Question

The half-life of a radioactive substance is 3200 years. Find the quantity \(Q(t)\) of the substance left at time \(t>0\) if \(Q(0)=20 \mathrm{~g}\).

Short answer

Answer: The formula for the quantity left at any given time t>0 is: Q(t) = 20 * e^((-ln(2)/3200) * t)

Step-by-step solution

Find the decay constant (k)

To find the decay constant \(k\), we can use the formula for the half-life, which is \(T_{1/2}=\dfrac{ln(2)}{k}\). Given the half-life of the substance is 3,200 years, we have: $$3200 = \dfrac{ln(2)}{k} \Rightarrow k = \frac{ln(2)}{3200}$$ We can now substitute this into our formula for \(Q(t)\).

Find the quantity left at time t (Q(t))

Now that we have the decay constant, we can find the quantity left at any given time \(t>0\) by plugging the values of \(Q(0)\) and \(k\) into the formula \(Q(t)=Q(0)e^{-kt}\): $$ Q(t) = 20{e} ^{-\frac{ln(2)}{3200} \times t}$$ This formula represents the quantity left of the radioactive substance at any given time \(t>0\).

Key concepts

Half-Life

In the world of radioactive decay, the term "half-life" represents a crucial concept. The half-life of a substance is the time it takes for half of the initial amount of the radioactive isotopes present to decay. For example, if you start with 100 grams of a radioactive substance, after one half-life, only 50 grams of the substance will remain. This process continues, with half of the remaining substance decaying every subsequent half-life period.
It's important to note that the half-life is a constant for a given substance. It does not depend on the initial quantity you start with. In our example, the substance has a half-life of 3200 years. Thus, it will consistently take 3200 years for any starting amount to halve.
Understanding the half-life helps in predicting how quickly a substance will decay and helps with calculating the decay constant, which we'll discuss next.

Decay Constant

The decay constant, often represented by the symbol \(k\), is a number that helps describe the rate at which a radioactive substance decays. This constant is related to the half-life through the formula \(T_{1/2} = \frac{\ln(2)}{k}\), where \(T_{1/2}\) is the half-life.
To find the decay constant of our substance with a half-life of 3200 years, we rearrange the formula:
\[ k = \frac{\ln(2)}{3200} \] This relation shows that the decay constant and half-life are inversely related. The smaller the decay constant, the longer it takes for the substance to decay, and vice versa. Knowing the decay constant is crucial for predicting how much of the radioactive substance will remain over various time periods.

Exponential Decay

Exponential decay is a mathematical model that describes how the quantity of a radioactive substance decreases over time. When a radioactive substance decays, it doesn't lose a fixed amount per unit of time. Instead, it loses a fixed proportion, leading to a rapid decrease initially.
The general equation for exponential decay is given by: \[ Q(t) = Q(0)e^{-kt} \] Here, \(Q(t)\) is the remaining quantity at time \(t\), \(Q(0)\) is the initial quantity, \(e\) is the base of natural logarithms, and \(k\) is the decay constant.
In our scenario, the exponential decay formula allows us to predict the future quantity of the substance based on the known initial quantity and decay constant, giving us a clear picture of the decay process over any time period.

Differential Equations

Differential equations play a vital role in modeling radioactive decay. They are equations involving derivatives, which represent how a quantity changes over time. In radioactive decay, the change in quantity of the substance over a small time interval is proportional to the amount present.
The standard form of the differential equation used in decay processes is: \[ \frac{dQ}{dt} = -kQ \] Here, \(\frac{dQ}{dt}\) represents the rate of change of the quantity \(Q\) over time \(t\), and \(k\) is the decay constant. The negative sign indicates that the quantity is decreasing.
Solving this differential equation with given initial conditions, such as \(Q(0) = 20 \, \text{g}\), leads us to the exponential decay formula. Thus, differential equations provide the foundation for understanding and predicting the behavior of decaying substances.