If 12 % of a radioactive substance decays in 7 hours, what is the half-life of the substance?
Carry out all calculations exactly, then round the final answer to three decimal places.
The half life is what hours?
an interesting thought, carrying out log calculations exactly. Would require an infinite amount time adding up an infinite series.
But more practical, not exact, but using approximations from ones calculator.. if .12 decays,then .88 remains..
.88=e^(t*ln(.5)/th)
t=7hrs, solve for th
take ln of each side
ln (.88)=-ln(.5)*7 / th
thalflife= 7(ln(.5)/ln.88
To find the half-life of the substance, we need to determine how long it takes for half of the substance to decay.
Given that 12% of the substance decays in 7 hours, we can calculate the fraction of the substance remaining after 7 hours:
Remaining fraction = 1 - 0.12 = 0.88
Now, we need to find the time it takes for the remaining fraction to reach half, which is 0.5:
0.88 * (1/2) = 0.44
Let's call the half-life of the substance "t". Using the exponential decay formula:
0.88 * (1/2)^(7/t) = 0.44
Simplifying the equation:
(1/2)^(7/t) = 0.5
Taking the logarithm of both sides:
log[(1/2)^(7/t)] = log(0.5)
Using the logarithmic property: log(a^b) = b*log(a):
(7/t) * log(1/2) = log(0.5)
Recall that log(1/2) = -log(2):
(7/t) * (-log(2)) = log(0.5)
Rearranging the equation:
(7/t) = log(0.5) / (-log(2))
Simplifying the right side:
(7/t) = log(0.5) / log(2)
Now, let's evaluate this expression:
log(0.5) ≈ -0.301
log(2) ≈ 0.301
Substituting these values back into the equation:
(7/t) = -0.301 / 0.301
Dividing both sides by 7:
1/t = (-0.301 / 0.301) / 7
Simplifying the right side:
1/t = -1 / 7
Taking the reciprocal of both sides:
t = -7 / 1
Rounding the final answer to three decimal places:
t ≈ -7 hours
Since time cannot be negative, we discard the negative sign. Therefore, the half-life of the substance is approximately 7 hours.
To determine the half-life of a substance, we need to find the time it takes for half of the substance to decay. In this case, we are given that 12% of the substance decays in 7 hours.
To calculate the half-life, we can set up the following equation:
0.12 * C = C/2
Where C is the initial amount of the substance.
Simplifying the equation, we can multiply both sides by 2:
0.24 * C = C
Next, we divide both sides by C:
0.24 = 1
This tells us that 0.24 is equal to 1, which is not possible. It means there must be an error in our equation.
Let's try setting up the equation again:
0.12 * C = C * (1/2)
This equation correctly represents that 12% of the substance should equal half the initial amount.
Now, we can solve for C:
0.12 * C = C/2
Multiplying both sides by 2:
0.24 * C = C
Subtracting C from both sides:
0.24 * C - C = 0
Combining like terms:
0.24C - C = 0
Factoring out C:
C(0.24 - 1) = 0
Simplifying:
C(-0.76) = 0
Dividing both sides by -0.76:
C = 0 / -0.76
C = 0
This tells us that the initial amount of the substance is 0, which does not make sense. We have made an error in our calculations.
Upon reviewing the problem, it seems that our setup of the equation was incorrect. We should have set up the equation as follows:
(1 - 0.12) * C = C/2
Simplifying,
0.88 * C = C/2
Now, let's solve for C:
0.88 * C = C * (1/2)
0.88 = 1/2
C = (1/2) / 0.88
C = 0.5681
Therefore, the initial amount of the substance is approximately 0.5681.
Since we are looking for the half-life of the substance, we can use the formula:
t = (ln(2))/k
Where t represents the half-life and k is the decay constant. In this case, k can be found by taking the natural logarithm of the fraction of remaining substance after one half-life.
k = ln(1/2) / t
Substituting the values we have:
k = ln(1/2) / 7
Using a calculator, we find:
k ≈ -0.0995
Finally, we can substitute the value of k into the half-life formula:
t = ln(2) / k
t = ln(2) / -0.0995
Using a calculator, we find:
t ≈ 6.966
Rounded to three decimal places, the half-life of the substance is approximately 6.966 hours.