A radioactive substance has a half-life of 27 years. Find an expression for the amount of the substance at time t if 20 grams were present initially.
mIHOVIL NEJGEVEN
20 * (1/2)^(t/27)
To find the expression for the amount of the substance at any time t, we need to use the concept of exponential decay.
The equation for exponential decay is given by:
A(t) = A₀ * e^(-kt)
Where:
A(t) is the amount of substance remaining at time t
A₀ is the initial amount of the substance
e is the base of the natural logarithm system (~2.71828)
k is the decay constant, which is related to the half-life of the substance
t is the time elapsed
In this case, the half-life of the substance is given as 27 years. The decay constant (k) can be calculated using the formula:
k = ln(2) / half-life
Substituting the given values into the equation, we get:
k = ln(2) / 27
Now, we can substitute the given initial amount (A₀ = 20 grams) and the calculated decay constant (k) into the exponential decay equation to find the expression for the amount of the substance at time t:
A(t) = 20 * e^(-kt)
Replacing k with ln(2) / 27, the expression becomes:
A(t) = 20 * e^(-t * ln(2) / 27)
Thus, the expression for the amount of the substance at time t, given an initial amount of 20 grams and a half-life of 27 years, is:
A(t) = 20 * e^(-t * ln(2) / 27)