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Question: A nucleus of \(Ux_{1}\) has a half life of \(24.1\) days how long a sample of \(Ux_{1}\)will take to...

A nucleus of Ux1Ux_{1} has a half life of 24.124.1 days how long a sample of Ux1Ux_{1}will take to change to 90% of Ux1Ux_{1}

A

80 days

B

40 days

C

20 days

D

10 days

Answer

80 days

Explanation

Solution

: Here, λ=0.693T1/2=0.69324.1=0.02876day1\lambda = \frac{0.693}{T_{1/2}} = \frac{0.693}{24.1} = 0.02876day^{- 1}

N=N090%ofN0=N010N = N_{0} - 90\% ofN_{0} = \frac{N_{0}}{10}

As N=N0eλtN = N_{0}e^{- \lambda t}

N010=N0eλtor110=eλtor1=eλt\therefore\frac{N_{0}}{10} = N_{0}e^{- \lambda t}or\frac{1}{10} = e^{- \lambda t}or1 = e^{\lambda t}

Or loge10=λt\log_{e}{}10 = \lambda t

t=1λloge10=2.303log100.02870=2.303×10.02876=80days\therefore t = \frac{1}{\lambda}\log_{e}{}10 = \frac{2.303\log 10}{0.02870} = \frac{2.303 \times 1}{0.02876} = 80days