The Slow Drip: A Model Of Exponential Decay - Question of the Week 10/27-10/31

A vertical tube is filled to 170 cm with blue water, as seen in the photograph at the left below. The photograph at the center below shows the bottom end of the tube, connected to a much narrower capillary tube mounted horizontally as seen. A cap covers the end of the tube; when the cap is removed the water will begin to flow out of the tube into the pan beneath. A timer can be used to measure how long water takes to flow out.

The timer, initially set to zero in the picture at the left, is started at the same time the cap is removed and the water begins to flow out of the tube. The time taken for half of the water to flow out of the tube (the level has gone down from 170 cm to 85 cm) is about 150 seconds, as read on the timer in the photograph at the right.

The question this week regards how the water will continue to flow out of the tube as time continues to pass. For example, what will happen in another (approximately) 150 seconds?

• (a) All the water will be gone before an additional 150 seconds passes.
• (b) All the water will be gone just as an additional 150 seconds is reached.
• (c) Almost, but not quite, all of the water will be gone after an additional 150 seconds.
• (d) Only about half of the remaining water will be gone after an additional 150 seconds.
• (e) The water will stop flowing shortly after the initial 150 seconds.

The answer is (d): about half of the remaining water will be gone after an additional 150 seconds. In fact, after additional time intervals of about 150 seconds, the amount of water in the vertical tube is reduced by half, as seen in the sequence of photographs below.

Well, actually it is not quite that simple, but is approximately correct: the first half life is 150 seconds, the second 140 seconds, the third 130 seconds, and the fourth 130 seconds. So the average half-life for this experiment is about 140 seconds.

This is an example of exponential decay, as can be seen in the composite of these photographs shown below. Note that the curve joining each of the water surface positions is a lot like an exponential decay graph.

This is a water model of the decay of a radioactive material, in which after each "half-life" unit of time one half of the nuclei decay. Thus, if there is one unit of material (radioactive nuclei or water in the tube) at time t=0, then after one unit of time 1/2 of the material is left, after 2 units of time 1/2 of 1/2, or 1/4 of the material is left, after 3 time units 1/8 of the material is left, after 4 units of time 1/16 of the material is left, and so on.

It turns out that the mathematics for both of these cases is very nearly identical: the only assumption that must be made is that, in the case of the water tube, the rate at which the water flows out is proportional to the amount in the tube (which is proportional to the pressure), or in the case of radioactive nuclei, the number of decays is proportional to the number of radioactive nuclei in the sample at any given time. This basic assumption leads to the process of exponential decay and the concept of half-life.