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Physics LibreTexts

5.8: Exponential Change Model

Things grow (or decay) exponentially when the time rate of change in the quantity is proportional to the amount of “stuff” already present. Thus, bacteria grow exponentially because the more you have, the more they reproduce (until they run out of food, or their waste starts to poison the environment.) The decay of radioactive elements is exponential because as the number of radioactive atoms decreases, there are fewer available to decay, so less decay occurs. The key idea is that the rate of change depends on the amount present.

Mathematically, it looks like this:

\[\frac{dy}{dt} = ky\]

In English words: the rate of change in \(y\) is proportional to \(y\) , and the constant of proportionality is \(k\) . If \(k\) is positive, then \(y\) is getting bigger (e.g., growth of organisms or compound interest); if \(k\) is negative, then \(y\) is getting smaller (e.g., decay of voltage across a cell membrane or nuclear decay.) Again, this is the key to having exponential change. The rate of change is proportional to the amount present at that time.

The mathematical relationship shown above is a differential equation. A solution of this equation 5.8.1 that will give us a description of the time rate of change in \(y\) , or \(y\) as a function of  \(t\) is the following:

\[y(t) = y_0e^{±kt}\]

Again, positive  \(k\) indicates growth, negative \(k\) , decay. So for nuclear decay that you probably studied in chemistry:

\[N(t) = N_0e^{- \lambda t} \]

Where  \(N\) is the number of atoms, and   \(\lambda\) is the decay constant specific to a particular isotope. In biology, exponential growth is often expressed as:

\[N(t) = N_0e^{rt} ~~~Where~ r~ is~ the~ growth ~rate.\]

Before we move on to some new phenomena, let’s check out that these expressions for the number of nuclei that haven’t decayed and the number of bacteria actually “satisfy the differential equation.” That is, if we do what the differential equation says to do, namely take the time derivative, do we get back the expression times a constant? Let’s try it out.

The exponential function is unique among all functions in that to within some constants, when you differentiate it, you get back the original function. In this case, the derivative of \(e^{±ax}~ is~ ±ae^{±ax}\) . So, for the case of nuclear decay, when we differentiate \(N(t) = N_0e^{-\lambda t}\) with respect to \(t\), we get \(– \lambda \) times the original function. In each time interval equal to1/\(\lambda\) , the number of nuclei will have been reduced to 1/e of each previous value. Similarly for the bacterial growth. In each time period equal to 1/r, the population will have grown by the factor e.

Now let’s add two new exponential phenomena to the collection: the cooling of objects, and charging and discharging of electric capacitors.

It’s clear to anyone who’s been slow to finish a cup of coffee or tea that hot objects cool down to room temperature. What might not be so obvious without taking some data is that the rate of cooling depends on the temperature difference between the hot object and its environment. So the hotter the cup of coffee, and the colder the room, the faster heat will move from the coffee to the room. (This has the name “Newton’s Law of Cooling” in some textbooks.)

\[\Delta T(t) = \Delta T_0 e^{-kt} ~~~ where~~ \Delta T_0 = T_{object,~ initial} - T_{environment} \]

So as the hot object approaches the temperature of its environment, the rate of cooling decreases and asymptotically approaches zero. The temperature difference behaves exactly like the example of nuclear decay.

When we discussed electric circuits earlier in this chapter we limited ourselves to circuits with batteries, wires and resistors. It is now time to consider another circuit component, the capacitor.

Capacitors are useful because they can store electrical energy and release that stored energy quickly. (Batteries store energy too, they just let it trickle out over a relatively long time.) When you take a flash photograph, you may have noticed a high-pitched whine as the camera charged a capacitor. The capacitor then discharges a large burst of energy to light the flashbulb. Capacitors store energy by accumulating charge on two conducting plates, a net positive charge on one plate, a net negative charge on the other. This creates an electric field between the plates. (We will discuss fields in depth in Part 3.) You should remember that like charges repel each other, so it makes sense that as the charge builds up on each plate, it becomes increasingly difficult to add more charge. Or if you think about a capacitor that is already charged, at first there will be a large accumulation of charge pushing charges off the plates, and as the charges move, the “pressure” pushing them will decrease. Here is another situation where the change in a variable is related to the amount already there. It should come as no surprise, then, that the charge on a capacitor in a series circuit as a function of time can be described with an equation that looks a lot like those above:

\[Q(t) = Q_0e^{ \frac{-t}{RC}} \]

where C is the capacitance of the capacitor, and R is the total resistance in the circuit. RC is sometimes called the “time constant” of the circuit because it tells us how long it takes for the capacitor to charge or discharge its energy. In this case, the charge is reduced to 1/e of its value in a time equal to the time constant, RC.

In each of these phenomena (and in several others you will work with in discussion/lab) we can understand the change by applying the basic ideas of the exponential change model. The fact that each version of the equation looks a bit different can easily hide that fact that the ideas underlying how the system changes are the same. The advantage of understanding the underlying behavior makes it possible for you to recognize the general pattern, even though the symbols are different or the equation is written differently.