Sodium-24 has a half-life of approximately 15 hours. If only one-eighth of the sodium-24 remains, about how much time has passed?
a
15 hours
b
45 hours
c
30 hours
d
60 hours
The correct answer is c) 30 hours.
The half-life of sodium-24 is 15 hours. This means that after 15 hours, half of the initial amount of sodium-24 will remain, and the other half will have decayed.
If only one-eighth of the sodium-24 remains, it means that 7/8 of the initial amount has decayed.
Since each half-life is 15 hours, we can determine how many half-lives have passed by dividing 7/8 by 1/2:
(7/8)/(1/2) = (7/8) * (2/1) = 7/4
This means that 7/4 of a half-life has passed.
To find the time that has passed, we multiply the number of half-lives by the length of each half-life:
(7/4) * 15 hours = 105/4 hours = 26.25 hours ≈ 30 hours (rounded to the nearest hour)
To find out how much time has passed given that only one-eighth of Sodium-24 remains, we can use the concept of half-life.
First, let's determine the number of half-lives that have occurred. Since one-eighth is left, it means seven-eighths (8 - 1 = 7) have decayed.
The number of half-lives can be calculated using the formula:
Number of half-lives = log(base 1/2) (fraction remaining)
Number of half-lives = log(base 1/2) (1/8)
Using the logarithm property:
Number of half-lives = log(base 2) (1/8)
Since 2^3 = 8, we can rewrite the fraction as:
Number of half-lives = log(base 2) (2^-3)
Using the logarithm property:
Number of half-lives = -3 log(base 2) (2)
Number of half-lives = -3 * 1
Number of half-lives = -3
Therefore, three half-lives have occurred.
Given that the half-life of Sodium-24 is approximately 15 hours, we can multiply the number of half-lives by the half-life to find how much time has passed.
Time passed = Number of half-lives * Half-life
Time passed = 3 * 15 hours
Time passed = 45 hours
Therefore, approximately 45 hours have passed. Hence, the correct answer is option b) 45 hours.
To find out how much time has passed, we need to determine the number of half-lives that have occurred. Since the half-life of sodium-24 is 15 hours and only one-eighth of the original amount remains, this means that seven-eighths of the sodium-24 has decayed.
To solve this, we can set up the equation:
(1/2)^(n) = 1/8
Here, n represents the number of half-lives that have occurred.
To simplify, we can convert 1/8 to 2^(-3):
(1/2)^(n) = 2^(-3)
Next, we can equate the exponents:
n = - 3
Therefore, the number of half-lives that have occurred is -3. Since time cannot be negative, we know that something is amiss with the equation.
The mistake lies in the equation itself. The correct equation should be:
(1/2)^(n) = 7/8
Setting up this equation:
(1/2)^(n) = 7/8
To simplify, we can convert 7/8 to 2^(-1):
(1/2)^(n) = 2^(-1)
We can now equate the exponents:
n = - 1
Since n represents the number of half-lives and cannot be negative, we take the absolute value of -1:
n = 1
This means that one half-life has occurred. Multiply the half-life (15 hours) by the number of half-lives to find the total time that has passed:
15 hours * 1 = 15 hours
Thus, the correct answer is a) 15 hours.