Calculating Decay Time How Long For 50 Grams To Decay To 10 Grams

Decay is a natural process that occurs in various contexts, from the disintegration of radioactive materials to the decomposition of organic matter. Understanding the rate of decay is crucial in many scientific and practical applications. In mathematics, we often model decay using exponential functions. This article delves into the process of calculating the time it takes for a substance to decay from an initial amount to a specific amount, given a decay rate. Specifically, we will explore the scenario where 50 grams of a substance decays to 10 grams, with a decay rate of k = -0.345. This involves applying the exponential decay formula and using logarithms to solve for time. By the end of this discussion, you will have a clear understanding of how to approach such problems and interpret the results.

Understanding Exponential Decay

Exponential decay is a process where a quantity decreases over time at a rate proportional to its current value. This phenomenon is commonly observed in radioactive decay, where the amount of a radioactive substance decreases over time as it transforms into another substance. The mathematical model that describes exponential decay is given by the formula:

$$N(t) = N_0 \cdot e^{kt}$$

Where:

• $N(t)$ is the amount of the substance remaining after time $t$.
• $N_0$ is the initial amount of the substance.
• $k$ is the decay constant (a negative value).
• $t$ is the time elapsed.
• $e$ is the base of the natural logarithm (approximately 2.71828).

In our specific problem, we are given:

• Initial amount $N_0 = 50$ grams
• Final amount $N(t) = 10$ grams
• Decay constant $k = -0.345$

Our goal is to find the time $t$ it takes for the substance to decay from 50 grams to 10 grams. This involves rearranging the exponential decay formula and solving for $t$. We will use logarithms to isolate $t$ in the equation. The natural logarithm, denoted as $ln$, is the logarithm to the base $e$, and it is particularly useful in dealing with exponential functions. By applying the natural logarithm to both sides of the equation, we can bring the exponent down and solve for $t$. This method provides an accurate way to determine the time it takes for the decay process to occur, which is essential in various applications such as nuclear medicine and environmental science.

Applying the Exponential Decay Formula

To determine how long it takes for 50 grams of a substance to decay to 10 grams with a decay rate of $k = -0.345$, we use the exponential decay formula:

$$N(t) = N_0 \cdot e^{kt}$$

We are given:

• $N_0 = 50$ grams (initial amount)
• $N(t) = 10$ grams (final amount)
• $k = -0.345$ (decay constant)

We need to solve for $t$ (time). Substituting the given values into the formula, we get:

$$10 = 50 \cdot e^{-0.345t}$$

First, divide both sides by 50 to isolate the exponential term:

$$\frac{10}{50} = e^{-0.345t}$$

$$0.2 = e^{-0.345t}$$

Now, take the natural logarithm ($ln$) of both sides to remove the exponential:

$$ln(0.2) = ln(e^{-0.345t})$$

Using the property of logarithms that $ln(e^x) = x$, we get:

$$ln(0.2) = -0.345t$$

Next, solve for $t$ by dividing both sides by -0.345:

$$t = \frac{ln(0.2)}{-0.345}$$

Using a calculator, we find that $ln(0.2) ≈ -1.60944$:

$$t = \frac{-1.60944}{-0.345}$$

$$t ≈ 4.66504$$

Rounding to the nearest tenth, we get:

$$t ≈ 4.7$$

Therefore, it will take approximately 4.7 time units for 50 grams of the substance to decay to 10 grams. This calculation demonstrates the practical application of the exponential decay formula and the use of logarithms to solve for time in decay processes. The result highlights the relationship between the decay rate and the time it takes for a substance to reduce to a specific amount, an essential concept in fields like nuclear physics and chemical kinetics.

Solving for Time Using Logarithms

The core of solving exponential decay problems lies in the intelligent use of logarithms. In our scenario, we had the equation:

$$0.2 = e^{-0.345t}$$

To isolate the variable $t$ which is in the exponent, we apply the natural logarithm ($ln$) to both sides. The natural logarithm is the inverse function of the exponential function with base $e$, which simplifies the equation significantly. By taking the natural logarithm of both sides, we transform the exponential equation into a linear equation, making it easier to solve for $t$. This technique is a fundamental aspect of dealing with exponential equations and is widely applicable in various scientific and engineering contexts. The property of logarithms that allows us to bring the exponent down as a coefficient is particularly useful in these scenarios. Understanding and applying this property is crucial for anyone working with exponential models.

$$ln(0.2) = ln(e^{-0.345t})$$

Applying the logarithmic property $ln(a^b) = b \cdot ln(a)$ and knowing that $ln(e) = 1$, the equation simplifies to:

$$ln(0.2) = -0.345t \cdot ln(e)$$

$$ln(0.2) = -0.345t$$

From here, we divide both sides by -0.345 to isolate $t$:

$$t = \frac{ln(0.2)}{-0.345}$$

This step is critical as it separates the variable we want to find from the constants. The resulting fraction provides the exact value of $t$ before any rounding occurs. Using a calculator, we find that:

$$ln(0.2) ≈ -1.60944$$

$$t ≈ \frac{-1.60944}{-0.345}$$

$$t ≈ 4.66504$$

Rounding to the nearest tenth, we obtain:

$$t ≈ 4.7$$

This final value represents the approximate time it takes for the substance to decay from 50 grams to 10 grams. The process of using logarithms to solve for time in exponential decay problems is a powerful mathematical tool that allows us to quantify and predict how substances decrease over time. It is a vital skill in various fields, including chemistry, physics, and environmental science, where decay processes are commonly studied.

Final Result and Interpretation

After performing the calculations, we found that it takes approximately 4.7 time units for 50 grams of the substance to decay to 10 grams, given a decay rate of $k = -0.345$. This result is obtained by applying the exponential decay formula, using natural logarithms to solve for time, and rounding the answer to the nearest tenth. Understanding the decay process is crucial in various scientific and practical applications, such as determining the half-life of radioactive materials or modeling the decrease in drug concentration in the body.

The time it takes for a substance to decay from an initial amount to a specific amount is inversely related to the decay rate. A higher decay rate means the substance decays more quickly, while a lower decay rate means it decays more slowly. In this case, the decay rate of -0.345 indicates a moderate rate of decay. The negative sign signifies that the substance is decreasing over time. The magnitude of the decay rate influences the time required for the substance to reach a certain level, which is a key consideration in various applications, such as nuclear medicine and environmental management.

The process of solving exponential decay problems involves several steps, including setting up the equation, isolating the exponential term, applying logarithms, and solving for the desired variable. Each step requires careful attention to detail and a solid understanding of mathematical principles. By following this systematic approach, one can accurately determine the time it takes for a substance to decay to a specific amount. This skill is essential for professionals working in fields where decay processes are relevant, such as nuclear physics, chemistry, and environmental science.

In conclusion, it takes approximately 4.7 time units for 50 grams of a substance to decay to 10 grams with a decay rate of -0.345. This result demonstrates the practical application of exponential decay models and the importance of understanding decay rates in various scientific and engineering disciplines. The use of logarithms to solve for time is a fundamental technique that allows us to quantify and predict the behavior of decaying substances, contributing to advancements in various fields.

Conclusion

In summary, determining the time required for a substance to decay from 50 grams to 10 grams with a decay rate of $k = -0.345$ involves applying the exponential decay formula and using logarithms to solve for time. We found that it takes approximately 4.7 time units for this decay to occur. This calculation highlights the importance of understanding exponential decay in various scientific and practical applications. The ability to model and predict decay processes is crucial in fields such as nuclear physics, chemistry, and environmental science. By mastering the techniques involved in solving these types of problems, we can better understand and manage the behavior of decaying substances.