JEE Main 2021 · 16 March · Evening Shift · Physics · Nuclear Physics
B. 20min20min20min
N1=N0e−λt1{N}_{1} = {N}_{0}{e}^{- \lambda{t}_{1}}N1=N0e−λt1
N1N0=e−λt1\dfrac{{N}_{1}}{{N}_{0}} = {e}^{- \lambda{t}_{1}}N0N1=e−λt1
0.67=e−λt10 . 67 = {e}^{- \lambda{t}_{1}}0.67=e−λt1
ln(0.67)=−λt1\ln \left(0 . 67\right) = - \lambda{t}_{1}ln(0.67)=−λt1
N2=N0e−λt2{N}_{2} = {N}_{0}{e}^{- \lambda{t}_{2}}N2=N0e−λt2
N2N0=e−λt2\dfrac{{N}_{2}}{{N}_{0}} = {e}^{- \lambda{t}_{2}}N0N2=e−λt2
0.33=e−λt20 . 33 = {e}^{- \lambda{t}_{2}}0.33=e−λt2
ln(0.33)=−λt2\ln \left(0 . 33\right) = - \lambda{t}_{2}ln(0.33)=−λt2
ln(0.67)−ln(0.33)=λt1−λt2\ln \left(0 . 67\right) - \ln \left(0 . 33\right) = \lambda{t}_{1} - \lambda{t}_{2}ln(0.67)−ln(0.33)=λt1−λt2
λ(t1−t2)=ln(0.670.33)\lambda\left({t}_{1} - {t}_{2}\right) = \ln \left(\dfrac{0 . 67}{0 . 33}\right)λ(t1−t2)=ln(0.330.67)
λ(t1−t2)≅ln2\lambda\left({t}_{1} - {t}_{2}\right) ≅ \ln 2λ(t1−t2)≅ln2
t1−t2≃ln2λ=t1/2{t}_{1} - {t}_{2} ≃ \dfrac{\ln 2}{\lambda} = {t}_{1/2}t1−t2≃λln2=t1/2
Half life =t1/2=20= {t}_{1/2} = 20=t1/2=20 minutes.
