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Chapter 10: Problem 5

Radioactive substances decay at a rate proportional to the quantity present. Write a differential equation for the quantity, \(Q\), of a radioactive substance present at time \(t\). Is the constant of proportionality positive or negative?

Short Answer

Expert verified
The differential equation is \(\frac{dQ}{dt} = -kQ\), with a negative constant of proportionality.

Step by step solution

01

Identify the Relationship

Radioactive decay is a process in which a substance decreases its quantity over time. This means the rate of change of the substance is proportional to the current quantity present.
02

Express the Decay Proportionally

The rate of change of the quantity, \(\frac{dQ}{dt}\), is proportional to the quantity \(Q\) itself. This relationship can be expressed as: \[\frac{dQ}{dt} = kQ\]where \(k\) is the constant of proportionality.
03

Determine the Sign of Constant

Since the substance is decaying, the quantity is decreasing over time. This means that the change in quantity, \(\frac{dQ}{dt}\), must be negative to represent a decrease.
04

Adjust the Proportionality Constant

Make the proportionality constant, \(k\), negative to reflect the decay: \[\frac{dQ}{dt} = -kQ\]This differential equation properly models the decay process with \(k > 0\), ensuring that \(\frac{dQ}{dt}\) is negative.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radioactive Decay
Radioactive decay is a natural process by which unstable atomic nuclei lose energy by emitting particles or electromagnetic waves. This leads to a reduction in the quantity of the radioactive substance over time. Understanding radioactive decay is crucial in fields such as nuclear physics, medicine, and environmental science. Here鈥檚 how the process can be simplified:
  • It's naturally occurring and involves the transformation of one element into another.
  • The substance continuously loses some of its mass as radiation, which can be in the form of alpha, beta, or gamma radiation.
  • Radioactive decay continues until a stable form of the element is achieved, which means a non-radioactive element is left.
What鈥檚 important to recognize is that the time it takes to reach a stable form is determined by what is known as the half-life 鈥� the time it takes for half of the original quantity of a substance to decay.
Rate of Change
The rate of change in the context of radioactive decay describes how quickly the quantity of the substance decreases as time progresses. Mathematically, this is quantified using a derivative. The derivative of the quantity with respect to time, denoted as \( \frac{dQ}{dt} \), represents how the quantity changes over time.Key points about rate of change:
  • A negative rate of change, \( \frac{dQ}{dt} < 0 \), indicates a decrease in the quantity, reflective of decay.
  • In radioactive decay, the rate is directly proportional to the quantity present; that larger quantities tend to decay faster.
  • The unit of \( \frac{dQ}{dt} \) could be something like grams per year or atoms per second, depending on the context.
Thus, understanding the rate of change helps us predict how the quantity of the substance will diminish over specific time intervals.
Proportional Relationship
A proportional relationship in mathematics refers to a scenario where two quantities change at a constant rate relative to each other. In radioactive decay, this concept is embodied by the equation \( \frac{dQ}{dt} = -kQ \), where \( k \) is a positive constant.Let's break it down:
  • This equation shows that the rate of change of the quantity \( Q \) is proportional to \( Q \) itself.
  • The constant \( k \) ensures that the rate is always a consistent proportion of the quantity. This constant is always positive since we attach a negative sign in the equation to represent decay.
  • This means that as the quantity decreases, so does the rate at which it decreases, leading to an exponential decay pattern.
Remember that proportional relationships, like this one, are foundational in modeling natural phenomena, allowing us to predict future behaviors of systems accurately.

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Most popular questions from this chapter

Check that \(y=A+C e^{k t}\) is a solution to the differential equation $$ \frac{d y}{d t}=k(y-A) $$

A bank account earns \(7 \%\) annual interest compounded continuously. You deposit \(\$ 10,000\) in the account, and withdraw money continuously from the account at a rate of \(\$ 1000\) per year. (a) Write a differential equation for the balance, \(B\), in the account after \(t\) years. (b) What is the equilibrium solution to the differential equation? (This is the amount that must be deposited now for the balance to stay the same over the years.) (c) Find the solution to the differential equation. (d) How much is in the account after 5 years? (e) Graph the solution. What happens to the balance in the long run?

Money in an account earns interest at a continuous rate of \(8 \%\) per year, and payments are made continuously out of the account at the rate of \(\$ 5000\) a year. The account initially contains \(\$ 50,000\). Write a differential equation for the amount of money in the account, \(B\), in \(t\) years. Solve the differential equation. Does the account ever run out of money? If so, when?

Find particular solutions \(\frac{d Q}{d t}=0.3 Q-120, \quad Q=50\) when \(t=0\)

Create a system of differential equations to model the situations. You may assume that all constants of proportionality are \(1 .\) A population of fleas is represented by \(x\), and a population of dogs is represented by \(y\). The fleas need the dogs in order to survive. The dog population, however, is unaffected by the fleas.
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