To solve this problem, we need to model the decay using an exponential function. An exponential decay model can be described by the equation:

$M(t) = M_0 e^{-kt}$

where:

• $M(t)$ is the mass of the substance at time $t$.
• $M_0$ is the initial mass of the substance (what we are trying to find).
• $k$ is the decay constant.
• $t$ is the time in hours.

From the given data, we have:

Hours ($t$)	Mass ($M(t)$)
4	34.5
5	21.7
6	13.7
7	8.6
8	5.4

Step 1: Establish two equations using the data points

We can use two data points to create a system of equations to solve for $M_0$ and $k$. Let's use the data for $t = 4$ and $t = 5$.

1. At $t = 4$, $M(4) = 34.5$: $34.5 = M_0 e^{-4k}$

2. At $t = 5$, $M(5) = 21.7$: $21.7 = M_0 e^{-5k}$

Step 2: Solve for $k$

Divide the second equation by the first to eliminate $M_0$:

$\frac{21.7}{34.5} = \frac{M_0 e^{-5k}}{M_0 e^{-4k}}$

$\frac{21.7}{34.5} = e^{-5k + 4k}$

$\frac{21.7}{34.5} = e^{-k}$

Now, solve for $k$:

$e^{-k} = \frac{21.7}{34.5}$

$-k = \ln\left(\frac{21.7}{34.5}\right)$

$k = -\ln\left(\frac{21.7}{34.5}\right)$

Let's calculate $k$:

$k \approx -\ln\left(\frac{21.7}{34.5}\right) \approx -\ln(0.629) \approx 0.462$

Step 3: Solve for $M_0$

Use the value of $k$ and one of the original equations to find $M_0$. We'll use $t = 4$ and $M(4) = 34.5$:

$34.5 = M_0 e^{-4 \cdot 0.462}$

$34.5 = M_0 e^{-1.848}$

$34.5 = M_0 \times 0.157$

$M_0 = \frac{34.5}{0.157}$

$M_0 \approx 219.7$

Conclusion

The initial mass of the substance, $M_0$, that was exposed to the reagent was approximately 219.7 grams.

Would you like more details, or do you have any questions?

Here are 5 related questions you might consider:

1. How can we determine the half-life of the substance using the decay constant $k$?
2. What would the mass of the substance be after 10 hours?
3. How would the model change if the decay was not exponential but followed a different pattern, such as linear decay?
4. Can you determine the time at which only 1 gram of the substance remains?
5. How does the decay constant $k$ affect the rate of decay?

Tip: When dealing with exponential growth or decay, always try to express the relationships in terms of natural logarithms (base $e$) to simplify the calculations.