The radioisotope sodium-24 is used to determine the levels of electrolytes in the body. A 16-μg sample of sodium-24 decays to 2.0 μg in 45 h. What is the half-life, in hours, of sodium-24?

t = 15 hours

Explanation:

To get this, we need to write the general expression for the mass or concentration in life decay which is the following:

A = Ao * e^ (-kt) (1)

K is a constant decay and depends of the half life. The expression is:

k = ln2 / t₁/₂ (2)

In this question, we have the innitial mass and the final mass, and the time required to get that mass. So, with these data, we first solve for the constant k, and then, use this value to get the half life. In other words, we solve for k in (1) and then, solve for half life in (2):

2 = 16 * e (-45k)

2/16 = e (-45k)

0.125 = e (-45k)

ln (0.125) = - 45k

k = - ln (0.125) / 45

k = 0.0462 h⁻¹

Now that we have k, let's calculate half life:

t₁/₂ = ln2/k

t₁/₂ = ln2 / 0.0462

t₁/₂ = 15 h

This is the half life of sodium-24