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vesna_86 [32]
3 years ago
10

Chegg ) Alice owns 20 grams of a radioactive isotope that has a half-life of ln(4) years. (a) Find an equation for the mass m(t)

of the remaining isotope Alice owns after t years.
Physics
1 answer:
stepladder [879]3 years ago
8 0

Answer:

m(t)=20e^{-0.5t}

Explanation:

Given:

Initial mass of isotope (m₀) = 20 g

Half life of the isotope (t_{1/2}) = (ln 4) years

The general form for the radioactive decay of a radioactive isotope is given as:

m(t)=m_0e^{-kt}

Where,

m(t)\to mass\ after\ 't'\ years\\t\to years\ passed\\k\to rate\ of\ decay\ per\ year

So, the equation is: m(t)=20e^{-kt}

At half-life, the mass is reduced to half of the initial value.

So, at t=t_{1/2},m(t)=\frac{m_0}{2}. Plug in these values and solve for 'k'. This gives,

\frac{m_0}{2}=m_0e^{-k\times\ln 4}\\\\0.5=e^{-k\times\ln 4}\\\\Taking\ natural\ log\ on\ both\ sides,we\ get:\\\\\ln(0.5)=-k\times \ln 4\\\\k=\frac{\ln 0.5}{-\ln 4}=0.5

Hence, the equation for the mass remaining is given as:

m(t)=20e^{-0.5t}

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