An unknown radioactive substance has a half-life of 3.20 . If 34.8 of the substance is currently present, what mass was present 8.00 hours ago?

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To determine the mass of the unknown radioactive substance that was present 8.00 hours ago, we'll use the concept of half-life.

The half-life of a radioactive substance is the time it takes for half of the initial amount to decay. In this case, the half-life is 3.20 hours.

To solve this problem, we can use the formula for radioactive decay:

\[ N = N_0 \times (1/2)^{t/h} \]

Where:
- N is the current amount of the substance
- N0 is the initial (or starting) amount of the substance
- t is the elapsed time
- h is the half-life of the substance

We are given that the current amount, N, is 34.8% of the initial amount. Since percentages are relative to 100, we can convert 34.8% to a decimal by dividing it by 100:

\[ N = 34.8 / 100 \Rightarrow N = 0.348 \]

Plugging the given values into the formula, we have:

\[ 0.348 = N_0 \times (1/2)^{8/3.20} \]

To solve for the initial amount N0, we can rearrange the equation to isolate N0:

\[ N_0 = \frac{0.348}{(1/2)^{8/3.20}} \]

Calculating this expression will give us the initial mass of the substance 8.00 hours ago.