What is the half-life of the substance if a sample of a radioactive substance decayed to 97.5% of its original amount after a year? (b) How long would it take the sample to decay to 80% of its original amount? _______years??

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Answer 1 · 2015-03-29T22:35:29

(a). #t_(1/2)=27.39"a"#

(b). #t=8.82"a"#

#N_t=N_0e^(-lambda t)#

#N_t=97.5#

#N_0=100#

#t=1#

So:

#97.5=100e^(-lambda.1)#

#e^(-lambda)=(97.5)/(100)#

#e^(lambda)=(100)/(97.5)#

#lne^(lambda)=ln((100)/(97.5))#

#lambda=ln((100)/(97.5))#

#lambda=ln(1.0256)=0.0253"/a"#

#t_((1)/(2))=0.693/lambda#

#t_((1)/(2))=0.693/0.0253=color(red)(27.39"a")#

Part (b):

#N_t=80#

#N_0=100#

So:

#80=100e^(-0.0253t)#

#80/100=e^(-0.0235t)#

#100/80=e^(0.0253t)=1.25#

Taking natural logs of both sides:

#ln(1.25)=0.0253t#

#0.223=0.0253t#

#t=0.223/0.0253=color(red)(8.82"a")#