the isotope titanium - 44 decays into scandium - 44, with a half - life of 60 years. in a sample of titanium - 44 containing 2×10¹⁰ atoms, how many titanium - 44 and scandium - 44 atoms would there be in the sample after 240 years?
step 1: how many half - lives will occur? 1 half - life = ______ years
240 years ÷ ______ years/half - life = ____ half - lives
step 2: how many titanium - 44 atoms remain?
step 3: how many scandium - 44 atoms have been produced?
2×10¹⁰−1.25×10⁹ = ______×10¹⁰ atoms of scandium - 44

Question

Accepted Answer

# Explanation:
## Step1: Calculate the number of half - lives
The half - life of titanium - 44 is 60 years.
To find the number of half - lives in 240 years, we use the formula $n=\frac{t}{T}$, where $t = 240$ years (total time) and $T=60$ years (half - life).
$n=\frac{240}{60}=4$ half - lives.

## Step2: Calculate the remaining titanium - 44 atoms
The formula for the number of remaining atoms $N = N_0\times(\frac{1}{2})^n$, where $N_0 = 2\times10^{10}$ (initial number of atoms) and $n = 4$ (number of half - lives).
$$
\begin{align*}
N&=2\times 10^{10}\times(\frac{1}{2})^4\
&=2\times10^{10}\times\frac{1}{16}\
&=1.25\times 10^{9}
\end{align*}
$$
We can also fill the table:
- When $n = 0$, $N=2\times10^{10}$ (so $2\times10^{10}$, $10$ has exponent $10$)
- When $n = 1$, $N = 2\times10^{10}\times\frac{1}{2}=1\times10^{10}$ (so $1\times10^{10}$)
- When $n = 2$, $N=2\times10^{10}\times(\frac{1}{2})^2 = 0.5\times10^{10}=5\times10^{9}$
- When $n = 3$, $N=2\times10^{10}\times(\frac{1}{2})^3=0.25\times 10^{10}=2.5\times10^{9}$
- When $n = 4$, $N = 2\times10^{10}\times(\frac{1}{2})^4=1.25\times10^{9}$

## Step3: Calculate the number of scandium - 44 atoms
The number of scandium - 44 atoms $N_{Sc}=N_0 - N$
$$
\begin{align*}
N_{Sc}&=2\times10^{10}-1.25\times10^{9}\
&=20\times10^{9}-1.25\times10^{9}\
&=(20 - 1.25)\times10^{9}\
&=18.75\times10^{9}=1.875\times10^{10}
\end{align*}
$$

# Answer:
- Step 1: $1$ half - life $= 60$ years; $240$ years $\div60$ years/half - life $ = 4$ half - lives
- Step 2: The table:
| Number of Half - lives | Number of Titanium - 44 Atoms Remaining |
|----|----|
| $0$ | $2\times10^{10}$ |
| $1$ | $1\times10^{10}$ |
| $2$ | $5\times10^{9}$ |
| $3$ | $2.5\times10^{9}$ |
| $4$ | $1.25\times10^{9}$ |
- Step 3: $2\times10^{10}-1.25\times10^{9}=1.875\times10^{10}$ atoms of scandium - 44.

Number of Half - lives	Number of Titanium - 44 Atoms Remaining
\(1\)	\(1\times10^{10}\)
\(2\)	\(5\times10^{9}\)
\(3\)	\(2.5\times10^{9}\)
\(4\)	\(1.25\times10^{9}\)

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Question

Explanation:

Step1: Calculate the number of half - lives

Step2: Calculate the remaining titanium - 44 atoms

Step3: Calculate the number of scandium - 44 atoms

Answer: