The half life of radium is 3.82 days. In what time will the activity decay to (1/16) of its original value.
1/16 = (1/2)^n
0.0625 = 0.5^n
log 0.0625 = n log 0.5
-1.20412 = n * -0.30103
n = 4 half lives
4 * 3.82 = 15.3 days
To find the time it takes for the activity of radium to decay to (1/16) of its original value, we can use the formula for exponential decay:
Activity = Initial Activity * (1/2)^(t/h)
Where:
- Activity = current activity level
- Initial Activity = original activity level
- t = time passed
- h = half-life
We know that the half-life of radium is 3.82 days. Let's denote the time it takes for the activity to decay to (1/16) of its original value as T.
Given that the activity at time T is (1/16) of its initial value, we can substitute these values into the formula:
(1/16) * Initial Activity = Initial Activity * (1/2)^(T/3.82)
We can simplify this equation by canceling out the Initial Activity on both sides:
1/16 = 1/2^(T/3.82)
Now, let's solve for T:
1/2^(T/3.82) = 1/16
To get rid of the fractions, we can rewrite 1/16 as 2^(-4):
1/2^(T/3.82) = 2^(-4)
We can compare the exponents on both sides of the equation:
T/3.82 = -4
To isolate T, we can multiply both sides of the equation by 3.82:
T = -4 * 3.82
Calculating this value gives us:
T = -15.28
Since time cannot be negative, we discard the negative value of -15.28. Therefore, the time it takes for the activity of radium to decay to (1/16) of its original value is approximately 15.28 days.
To determine the time it will take for the activity of radium to decay to (1/16) of its original value, we need to know the number of half-lives it will take to reach this point.
The formula for calculating the number of half-lives is:
Number of half-lives = log (final activity / initial activity) / log(1/2)
In this case, the final activity is (1/16) of the initial activity (or 1/16 times the original value). So the final activity is 1/16.
Plugging in the values in the formula:
Number of half-lives = log (1/16) / log(1/2)
Using logarithm identities:
Number of half-lives = log (1/16) / log(2)
Calculating the numerator:
log (1/16) = log(1) - log (16) = 0 - log(2^4) = -4 log (2) = -4 * log (2) ≈ -4 * 0.3010 ≈ -1.204
Calculating the denominator:
log(2) ≈ 0.3010
Plugging in the values:
Number of half-lives ≈ -1.204 / 0.3010 ≈ -4
Since the number of half-lives cannot be negative, the absolute value of the number of half-lives is taken, which is 4.
Now, to find the time it will take for the activity to decay to (1/16) of its original value, we multiply the number of half-lives by the half-life of radium.
Time = Number of half-lives * Half-life
Time = 4 * 3.82 days ≈ 15.28 days
Therefore, it will take approximately 15.28 days for the activity of radium to decay to (1/16) of its original value.