In 2007, A Sample Of A Radioactive Substance Had A Mass Of 550 Milligrams. Since Then, The Sample Has Decayed By $5.7 \%$ Each Year.Let $t$ Be The Number Of Years Since 2007. Let \$y$[/tex\] Be The Mass Of The

by ADMIN 215 views

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 analyze the decay of a sample of a radioactive substance that had a mass of 550 milligrams in 2007. The sample has decayed by 5.7% each year since then. We will use mathematical models to describe the decay process and determine the mass of the sample at any given time.

Mathematical Model

Let's assume that the mass of the sample at time t is denoted by y(t). We can model the decay process using the following differential equation:

dy/dt = -ky

where k is a constant that represents the decay rate. In this case, the decay rate is 5.7% per year, which can be expressed as a decimal:

k = 0.057

The initial condition is y(0) = 550, which represents the mass of the sample in 2007.

Solving the Differential Equation

To solve the differential equation, we can use the following method:

y(t) = y(0)e^(-kt)

where e is the base of the natural logarithm. Plugging in the values, we get:

y(t) = 550e^(-0.057t)

This equation represents the mass of the sample at any given time t.

Graphical Representation

To visualize the decay process, we can plot the mass of the sample over time. Using a graphing calculator or software, we can create a graph of y(t) = 550e^(-0.057t). The graph shows an exponential decay curve, with the mass of the sample decreasing rapidly at first and then slowing down over time.

Calculating the Mass at a Given Time

To calculate the mass of the sample at a given time, we can plug in the value of t into the equation y(t) = 550e^(-0.057t). For example, if we want to find the mass of the sample in 2010, we can plug in t = 3:

y(3) = 550e^(-0.057(3)) = 550e^(-0.171) ≈ 492.5 milligrams

Conclusion

In this article, we analyzed the decay of a sample of a radioactive substance that had a mass of 550 milligrams in 2007. We used a mathematical model to describe the decay process and determined the mass of the sample at any given time. The results show an exponential decay curve, with the mass of the sample decreasing rapidly at first and then slowing down over time. We can use this model to calculate the mass of the sample at any given time, providing valuable information for scientists and researchers working with radioactive materials.

References

Mathematical Derivations

Derivation of the Differential Equation

The differential equation dy/dt = -ky can be derived from the following assumptions:

  • The mass of the sample at time t is denoted by y(t).
  • The decay rate is constant and denoted by k.
  • The mass of the sample decreases exponentially over time.

Using the chain rule, we can write:

dy/dt = d/dt (y(0)e^(-kt)) = y(0)e^(-kt) (-k) = -ky

This equation represents the differential equation for the decay process.

Derivation of the Solution

To solve the differential equation, we can use the following method:

y(t) = y(0)e^(-kt)

This equation represents the solution to the differential equation. Plugging in the values, we get:

y(t) = 550e^(-0.057t)

Q: What is radioactive decay?

A: 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.

Q: What is the difference between radioactive decay and other types of decay?

A: Radioactive decay is a unique process that occurs in unstable atoms, whereas other types of decay, such as chemical decay, occur in molecules or compounds.

Q: How does the decay rate affect the mass of the sample?

A: The decay rate is a measure of how quickly the sample decays. A higher decay rate means that the sample will decay more quickly, resulting in a lower mass over time.

Q: Can the decay rate be affected by external factors?

A: Yes, the decay rate can be affected by external factors such as temperature, pressure, and radiation. However, in the case of radioactive decay, the decay rate is typically constant and unaffected by external factors.

Q: How can the mass of the sample be calculated at a given time?

A: The mass of the sample at a given time can be calculated using the equation y(t) = y(0)e^(-kt), where y(0) is the initial mass, k is the decay rate, and t is the time.

Q: What is the significance of the half-life in radioactive decay?

A: The half-life is the time it takes for the sample to decay to half of its initial mass. The half-life is a measure of the stability of the sample and is used to determine the decay rate.

Q: Can the half-life be affected by external factors?

A: No, the half-life is a constant property of the sample and is unaffected by external factors.

Q: How can the half-life be calculated?

A: The half-life can be calculated using the equation t1/2 = ln(2)/k, where k is the decay rate.

Q: What is the relationship between the half-life and the decay rate?

A: The half-life is inversely proportional to the decay rate. A higher decay rate results in a shorter half-life, while a lower decay rate results in a longer half-life.

Q: Can the decay rate be affected by the presence of other substances?

A: Yes, the decay rate can be affected by the presence of other substances, such as catalysts or inhibitors. However, in the case of radioactive decay, the decay rate is typically constant and unaffected by the presence of other substances.

Q: How can the mass of the sample be measured over time?

A: The mass of the sample can be measured over time using a variety of techniques, such as weighing the sample or using a mass spectrometer.

Q: What are some common applications of radioactive decay?

A: Radioactive decay has a wide range of applications, including medicine, industry, and scientific research. Some common applications include:

  • Medical imaging: Radioactive decay is used to create images of the body and diagnose diseases.
  • Industrial processes: Radioactive decay is used to measure the thickness of materials and detect defects.
  • Scientific research: Radioactive decay is used to study the properties of materials and understand the behavior of atoms.

Q: What are some safety precautions that should be taken when working with radioactive materials?

A: When working with radioactive materials, it is essential to take safety precautions to prevent exposure to radiation. Some common safety precautions include:

  • Wearing protective clothing and gloves
  • Using a lead shield to block radiation
  • Following proper handling and storage procedures
  • Monitoring radiation levels and taking regular breaks to avoid exposure.