Answer: The simple solution to this problem is to recognize that the half-life of the radioactive atom is 2 months. This means that after 2 months, half of the radioactive material will decay and half will remain.
So, after 2 months, 1/2 g of the element will remain.
After another 2 months (4 months total), half of the remaining 1/2 g will decay and half will remain. This means that after 4 months, 1/4 g of the element will remain.
After another 2 months (6 months total), half of the remaining 1/4 g will decay and half will remain. This means that after 6 months, 1/8 g of the element will remain.
Therefore, the answer is (c) 1/8.
Solution: The decay of a radioactive element follows an exponential decay function, which can be expressed as:
N(t) = N₀ * e^(-λt)
where:
N(t) is the amount of the radioactive element remaining after time t
N₀ is the initial amount of the radioactive element
λ is the decay constant
e is the mathematical constant (approx. 2.71828)
The half-life (t½) of a radioactive element is the time it takes for half of the initial amount of the element to decay. The relationship between t½ and λ is:
t½ = ln(2) / λ
From the given information, the half-life (t½) of the radioactive element is 2 months. So we can calculate the decay constant λ as:
λ = ln(2) / t½ = ln(2) / 2 = 0.3466 (approx.)
Now we can use the exponential decay function to find the amount of the radioactive element remaining after 6 months:
N(6) = N₀ * e^(-λ6) = 1 * e^(-0.34666) = 0.125
Therefore, the amount of the radioactive element remaining after 6 months is 0.125 g or 1/8 of the initial amount.
So, the answer is (c) 1/8