Calculating the half-life of a substance involves determining the time it takes for half of that substance to decay or be eliminated. The process can vary depending on whether you're working with radioactive decay or other types of decay (such as pharmacological elimination). Here are the general steps for calculating half-life from different contexts:
1. Radioactive Decay:
If you are calculating the half-life of a radioactive isotope, you can use the following method:
A. Determine Initial and Remaining Quantity
- Measure the initial quantity of the radioactive substance, \(N_0\).
- Measure the quantity remaining after some time, \(N_t\).
B. Use the Half-Life Formula
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The relationship between initial quantity, remaining quantity, and time can be expressed as:
\[ N_t = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
where:
- \(N_t\) = remaining quantity at time \(t\)
- \(N_0\) = initial quantity
- \(t_{1/2}\) = half-life
- \(t\) = total time elapsed
C. Solve for Half-Life
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Rearranging the formula to solve for half-life:
\[ t_{1/2} = \frac{t}{\log_2 \left( \frac{N_0}{N_t} \right)} \]
2. Pharmacological Context:
In pharmacology, the half-life refers to the time it takes for the concentration of the drug in the bloodstream to reduce to half its initial value. The steps are similar but are usually based on elimination rates:
A. Determine Initial Concentration
- Measure the initial drug concentration in the bloodstream.
B. Measure Concentration Over Time
- Conduct periodic measures of concentration until it stabilizes at lower levels.
C. Plot Data
- Consider plotting drug concentration over time on a graph. The resulting curve is often exponential.
D. Use the Exponential Decay Formula
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A simplified version of the decay can be expressed as:
\[ C_t = C_0 e^{-kt} \]
where \(C_t\) is concentration at time \(t\), \(C_0\) is initial concentration, and \(k\) is the elimination rate constant.
E. Solve for Half-Life
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The half-life can be derived using the elimination rate constant \(k\):
\[ t_{1/2} = \frac{\ln(2)}{k} \]
3. Example Calculation:
If a radioactive substance has an initial amount \(N_0 = 100 g\) and after 3 hours \(N_t = 25 g\), the half-life can be calculated as follows:
- Given \(N_t = 25 g\), \(N_0 = 100 g\), and \(t = 3\) hours.
- Using the formula:
\[ t_{1/2} = \frac{t}{\log_2 \left( \frac{N_0}{N_t} \right)} = \frac{3}{\log_2 \left( \frac{100}{25} \right)} = \frac{3}{\log_2(4)} = \frac{3}{2} = 1.5 \text{ hours} \]
With these steps, you can calculate half-life for various contexts. Always ensure you have the appropriate information and context for accurate calculations!
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