Chapter 10: Problem 2

Using only symbolic variables and constants, write an expression that defines that time necessary for \(95 \%\) of a radioactive isotope to decay. Hint: interpret this to mean that we seek an expression for t when \(\mathrm{N} / \mathrm{N}_{0}=0.05\)

Short Answer

Expert verified

The time necessary for 95% of a radioactive isotope to decay is given by the expression \(t = \ln(0.05)/-k\)

Step by step solution

Write the equation for nuclear decay

Write the equation for nuclear decay, which is \(N = N_0 e^{-kt}\). This models the decay of radioactive isotopes.

Solve for decay

Plug in the given condition that \(N/N_0 = 0.05\). This gives the equation \(0.05 = e^{-kt}\).

Solve for t

To solve for \(t\), first take the natural log on both sides to remove the exponential on the right-hand side. This yields \(\ln(0.05) = -kt\). Then, solve for \(t\) by dividing both sides by \(-k\). This gives us \(t = \ln(0.05)/-k\).

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Using only symbolic variables and constants, write an expression that defines that time necessary for \(95 \%\) of a radioactive isotope to decay. Hint: interpret this to mean that we seek an expression for t when \(\mathrm{N} / \mathrm{N}_{0}=0.05\)

Short Answer

Step by step solution

Write the equation for nuclear decay

Solve for decay

Solve for t

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