#n = 4#

So, you know that it takes #9# years for #4# half-lives to pass, which means that one half-life, #t_"1/2"#, is

#t_"1/2" = "9 years"/4 = color(green)(|bar(ul(color(white)(a/a)color(black)("2.25 years")color(white)(a/a)|)))#