Problem 20 - Algebra 2 and Trigonometry

Question

The half-life of einsteinium is 276 days. a. To five decimal places, what is the decay constant of einsteinium? Assume the exponential decay occurs continuously. b. After how many days will 2.5 grams of einsteinium remain of a sample of 20 grams?

Step-by-Step Solution

Answer

The decay constant of einsteinium is approximately 0.002512 per day. It will take about 824.62 days for 2.5 grams to remain from 20 grams.

Step 1: Understand Half-Life and Decay Constant

The half-life of a substance is the time it takes for half of the sample to decay. The decay constant \( \lambda \) is related to the half-life \( T_{1/2} \) by the formula \( T_{1/2} = \frac{\ln(2)}{\lambda} \). We can use this relationship to find the decay constant.

Step 2: Calculate Decay Constant

Rearrange the formula to solve for \( \lambda \): \( \lambda = \frac{\ln(2)}{T_{1/2}} \). Substitute \( T_{1/2} = 276 \) days into the formula to find \( \lambda \): \[ \lambda = \frac{\ln(2)}{276} \approx 0.002512 \] Therefore, the decay constant is approximately \( 0.002512 \) per day.

Step 3: Use Exponential Decay Formula to Find Time

Use the formula for exponential decay: \( N(t) = N_0 e^{-\lambda t} \), where \( N(t) \) is the remaining quantity of the substance, \( N_0 \) is the initial quantity, and \( t \) is the time in days. We need to find \( t \) such that \( N(t) = 2.5 \) grams when \( N_0 = 20 \) grams.

Step 4: Solve for Time

Plug the known values into the decay formula: \( 2.5 = 20 e^{-0.002512 t} \). Divide both sides by 20: \( 0.125 = e^{-0.002512 t} \). Take the natural logarithm of both sides: \( \ln(0.125) = -0.002512 t \). Solve for \( t \): \[ t = \frac{\ln(0.125)}{-0.002512} \approx 824.615 \] Thus, it will take approximately 824.62 days for 2.5 grams of einsteinium to remain.

Key Concepts

Half-Life

The half-life of a substance is a crucial concept in understanding how materials decay over time. It describes the amount of time required for half of the substance to transform or degrade. In other words, if you start with a certain quantity of a material, after one half-life, you'll have half of it left. This provides a predictable timeline that helps scientists and researchers determine how substances will diminish over time.

For einsteinium, its half-life is 276 days. This means that if you have a sample of einsteinium, after 276 days, only half of that sample will remain unchanged. Half-life doesn't just apply to radioactive materials; it can also describe the decay of other mean values such as medicine effects or population decrease.

Decay Constant

The decay constant, often denoted by the Greek letter \( \lambda \), is a parameter that represents the rate at which a substance undergoes exponential decay. It is directly related to the half-life of the substance.

Mathematically, the relationship between half-life \( T_{1/2} \) and decay constant \( \lambda \) is described by the formula:

• \[ T_{1/2} = \frac{\ln(2)}{\lambda} \]

To find the decay constant when you know the half-life, you rearrange the formula as follows:

• \[ \lambda = \frac{\ln(2)}{T_{1/2}} \]

Using this formula, for einsteinium with a half-life of 276 days, the decay constant is roughly \( 0.002512 \) per day. This small number reflects how fast (or slow) the decay process occurs.

Exponential Decay Formula

The exponential decay formula provides a way to calculate how much of a substance remains after a certain period of time has passed. The formula is expressed as:

• \[ N(t) = N_0 e^{-\lambda t} \]

In this formula:

• \( N(t) \) is the remaining quantity of the substance at time \( t \).
• \( N_0 \) is the initial quantity of the substance you started with.
• \( \lambda \) is the decay constant.
• \( t \) is the time that has elapsed.

For instance, if you know the decay constant and want to determine the time it takes for the substance to reduce to a specific amount, you can rearrange the formula to solve for \( t \). This involves taking the natural logarithm, which we'll explore next.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical function that is extremely useful in solving exponential decay problems. It helps to "undo" the exponential component in equations, making it possible to isolate variables like time.

For instance, in solving for time \( t \) in our exponential decay formula, we take the natural logarithm of both sides to make the exponent manageable:

• If you have an equation like \( e^{-\lambda t} = 0.125 \), taking the \( \ln \) of both sides would give: \( \ln(0.125) = -\lambda t \).

This process allows us to solve for \( t \) by dividing by \( -\lambda \), simplifying the equation to find how many days it takes for a certain decay to occur. Understanding how to use \( \ln \) is key in navigating equations involving exponential functions.