24.117 CP - Chemistry: Molecular Nature Of Matter And Change

Question

A bone sample containing$strontium-90 \left(t_{1/2}=29yr\right)$  emits$7.0\times {10}^4{\beta }^{-}$  particles per month. How long will it take for the emission to decrease todata-custom-editor="chemistry" $1.0\times {10}^4$  particles per month?

Step-by-Step Solution

Answer

t = 81.413 years will it take for the emission to decrease to $1.0\times {10}^4$ particles per month.

Step 1: Definition of Emission spectrum

The range of wavelengths released by an atom or compound driven by heat or electric current is referred to as an emission spectrum.

Step 2: Determining time for emission

We must use the half-life formula to find the constant (k) in this situation.

$t_{\frac{1}{2}} = \frac{\ln 2}{k}=k = \frac{\ln 2}{t_{\frac{1}{2}}}$

$k = \frac{\ln 2}{t_{\frac{1}{2}}} = \frac{\ln 2}{29yr} = \frac{0.0239016}{yr}$

We can solve for the value of t by using the expression for calculating the number of nuclei left.

$\ln \frac{N_0}{N_t} = kt$

$\ln \left(\frac{7x{10}^4}{1x{10}^4}\right) = \frac{0.0239016}{yr}\left(t\right)$

$t = 81.4134 years$