The Ancient Cosmic Clock · AI Generator #GEN-C9DFE2AF

Physics · modern-physics

Problem

Archaeologists have discovered an ancient artifact containing a mysterious crystalline substance. Initial analysis reveals it contains equal amounts of two isotopes:

^{87}

Rb (rubidium-87) and

^{87}

Sr (strontium-87). The artifact also contains a thin metallic layer that exhibits X-ray fluorescence. When bombarded with high-energy electrons, this metallic layer emits characteristic X-rays with a frequency of

f = 1.24 \times 10^{18}\;\text{Hz}

. Modern rubidium ore contains

^{87}

Rb and

^{87}

Sr in a ratio of 2.61:1 by number of atoms. It is known that

^{87}

Rb undergoes beta-minus decay to form stable

^{87}

Sr with a half-life of

T_{1/2} = 4.88 \times 10^{10}\;\text{years}

. The ancients somehow knew that this artifact would serve as a "cosmic clock" - they inscribed on it a symbol representing the atomic number of the metal in the fluorescent layer. (a) What is the atomic number

Z

of the metal in the fluorescent layer? (b) How old is this artifact? Express your answer in billions of years. Given: Moseley's law constant

R = 1.097 \times 10^7\;\text{m}^{-1}

, speed of light

c = 3.00 \times 10^8\;\text{m/s}

, Planck's constant

h = 6.626 \times 10^{-34}\;\text{J·s}

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