What is the mass (in mg) of the sample? And what is the mass 6 weeks after the start?

Question

The half life of Palladium-100 is 4 days. After 20 days a sample of Palldium-100 has been reduced to a mass of 3 mg.

Answer 1

With the given information, the equation for mass vs time is:

m = m₀n *2^-x/4 where

x    = number of days since the original mass of palladium started to decay

m_o = original mass on day 0.

m   = mass after x days

Thus, after 20 days,

m  = m₀*2^-20/4

     = m₀*2^-5

     = m₀/32

     = 3 mg (given) therefore:

m₀ = 32 * 3

     = 96 mg.

After 6 weeks (6*7 = 42 days),

m = 96*2^-42/4

   = 96*2^-10.5

   = 0.0663 mg

Note: Most scientific calculators can calculate 2^-10.5 directly, but if yours doesn’t, or if you don’t have one, logarithms and log tables (or a calculator) can be used, along with the formula:

log_ax = log_bx/log_ba   where a=2 (for this problem), and b is usually 10 or e.

For this problem, that translates to

Log₁₀x = log₂x * log₁₀2

           = (-10.5)*0.30103

           = -3.1608

x = 10^-3.1608

   = 0.000691

So, the 6-week mass = 96*0.000691 = 0.0663 mg

Answer 2

fraction remaining = 0.5ⁿ where n is the number of half lives that have elapsed.

number of half lives elapsed = 20 days x 1 half live/4 days = 5 half lives

fraction remaining = 0.5⁵ = 0.03125

original mass: 0.03125 x = 3 mg and x = 96 mg

6 weeks = 6 wks x 7 days/wk = 42 days

number of half lives = 42 days/4 days = 10.5 half lives

fraction remaining = 0.5^10.5 = 0.000691

mass remaining after 6 weeks = 0.000691 x 96 mg = 0.066 mg