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

$m(t) = m_0 e^{-kt}$

Where:

• $m_0$ is the initial mass of the substance ($2000 \, \text{g}$),
• $k$ is the decay constant,
• $t$ is the time elapsed,
• $m(t)$ is the mass at time $t$.

Step 1: Calculate the decay constant $k$

The half-life formula is: $t_{1/2} = \frac{\ln(2)}{k}$ Here, the half-life $t_{1/2} = 7 \, \text{h}$, so: $k = \frac{\ln(2)}{7} \approx 0.09902 \, \text{h}^{-1}$

Step 2: Determine the mass after $t = 1.5 \, \text{days}$

Convert $t$ to hours ($1.5 \, \text{days} = 36 \, \text{hours}$): $m(36) = 2000 e^{-0.09902 \cdot 36}$ $m(36) \approx 2000 e^{-3.56472} \approx 2000 \cdot 0.02847 \approx 56.94 \, \text{g}$

Step 3: Compute the instantaneous rate of change

The instantaneous rate of change is given by the derivative of $m(t)$: $\frac{dm}{dt} = -k m_0 e^{-kt}$ Substitute $m_0 = 2000$, $k = 0.09902$, and $t = 36$: $\frac{dm}{dt} = -0.09902 \cdot 2000 e^{-0.09902 \cdot 36}$ $\frac{dm}{dt} \approx -0.09902 \cdot 2000 \cdot 0.02847 \approx -5.63 \, \text{g/h}$

Final Answer:

The instantaneous rate of change is approximately: $\boxed{-5.6 \, \text{g/h}}$

Let me know if you need more details or clarifications!

Relative Questions:

1. How is the decay constant $k$ derived from the half-life?
2. Why do we use exponential decay models for radioactive substances?
3. What is the significance of the instantaneous rate of change in this context?
4. How would the result change if the half-life were longer?
5. Can this approach be extended to other decay processes, like carbon dating?

Tip:

Always double-check units when solving decay problems, especially when converting between time units like hours and days!