Question #292624

A certain radioactive material is known to decay at a rate proportional to the amount



present. If the initially there is 50 milligrams of the material present and after two



hours it is observed that the material has lost 10% of its original mass. Find



a) An expression for the mass of the material remaining at any time t



b) The mass of the material after 4 hours



c) The time rate at which the material has decayed to one half of its initial mass

1
Expert's answer
2022-02-01T16:46:47-0500

Let A(t)=A=A(t)=A= the amount of radioactive material.

Given dAdt=kA,\dfrac{dA}{dt}=-kA, where kk is the proportionality constant.


dAA=kdt\dfrac{dA}{A}=-kdt

Integrate


dAA=kdt\int\dfrac{dA}{A}=-\int kdt

Let A(0)=A0=A(0)=A_0= the initialamount of radioactive material.

A=A0ektA=A_0e^{-kt}

Given A0=50 mg,A(2)=0.9A0A_0=50\ mg, A(2)=0.9A_0

Substitute


0.9A0=A0ek(2)0.9A_0=A_0e^{-k(2)}

2k=ln(0.9)-2k=\ln(0.9)

k=12ln(0.9) h1k=-\dfrac{1}{2}\ln (0.9) \ h^{-1}


a)


A(t)=50e12ln(0.9)tA(t)=50e^{\frac{1}{2}\ln (0.9)t}

A(t)=50(0.9)t/2A(t)=50(0.9)^{t/2}

b)


A(4)=50(0.9)4/2A(4)=50(0.9)^{4/2}

A(4)=40.5 mgA(4)=40.5\ mg

c)


A(t1)=12A0A(t_1)=\dfrac{1}{2}A_0

12A0=A0(0.9)t1/2\dfrac{1}{2}A_0=A_0(0.9)^{t_1/2}

(0.9)t1/2=2(0.9)^{-t_1/2}=2

t1=2ln2ln0.9t_1=-\dfrac{2\ln2}{\ln 0.9}

t113.1576 hourt_1\approx13.1576\ hour


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