In 6.20 hours, a 100-gram sample of11247Ag decaysto 25.0 grams. What is the half-life of11247Ag?

25 = 100 * (1/2)^2

So that means that 6.20 hours is two half-lives.

To find the half-life of the isotope Ag-11247, we can use the formula:

N = N₀ * (1/2)^(t / T₁/₂),

where:
N is the final amount of the sample,
N₀ is the initial amount of the sample,
t is the time in hours,
T₁/₂ is the half-life of Ag-11247 in hours.

We are given the following information:
N₀ = 100 grams,
N = 25 grams,
t = 6.20 hours.

Now we can rearrange the formula to solve for T₁/₂:

(1/2)^(t / T₁/₂) = N / N₀.

Substituting the given values:

(1/2)^(6.20 / T₁/₂) = 25 / 100.

Let's solve this equation step-by-step to find the value of T₁/₂:

1. Take the logarithm of both sides of the equation to eliminate the exponent:

log(base 2)[(1/2)^(6.20 / T₁/₂)] = log(base 2)[25 / 100].

2. Use the power rule of logarithms to bring down the exponent:

(6.20 / T₁/₂) * log(base 2)(1/2) = log(base 2)(25 / 100).

3. Simplify the logarithm:

(6.20 / T₁/₂) * (-1) = log(base 2)(25 / 100).

4. Multiply both sides by (-T₁/₂) / (6.20) to isolate T₁/₂:

T₁/₂ = -6.20 / [(log(base 2)(25 / 100)) * (-1)].

5. Calculate the value inside the brackets:

T₁/₂ = -6.20 / (log(base 2)(0.25)).

Let's evaluate this expression using a calculator:

T₁/₂ ≈ -6.20 / (-2) ≈ 3.10 hours.

Therefore, the half-life of Ag-11247 is approximately 3.10 hours.

To determine the half-life of a radioactive substance, you can use the formula:

t₁/₂ = (ln(2)) / k

Where:
t₁/₂ is the half-life of the substance,
ln(2) is the natural logarithm of 2 (approximately 0.693), and
k is the decay constant.

In this case, we are given the initial mass (100 grams), the final mass (25 grams), and the time (6.20 hours). We need to find the decay constant, k, in order to calculate the half-life.

First, we calculate the fraction of the substance remaining after the given time:

(Remaining mass) / (Initial mass) = e^(-kt)

Where e is Euler's number (approximately 2.718) and t is the time given (6.20 hours). Rearranging the equation gives:

k = - (ln((Remaining mass) / (Initial mass))) / t

Plugging in the values, we can calculate the decay constant, k:

k = - (ln(25 / 100)) / 6.20

Next, we substitute this value of k into the half-life formula to find the half-life, t₁/₂:

t₁/₂ = (ln(2)) / k

Calculating this equation will give us the half-life of the substance.