Exponential processes

Learning outcomes Exponential processes describe radioactive decay as an exponential process describe capacitor charge and discharge as exponential processes, including the time constant, RC calculate the decay constant and predict later activity of a radioactive sample, using and solve problems using and

Teaching challenges Showing how logarithms work Getting from the probability of radioactive decay to and the idea of equal ratios in equal times Conveying the meaning of a time constant for a capacitor (Why does the product RC give a time?) Developing confidence in solving problems with exponential equations

Exponential processes What do the following have in common? popping popcorn the head on a glass of beer sand falling through a hourglass timer air leaking from a tyre and these? cell growth in a foetus rabbit populations (good food supply, no disease or predators) time series data on Internet traffic nuclear chain reaction sand falling through hourglass time NOT exponential, as constriction passes only few grains at a time. Water draining from a burette probably IS exponential, provided flow is unimpeded at stopcock.

Growth and decay amount changes with time rate of increase proportional to amount rate of decrease proportional to amount A story: Doubling rice grains TAP question sheet: Exponential changes ‘Doubling rice grains shorten Doubling rice grains’ story – to save time.

Law of exponents In general, Here x is called the ‘base’ of a number system.

Logarithms Logarithms use base 10. log 100 = y where 10y = 100 What is log 203? log 1505? log 5? log1? log(.01)? Using logarithms changes multiplication to addition. What is 203 x 1505? Log and exponent (power) are inverse operations. What is log (102.5)? log (10y)?

Natural logarithms Base is the number e. 1 Make a table and evaluate e-x for integers in the range 2 Draw a graph of e-x against x and find gradients at x = 1, 2, 3, 4 If then ln and e are inverse operations. Each table allocates finding of gradients one per person, results pooled and discussed.

Exponential function e is the number (approximately 2.718281828) such that the function ex equals its gradient for all values of x. ex is sometimes written exp(x).

Radioactive decay Radioactive decay is a chance process. Activity, A (number of atoms in a sample disintegrating per second, dN/dt) is directly proportional to the number of atoms present, N.  is a constant characteristic of the atom (decay constant). If N0 atoms present, Handout: Smoothed-out radioactive decay (Adv Phys OHT) Question sheet: Radioactive decay with exponentials (Adv Phys 90S)

Half life

Discharging a capacitor At time t, during discharge of capacitor C, the p.d. across the capacitor is V and the charge on it is Q, then Q = VC. The discharge current I at time t equals the rate of loss of charge [- sign indicating Q decreases as t increases] V=IR so [RC is the decay constant] or which has solution

Charging a capacitor in series with a resistor R which has solution

Charging and discharging C Graphs for Q, I and V against t all have the same general shape. How are they related? I graph is the gradient of the Q graph (since I = dQ/dt). V = Q/C

An exponential relationship? Two tests for an exponential graph: 1 Half-life of radioactive materials is one instance of a more general pattern. In general, measured values have equal ratios in equal intervals of time. 2 Re-plot as log y against x (or ln y against x). An exponential relationship gives a straight line. e.g.

Charging and discharging C TAP experiment Analysing the discharge of a capacitor Question sheets: Charging capacitors Discharge and time constants

Exponentials – more examples Carbon-dating an archaeological artefact, by comparing C-14 in a sample with atmospheric C-14 Estimating the age of the Earth, from ratio of U-235 (t1/2 = 7.04 x108 y) to U-238 (t1/2 = 4.47x109 y) Intensity of -rays with thickness of absorber Calculating atmospheric pressure at altitude h Decay of amplitude for a vibration

Endpoints Further reading Helen Reynolds, Wherefore ? in Simon Carson (ed) (1999) Physics in mathematical mood IOP (booklet freely available from the National STEM Centre eLibrary)