Computational geology 26 mathematics of readioactivity - when the earth got old

Journal of Geoscience Education, Sep 2003 by Vacher, H L

Keywords: Education - geoscience; education - undergraduate; miscellaneous and mathematical geology.

Topics this issue

Mathematics: Integration of first-order rate equation; exponential vs. linear decay; half-life; log function; series expansions.

Geology: Radioactivity; radiometric dates; Th-series, U-series, and lead isotopes; age of the Earth..

INTRODUCTION

The Earth aged from 100 million years at the turn of the twentieth century to over a billion years a years later. The 100-million-year figure was calculated by William Thomson (1824-1907). The mathematics leading to the order-of-magnitude increase was due to Ernst Rutherford (1872-1937).

William Thomson was made Baron Kelvin of Largs (Lord Kelvin) in 1892, the name Kelvin being taken from a river near Glasgow. One of the principal architects of thermodynamics, Kelvin insisted strenuously that geologists' concept of geologic time must comport with fundamental laws of physics (Burchfield, 1975).

The New Zealand-born Ernst Rutherford was made Baron of Nelson (his birthplace) in 1931. One of the main players in the birth of atomic physics, Lord Rutherford conceived the modern science of geochronology (Harper, 1975, p. 2). Rutherford brought physics and geology together on the issue of geologic time.

LORD KELVIN

Kelvin was widely regarded as the foremost physicist of his day (Burchfield, 1975). A boy-wonder, he entered the University of Glasgow at 11. At age 22, he became a professor at that university after studying at Cambridge and Paris. At the time of his appointment at Glasgow, he had already published 26 papers (Albritton, 1980, p. 178). His first lecture was "Age of the Earth and its Limitations as Determined from the Distribution and Movement of Heat within it." This was in 1846, 16 years after Lyell's Principles of Geology, and 13 years before Darwin's Origin of Species.

In 1852, Kelvin published the second law of thermodynamics. His paper included the following statement about the impossibility of the endless cycles and perpetual status quo of the earth's condition envisioned by early uniformitarians (Kelvin, 1852, quoted in Burchfield, 1990, p. 22)

Within a finite period of time past, the earth must have been, and within a finite period of time to come the earth must again be, unfit for the habitation of man as at present constituted, unless operations have been, or are to be performed which are impossible under the laws to which the known operations going on at the present in the material world are subject.

In other words, the Earth must be running down (cooling off); otherwise, it would violate his second law of thermodynamics.

Kelvin had a few different strategies for estimating a finite limit on the Earth's age. The one that seemed most unassailable for getting an actual number was the one he introduced in his inaugural lecture in 1846. He reconsidered it in an influential paper read on April 28, 1862: "On the secular Cooling of the Earth" (Kelvin, 1863; Burchfield,1975). In these works, he applied the mathematical theory of heat conduction published by J. Fourier in 1822 to the question of how long it would take the Earth to cool from its initial temperature of solidification, given its thermal diffusivity and present-day geothermal gradient and assuming no internal source of heat (Dalrymple, 1991; Turcotte and Schubert, 1982, Equation 4-118). His answer in 1862 was 98 million years. With the uncertainties, the range of possibilities was 20-400 million.

Kelvin repeatedly returned to the problem (Burchfield, 1975), reducing and sharpening his estimate. In a paper read in 1868 (Kelvin, 1871; Burchfield, 1975), he concluded that 100 million is the maximum possible. His last word on the subject, a famous lecture entitled "The Age of the Earth as an Abode Fitted for Life" in 1897 (Kelvin. 1899; Burchfield, 1975; Albritton, 1980; Dalrymple, 1991), was that the possible age of the Earth is 20-40 million years, and probably closer to 20 million than to 40 million. Geologists and biologists, recognizing the huge uncertainties in quantifying geological and biological rates, largely accommodated the notion of 108 years, although they rebelled against the notion of 20 million years (Burchfield, 1975; Albritton, 1980; Morris, 1985).

LORD RUTHERFORD

In the Fall of 1895, two years before Kelvin's "Abode fitted for life" speech, Ernst Rutherford arrived at the Cavendish Laboratory of Cambridge University to work with J. J. Thomson (1856-1940). A month later, on November, 5, 1895, William Roentgen (1845-1923) discovered X-rays from cathode-ray tubes. A few months later in early 1896, Henri Becquerel (1852-1908) discovered a similar radiation from uranium salts. Shortly after that, in 1897, J. J. Thomson showed that the cathode rays are bent by a magnetic field and, specifically, that they consist of negatively-charged particles with mass a thousand times smaller than that of an atom. This was proof of the existence of electrons, the first subatomic particle. Atomic physics was underway.