To determine the decay constant [tex]\( a \)[/tex] for the exponential decay formula [tex]\( A(t) = A_0 \cdot a^t \)[/tex], given the half-life of gold-194, we follow these steps:
1. Understand the half-life concept:
The half-life [tex]\( t_{\text{half}} \)[/tex] is the time required for half of the substance to decay. For gold-194, [tex]\( t_{\text{half}} \)[/tex] is given as 1.6 days.
2. Set up the relationship at half-life:
At the half-life [tex]\( t = t_{\text{half}} \)[/tex], the remaining amount [tex]\( A(t) \)[/tex] is half of the initial amount [tex]\( A_0 \)[/tex]. This can be written mathematically as:
[tex]\[
A(t_{\text{half}}) = A_0 \cdot a^{t_{\text{half}}}
\][/tex]
Since [tex]\( A(t_{\text{half}}) = \frac{1}{2} A_0 \)[/tex], we substitute this into the equation:
[tex]\[
\frac{1}{2} A_0 = A_0 \cdot a^{t_{\text{half}}}
\][/tex]
3. Simplify the equation:
Cancel [tex]\( A_0 \)[/tex] from both sides of the equation (assuming [tex]\( A_0 \neq 0 \)[/tex]):
[tex]\[
\frac{1}{2} = a^{t_{\text{half}}}
\][/tex]
Substitute [tex]\( t_{\text{half}} = 1.6 \)[/tex]:
[tex]\[
\frac{1}{2} = a^{1.6}
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
To isolate [tex]\( a \)[/tex], we take the 1.6th root of both sides:
[tex]\[
a = \left( \frac{1}{2} \right)^{\frac{1}{1.6}}
\][/tex]
5. Calculate the value:
[tex]\[
a \approx 0.64842
\][/tex]
Thus, the decay constant [tex]\( a \)[/tex] rounded to six decimal places is [tex]\( 0.64842 \)[/tex].
