A radioactive element has a half life of 6 minutes if initial count rate is 824per minute the time taken by it to reach a count rate of 206 per minute is a) 20min b) 15min c) 12min?

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Answer 1 · 2018-03-26T05:34:12

selection c

The equation for half-life decay is:

#A(t) = A(0)(1/2)^(t/t_(1/2))#

Where #A(0)# is the original amount, #A(t)# is the amount at time t, and #t_(1/2)# is the half-life.

We are given: #A(0) = 824" min"^-1#, #A(t) = 206" min"^-1# and #t_(1/2) = 6" min"#:

#206" min"^-1 = 824" min"^-1(1/2)^(t/(6" min"))#

Divide both sides by #824" min"^-1#:

#206/824 = (1/2)^(t/(6" min"))#

Use the natural logarithm on both sides:

#ln(206/824) = ln((1/2)^(t/(6" min")))#

Use the property of logarithms that allow one to bring the exponent outside of the argument as a coefficient:

#ln(206/824) = (t/(6" min"))ln(1/2)#

Flip the equation:

#(t/(6" min"))ln(1/2) = ln(206/824)#

Divide both sides by #ln(1/2)#:

#t/(6" min") = ln(206/824)/ln(1/2)#

Multiply both sides by #6" min"#:

#t = ln(206/824)/ln(1/2)(6" min")#

#t = 12" min"#

This is selection c