In Pursuit of the Unknown by Ian Stewart Read Free Book Online

remained critical and much more work would be needed before the reactors could be considered fully under control, but claimed the leakage had been stopped.
    I don’t want to analyse the merits or otherwise of nuclear power here, but I do want to show how the logarithm answers a vital question: if you know how much radioactive material has been released, and of what kind, how long will it remain in the environment, where it could be hazardous?
    Radioactive elements decay; that is, they turn into other elements through nuclear processes, emitting nuclear particles as they do so. It is these particles that constitute the radiation. The level of radioactivity falls away over time just as the temperature of a hot body falls when it cools:exponentially. So, in appropriate units, which I won’t discuss here, the level of radioactivity N ( t ) at time t follows the equation
    N ( t ) = N 0 e −kt
    where N 0 is the initial level and k is a constant depending on the element concerned. More precisely, it depends on which form, or isotope, of the element we are considering.
    A convenient measure of the time radioactivity persists is the half-life, a concept first introduced in 1907. This is the time it takes for an initial level N 0 to drop to half that size. To calculate the half-life, we solve the equation

    by taking logarithms of both sides. The result is

    and we can work this out because k is known from experiments.
    The half-life is a convenient way to assess how long the radiation will persist. Suppose that the half-life is one week, for instance. Then the original rate at which the material emits radiation halves after 1 week, is down to one quarter after 2 weeks, one eighth after 3 weeks, and so on. It takes 10 weeks to drop to one thousandth of its original level (actually 1/1024), and 20 weeks to drop to one millionth.
    In accidents with conventional nuclear reactors, the most important radioactive products are iodine-131 (a radioactive isotope of iodine) and caesium-137 (a radioactive isotope of caesium). The first can cause thyroid cancer, because the thyroid gland concentrates iodine. The half-life of iodine-131 is only 8 days, so it causes little damage if the right medication is available, and its dangers decrease fairly rapidly unless it continues to leak. The standard treatment is to give people iodine tablets, which reduce the amount of the radioactive form that is taken up by the body, but the most effective remedy is to stop drinking contaminated milk.
    Caesium-137 is very different: it has a half-life of 30 years. It takes about 200 years for the level of radioactivity to drop to one hundredth of its initial value, so it remains a hazard for a very long time. The main practical issue in a reactor accident is contamination of soil and buildings. Decontamination is to some extent feasible, but expensive. For example, the soil can be removed, carted away, and stored somewhere safe. But this creates huge amounts of low-level radioactive waste.
    Radioactive decay is just one area of many in which Napier’s and Briggs’s logarithms continue to serve science and humanity. If you thumb through later chapters you will find them popping up in thermodynamics and information theory, for example. Even though fast computers have now made logarithms redundant for their original purpose, rapid calculations, they remain central to science for conceptual rather than computational reasons.
    Another application of logarithms comes from studies of human perception: how we sense the world around us. The early pioneers of the psychophysics of perception made extensive studies of vision, hearing, and touch, and they turned up some intriguing mathematical regularities.
    In the 1840s a German doctor, Ernst Weber, carried out experiments to determine how sensitive human perception is. His subjects were given weights to hold in their hands, and asked when they could tell that one weight felt heavier than another. Weber could