A radioactive substance is found to register 1000 counts per second on a Geiger counter. Twenty-four hours later it registers 125 counts per second. What is its half-life?
To determine the half-life of the radioactive substance, we can use the formula:
N = N₀ * (1/2)^(t / T)
Where:
N is the final count rate (125 counts per second)
N₀ is the initial count rate (1000 counts per second)
t is the time passed (24 hours)
T is the half-life we are trying to find
Let's substitute the values into the equation and solve for T:
125 = 1000 * (1/2)^(24 / T)
Divide both sides of the equation by 1000:
125 / 1000 = (1/2)^(24 / T)
Simplify the left side:
0.125 = (1/2)^(24 / T)
Take the logarithm of both sides of the equation:
log(0.125) = log((1/2)^(24 / T))
Using the power rule of logarithms:
log(0.125) = (24 / T) * log(1/2)
Since log(1/2) can be simplified to -(log2), we have:
log(0.125) = -(24 / T) * log2
Now, we can solve for T:
T = -24 / (log(0.125) / log2)
Calculating the values we get:
T ≈ -24 / (-3 / 0.301)
T ≈ 24 / (3 / 0.301)
T ≈ 8 / (1 / 0.301)
T ≈ 8 * 0.301
T ≈ 2.408
Therefore, the half-life of the radioactive substance is approximately 2.408 hours.
To determine the half-life of the radioactive substance, we can follow these steps:
Step 1: Calculate the decay constant (λ)
The decay constant (λ) is a measure of how quickly the substance decays. We can find it using the formula:
λ = ln(2) / T
Where ln denotes the natural logarithm and T is the time interval between two measurements.
In this case, the time interval is 24 hours, or in seconds, 24 hours × 60 minutes/hour × 60 seconds/minute = 86,400 seconds.
Therefore:
λ = ln(2) / 86,400 seconds
Step 2: Calculate the half-life (T½)
The half-life (T½) is the time it takes for half of the radioactive substance to decay. We can calculate it using the formula:
T½ = ln(2) / λ
Using the value of λ we calculated in Step 1:
T½ = ln(2) / (ln(2) / 86,400 seconds)
Simplifying:
T½ = 86,400 seconds
Hence, the half-life of the radioactive substance is 86,400 seconds, which is equivalent to 24 hours.
125 is 1000/8, so that would be 3 half-lives.
24/3 = 8 hours