Final answer:
After 180 years, approximately 47.41 grams of the original 50 grams of radium will be present. The calculation is done using the exponential decay formula considering the half-life of radium to be 1690 years.
Step-by-step explanation:
To determine how much radium will be present after 180 years, we need to understand the concept of half-life. The half-life of a radioactive isotope is the time it takes for half of the substance to decay into its daughter elements. Since the half-life of radium is 1690 years, and the time elapsed is only 180 years, it is a fraction of the half-life period, thus not much decay has occurred. We can use the formula for exponential decay:
N = N_0(1/2)^(t/T)
Where:
- N is the remaining quantity of the substance after time t,
- N_0 is the initial quantity of the substance,
- t is the time elapsed, and
- T is the half-life of the substance.
For 50 grams of radium with a half-life of 1690 years, after 180 years, the calculation would be:
N = 50g(1/2)^(180/1690)
To find the remaining amount of radium, we need to calculate the exponent first:
(1/2)^(180/1690) = (1/2)^(0.1065) ≈ 0.94823
Now we multiply this by the initial amount:
N ≈ 50g * 0.94823 ≈ 47.41 grams
Therefore, after 180 years, approximately 47.41 grams of radium will be present.