Question

A radioactive substance with an initial mass of 800 milligrams decays at a rate of 2% per hour. Determine the mass remaining after 10 hours and the half-life of the substance.

Answer 1

To determine the mass of the radioactive substance remaining after 10 hours and its half-life, we can use the exponential decay model.

1. **Calculate the mass remaining after 10 hours**:

The formula for the remaining mass \( m(t) \) of a substance after time \( t \) is given by:

\[
m(t) = m_0 \cdot e^{-kt}
\]

where:
- \( m_0 \) is the initial mass (800 mg),
- \( k \) is the decay constant,
- \( t \) is the time (10 hours),
- \( e \) is the base of the natural logarithm.

First, we need to calculate the decay constant \( k \). Since the decay percentage is given as 2% per hour, we have:

\[
k = 0.02 \text{ per hour}
\]

Now we can plug in the values to find the mass after 10 hours:

\[
m(10) = 800 \cdot e^{-0.02 \cdot 10}
\]

Calculating the exponent:

\[
-0.02 \cdot 10 = -0.2
\]

Now, calculate \( e^{-0.2} \):

\[
e^{-0.2} \approx 0.8187
\]

So,

\[
m(10) = 800 \cdot 0.8187 \approx 654.96 \text{ mg}
\]

Thus, the mass remaining after 10 hours is approximately **655 mg**.

2. **Determine the half-life of the substance**:

The half-life \( t_{1/2} \) is the time taken for the substance to decay to half of its initial mass. The relationship between half-life and decay constant is given by:

\[
t_{1/2} = \frac{\ln(2)}{k}
\]

where \( \ln(2) \approx 0.693 \). Substituting \( k = 0.02 \):

\[
t_{1/2} = \frac{0.693}{0.02} \approx 34.65 \text{ hours}
\]

Therefore, the half-life of the substance is approximately **34.65 hours**.

In summary:
- The mass remaining after 10 hours is approximately **655 mg**.
- The half-life of the substance is approximately **34.65 hours**.