Element X Is A Radioactive Isotope Such That Its Mass Decreases By \[$8\%\$\] Every Year. If An Experiment Starts Out With 460 Grams Of Element X, Write A Function To Represent The Mass Of The Sample After \[$t\$\] Years, Where The

Mar 1, 2025 by ADMIN 232 views

**Modeling Radioactive Decay with a Function**

Introduction

Radioactive decay is a process in which unstable atoms lose energy through radiation. This process is characterized by a decrease in the mass of the radioactive substance over time. In this article, we will explore how to model this process using a mathematical function. Specifically, we will consider a radioactive isotope that loses 8% of its mass every year.

The Problem

We are given that an experiment starts with 460 grams of Element X, a radioactive isotope that loses 8% of its mass every year. We want to write a function that represents the mass of the sample after t years.

Understanding Radioactive Decay

Radioactive decay is an exponential process. This means that the rate of decay is proportional to the amount of the substance present. Mathematically, this can be represented as:

dM/dt = -kM

where M is the mass of the substance, t is time, and k is a constant of proportionality.

Solving the Differential Equation

To solve this differential equation, we can use separation of variables. We can rewrite the equation as:

dM/M = -kdt

Integrating both sides, we get:

∫(dM/M) = -k∫dt

Evaluating the integrals, we get:

ln|M| = -kt + C

where C is a constant of integration.

Applying the Initial Condition

We are given that the initial mass of the sample is 460 grams. This means that at t = 0, M = 460. We can use this information to find the value of C.

ln|460| = -k(0) + C ln(460) = C

Finding the Function

Now that we have found the value of C, we can rewrite the equation as:

ln|M| = -kt + ln(460)

We can exponentiate both sides to get:

|M| = e^(-kt + ln(460))

|M| = 460e^(-kt)

Since mass cannot be negative, we can drop the absolute value sign:

M(t) = 460e^(-kt)

Finding the Value of k

We are given that the mass decreases by 8% every year. This means that after one year, the mass will be 92% of the initial mass. We can use this information to find the value of k.

M(1) = 460e^(-k(1)) 0.92(460) = 460e^(-k)

Dividing both sides by 460, we get:

0.92 = e^(-k)

Taking the natural logarithm of both sides, we get:

ln(0.92) = -k

k = -ln(0.92)

The Final Function

Now that we have found the value of k, we can write the final function:

M(t) = 460e^(-(-ln(0.92))t) M(t) = 460e^(ln(0.92)t) M(t) = 460(0.92)^t

Conclusion

In this article, we have modeled the radioactive decay of Element X using a mathematical function. We have shown that the mass of the sample decreases exponentially over time, and have written a function to represent this process. The function is given by:

M(t) = 460(0.92)^t

This function can be used to calculate the mass of the sample at any time t.

Code Implementation

Here is a Python code implementation of the function:

import math

def calculate_mass(t):
    """
    Calculate the mass of the sample after t years.

    Parameters:
    t (float): Time in years.

    Returns:
    float: Mass of the sample.
    """
    k = -math.log(0.92)
    mass = 460 * (0.92)**t
    return mass

# Test the function
t = 1
mass = calculate_mass(t)
print(f"Mass after {t} years: {mass:.2f} grams")

Example Use Cases

This function can be used in a variety of scenarios, such as:

Calculating the mass of a radioactive sample after a certain period of time.
Modeling the decay of a radioactive substance over time.
Predicting the mass of a sample after a certain number of years.

Introduction

In our previous article, we explored the concept of radioactive decay and developed a mathematical function to model the process. We introduced the Element X function, which represents the mass of a sample of Element X after a certain period of time. In this article, we will answer some frequently asked questions about the Element X function and radioactive decay.

Q: What is the significance of the 8% decrease in mass every year?

A: The 8% decrease in mass every year is a key characteristic of the Element X function. This means that after one year, the mass of the sample will be 92% of the initial mass. This rate of decay is a fundamental property of the Element X isotope and is used to model its behavior over time.

Q: How does the Element X function account for the exponential nature of radioactive decay?

A: The Element X function is an exponential function, which means that it accounts for the exponential nature of radioactive decay. The function is given by:

M(t) = 460(0.92)^t

This function shows that the mass of the sample decreases exponentially over time, with a rate of decay proportional to the amount of the substance present.

Q: Can the Element X function be used to model other types of radioactive decay?

A: While the Element X function is specifically designed to model the decay of Element X, the underlying principles of radioactive decay are more general. The function can be adapted to model other types of radioactive decay by changing the rate of decay (k) and the initial mass (M0).

Q: How accurate is the Element X function in predicting the mass of a sample over time?

A: The accuracy of the Element X function depends on the assumptions made in its development. The function assumes that the rate of decay is constant and that the sample is not subject to external influences that could affect its decay rate. In reality, the decay rate may vary over time due to factors such as changes in temperature, pressure, or the presence of other substances.

Q: Can the Element X function be used to predict the half-life of Element X?

A: Yes, the Element X function can be used to predict the half-life of Element X. The half-life is the time it takes for the mass of the sample to decrease by half. To find the half-life, we can set the mass to half of the initial mass and solve for time:

M(t) = 0.5M0

Substituting this into the Element X function, we get:

0.5M0 = 460(0.92)^t

Solving for t, we get:

t = ln(0.5)/ln(0.92)

This gives us the half-life of Element X.

Q: How can the Element X function be used in real-world applications?

A: The Element X function has a wide range of applications in fields such as nuclear physics, chemistry, and materials science. Some examples include:

Modeling the decay of radioactive substances in nuclear reactors or medical applications.
Predicting the behavior of radioactive waste over time.
Developing new materials with specific properties, such as high-energy density or long half-life.

Conclusion

In this article, we have answered some frequently asked questions about the Element X function and radioactive decay. We have shown that the function is a powerful tool for modeling the behavior of radioactive substances over time and has a wide range of applications in various fields.

Code Implementation

Here is a Python code implementation of the Element X function and the half-life calculation:

import math

def calculate_mass(t):
    """
    Calculate the mass of the sample after t years.

    Parameters:
    t (float): Time in years.

    Returns:
    float: Mass of the sample.
    """
    k = -math.log(0.92)
    mass = 460 * (0.92)**t
    return mass

def calculate_half_life():
    """
    Calculate the half-life of Element X.

    Returns:
    float: Half-life of Element X.
    """
    k = -math.log(0.92)
    half_life = math.log(0.5)/k
    return half_life

# Test the functions
t = 1
mass = calculate_mass(t)
half_life = calculate_half_life()
print(f"Mass after {t} years: {mass:.2f} grams")
print(f"Half-life of Element X: {half_life:.2f} years")

Example Use Cases

This code can be used to calculate the mass of a sample of Element X after a certain period of time and to predict the half-life of the substance.