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where t is age (time), X is the decay constant, ln is the natural logarithm (which of a given number is the exponent that must be assigned to e (about 2.71) to derive that same number), d is amount of the daughter element (which is how much is measurable now), and p is the parent element (also how much is measurable now). The application of a logarithmic function is necessary because radioactive decay is exponential, which means that a radioactive element decays at a gradually more rapid rate with time, which contrasts it with an arithmetic rate, which is simply subtraction or addition with time. This mathematical distinction is important in the understanding of radiometric age dating. Because time is needed for a parent element to decay to a daughter element, a high ratio of daughter to parent (d/p) indicates a greater amount of time than a comparatively low ratio, thus the rate of increase for a daughter element is directly proportional to the rate of decrease for a parent element.

Decay constants are calculated by using a lot of math, beginning with the following equation:

where X is the decay constant and t1/2 is the half-life of the element. Half-life is the amount of time needed for half of the parent element to have decayed. Halflife can also be expressed by knowing that it represents half of the original radioactive element:

For a specific radioactive element that underwent decay, the decay constant represents the measured number of atoms that decay per second, compared to the number of atoms that are still in the rock sample:

where (dN/dt) is the rate of change in number of atoms (dN) in proportion to rate of the change in time (dt), and N is the number of atoms now present. The measurement of the number of alpha particles emitted by an atom per second will give an indication of the first value through a standard number of atoms, such as in one Avogadro (Av) of a sample, 6.02 x 1023 atoms. For example, using 1.0 Av of 238U, which has a measured alpha particle emission rate of 2.96 x 106 particles per second, the decay constant is calculated as:

Step 1. X= (2.96 x 106 a/s)/6.02 x 1023 Step 2. = 4.92 x 10-18 s-1

The half-life can be solved for our example of 238U, because the decay constant is known:

Step 1. t1/2 = 0.693/4.92 x 10-18 s Step 2. = (0.693/4.92) x 1018 s Step 3. = 1.41 x 1017s

The number of seconds in a year is about 31,557,000 (give or take a thousand seconds), thus to convert the calculated number of seconds to years:

Step 4. t1/2 = 1.41 x 1017 s/3.1557 x 107 s/y Step 5. = 4.4 68 x 109 y

Therefore, the half-life for 238U is nearly 4.5 billion years, which means in that time about one-half of the original amount of 238U in a rock will have been lost through decay, then in another 4.5 billion years, only half of that half (25%) is left, and so on (Fig. 4.6). Once the approximate number of atoms for each element (parent and daughter) is counted from a rock sample, the ratio of one to the other is calculated, which is an indicator of the number of half-lives that have passed and then can be used to calculate the age of a rock. A device called a mass

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billion years 4.5 13 5 IB 0

FIGURE 4.6 Exponential loss of a parent element (238U) through time, showing changes in ratio with relation to the daughter element (206Pb) with each half-life. Note that the plot follows a curved line, not a straight line.

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billion years 4.5 13 5 IB 0

FIGURE 4.6 Exponential loss of a parent element (238U) through time, showing changes in ratio with relation to the daughter element (206Pb) with each half-life. Note that the plot follows a curved line, not a straight line.

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