Radium-226 is radioactive and has a half life of 1600. years. Calculate the activity of a 1.8mg sample of radium-226. Give your answer in becquerels and in curies. Be sure your answer has the correct number of significant digits.

To solve this problem, we need to use the formula for radioactive decay, which is:

A = λN

where: A is the activity (in decays per unit time, i.e., becquerels), λ is the decay constant, N is the number of atoms.

First, we need to find the decay constant (λ). The decay constant is related to the half-life (T) of the substance by the formula:

λ = ln(2) / T

where ln(2) is the natural logarithm of 2 (approximately 0.6931), and T is the half-life. Substituting the given half-life (T = 1600 years), we get:

λ = 0.6931 / 1600 years = 4.33 x 10^-4 years^-1

Next, we need to find the number of atoms (N). We can do this by converting the mass of the sample to moles (using the molar mass of radium-226, which is 226 g/mol), and then converting moles to atoms (using Avogadro's number, which is approximately 6.022 x 10^23 atoms/mol). The given mass of the sample is 1.8 mg, which is 0.0018 g. So:

N = (0.0018 g / 226 g/mol) x (6.022 x 10^23 atoms/mol) = 4.8 x 10^18 atoms

Now we can calculate the activity (A):

A = λN = (4.33 x 10^-4 years^-1) x (4.8 x 10^18 atoms) = 2.08 x 10^15 decays/year

To convert this to becquerels (decays per second), we divide by the number of seconds in a year (approximately 3.15 x 10^7 seconds/year):

A = 2.08 x 10^15 decays/year / 3.15 x 10^7 seconds/year = 6.6 x 10^7 Bq

To convert this to curies (Ci), we use the fact that 1 Ci = 3.7 x 10^10 Bq:

A = 6.6 x 10^7 Bq / 3.7 x 10^10 Bq/Ci = 1.78 x 10^-3 Ci

So, the activity of a 1.8 mg sample of radium-226 is approximately 6.6 x 10^7 Bq or 1.78 x 10^-3 Ci.