To determine the initial mass of the substance, we can use the information provided and the fact that the decay follows an exponential curve. The general form of an exponential decay function is:

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

where:

• $M(t)$ is the mass at time $t$,
• $M_0$ is the initial mass (the value we want to find),
• $k$ is the decay constant,
• $t$ is the time in hours.

Given the data, we can use two points to determine the decay constant $k$, and then find $M_0$.

Step 1: Set up two equations using the given data

We can use the data points at $t = 4$ and $t = 5$:

$34.5 = M_0 e^{-4k} \quad \text{(1)}$ $21.7 = M_0 e^{-5k} \quad \text{(2)}$

Step 2: Divide the two equations to eliminate $M_0$

Dividing equation (2) by equation (1):

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

$e^{-k} = \frac{21.7}{34.5} \approx 0.629$

Step 3: Solve for $k$

Taking the natural logarithm on both sides:

$-k = \ln(0.629)$

$k \approx -\ln(0.629) \approx 0.464$

Step 4: Substitute $k$ back into one of the original equations

We use equation (1):

$34.5 = M_0 e^{-4(0.464)}$

$34.5 = M_0 e^{-1.856}$

$M_0 \approx \frac{34.5}{e^{-1.856}} \approx \frac{34.5}{0.156} \approx 220.8 \, \text{grams}$

Thus, the initial mass $M_0$ of the substance exposed to the reagent was approximately 220.8 grams.

Would you like more details on any of these steps, or do you have any other questions?

Related Questions:

1. How does the value of the decay constant $k$ affect the rate of decay?
2. What happens to the mass of the substance as time goes on according to the exponential decay model?
3. How would the result change if the mass at $t = 4$ hours was different?
4. Can this method be used to predict the time it takes for the substance to completely decay?
5. How does this method compare to other types of decay models, such as linear decay?

Tip:

When dealing with exponential decay problems, using the logarithmic transformation can simplify solving for variables like the decay constant.