To solve this problem, we will use the exponential decay model, which is commonly used to describe the decay of substances over time. The general formula for exponential decay is:

$C(t) = C_0 e^{-kt}$

Where:

• $C(t)$ is the concentration at time $t$,
• $C_0$ is the initial concentration (at time $t = 0$),
• $k$ is the decay constant, and
• $t$ is time.

Given:

• $C(2.5) = 56$ mg/dL (concentration after 2.5 hours)
• $C(7) = 36$ mg/dL (concentration after 7 hours)

1. Finding the Decay Constant ($k$)

To find $k$, we can use the concentrations at two different times. We have:

$56 = C_0 e^{-2.5k}$ $36 = C_0 e^{-7k}$

We can divide these equations to eliminate $C_0$:

$\frac{56}{36} = \frac{C_0 e^{-2.5k}}{C_0 e^{-7k}}$

$\frac{56}{36} = e^{7k - 2.5k}$

$\frac{56}{36} = e^{4.5k}$

$\frac{56}{36} = e^{4.5k}$

$1.5556 = e^{4.5k}$

$\ln(1.5556) = 4.5k$

$k = \frac{\ln(1.5556)}{4.5}$

Now, we calculate the value of $k$:

$k \approx \frac{0.4417}{4.5}$

$k \approx 0.0982 \, \text{per hour}$

2. Finding the Initial Concentration ($C_0$)

Using $k$ and one of the equations, we can find $C_0$:

$56 = C_0 e^{-2.5 \times 0.0982}$

$C_0 = \frac{56}{e^{-0.2455}}$

$C_0 = \frac{56}{0.7824}$

$C_0 \approx 71.6 \, \text{mg/dL}$

3. Exponential Decay Model

The model can now be written as:

$C(t) = 71.6 e^{-0.0982t}$

4. Finding the Half-Life

The half-life $t_{1/2}$ is the time required for the concentration to reduce to half of its initial value. We use the equation:

$\frac{1}{2}C_0 = C_0 e^{-kt_{1/2}}$

$\frac{1}{2} = e^{-kt_{1/2}}$

$\ln\left(\frac{1}{2}\right) = -kt_{1/2}$

$-\ln(2) = -kt_{1/2}$

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

$t_{1/2} = \frac{\ln(2)}{0.0982}$

$t_{1/2} \approx 7.05 \, \text{hours}$

Summary

• Decay constant $k$: $0.0982$ per hour
• Initial concentration $C_0$: $71.6$ mg/dL
• Exponential decay model: $C(t) = 71.6 e^{-0.0982t}$
• Half-life $t_{1/2}$: $7.05$ hours

Would you like further details on any of these calculations or have any additional questions?

Here are some questions you might consider:

1. How do you derive the exponential decay formula?
2. What are some real-world applications of exponential decay?
3. How is the half-life concept applied in pharmacology?
4. Can the rate of decay be influenced by external factors?
5. How would you graph the decay of the concentration over time?

Tip: Understanding the derivation of the exponential decay formula can help in applying the concept to various real-world scenarios, such as pharmacokinetics and radioactive decay.