Atree falls in a forest. how many years must pass before the 14c activity in 1.03 g of the tree's carbon drops to 1.02 decay per hour? (carbon-14 has an abundance of 1.3 parts per trillion of carbon-12.)

t = 5.59x10⁴ y

Explanation:

To calculate the time for the ¹⁴C drops to 1.02 decays/h, we need to use the next equation:

   (1)

where : is the number of decays with time, A₀: is the initial activity, λ: is the decay constant and t: is the time.

To find A₀ we can use the following equation:  

  (2)

where N₀: is the initial number of particles of ¹⁴C in the 1.03g of the trees carbon

From equation (2), the N₀ of the ¹⁴C in the trees carbon can be calculated as follows:        

where : is the tree's carbon mass, : is the Avogadro's number and : is the ¹²C mass.  

Similarly, from equation (2) λ is:

where t 1/2: is the half-life of ¹⁴C= 5700 years

So, the initial activity A₀ is:  

Finally, we can calculate the time from equation (1):

I hope it helps you!