The unstable isotope 40K is used for dating rock samples. Its half-life is 1.28x10^9 y. (a) How many decays occur per second in a sample containing 1.63x10^-6 g of 40K? (b) What is the activity of the sample in curies?

Verified step by step guidance

1. Step 1: Calculate the number of moles of 40K in the sample. Use the formula: number of moles = mass of the sample (in grams) / molar mass of 40K. The molar mass of 40K is approximately 40 g/mol.
2. Step 2: Convert the number of moles of 40K to the number of atoms. Use Avogadro's number (approximately 6.022 x 10^23 atoms/mol) for this conversion. The formula is: number of atoms = number of moles x Avogadro's number.
3. Step 3: Calculate the decay constant (\(\lambda\)) using the half-life formula: \(\lambda = \frac{\ln(2)}{t_{1/2}}\), where \(t_{1/2}\) is the half-life of the isotope.
4. Step 4: Calculate the activity (A) of the sample in decays per second using the formula: A = \(\lambda\) x number of atoms. This will give you the number of decays per second.
5. Step 5: Convert the activity from decays per second to curies. Use the conversion factor: 1 curie = 3.7 x 10^10 decays per second. The formula is: activity in curies = activity in decays per second / 3.7 x 10^10.

Key Concepts

Half-life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. For 40K, with a half-life of 1.28 billion years, this concept is crucial for understanding the rate of decay and how it relates to the amount of the isotope present over time. It allows us to calculate the remaining quantity of the isotope after a certain period and is fundamental in radiometric dating.

Radioactive Decay

Radioactive decay is a random process by which unstable atomic nuclei lose energy by emitting radiation. This process can be quantified by the decay constant, which is related to the half-life. Understanding decay is essential for calculating the number of decays per second in a sample, as it directly influences the activity and stability of the isotope.

Activity (in Curies)

Activity is a measure of the rate at which a radioactive sample decays, typically expressed in units such as Curies (Ci). One Curie is defined as 3.7 x 10^10 decays per second. Calculating the activity of a sample involves determining the number of decays per second, which is essential for applications in dating and understanding the sample's radioactivity.