if the half life of a radioactive isotope is 80 hours how long would it take for 75% of the isotope to decay
if 75% has decayed, 25% is left
1/4 = (1/2)^2
so it takes two half-lives, or 160 hours
To calculate the time it would take for 75 % of the isotope to decay, we need to use the concept of half-life.
The half-life is the time it takes for half of the radioactive substance to decay. In this case, the half-life of the isotope is given as 80 hours.
We can use the following formula to calculate the time it would take for a certain percentage of the radioactive substance to decay:
\[ \text{{Time}} = \text{{Half-life}} \times \left( \frac{{\log (1/\text{{percentage left}})}}{{\log 2}} \right) \]
Substituting the given values, the calculation becomes:
\[ \text{{Time}} = 80 \times \left( \frac{{\log (1/0.75)}}{{\log 2}} \right) \]
Now, let's calculate it step-by-step:
Step 1: Calculate the percentage left after decay:
\[ \text{{Percentage left}} = 1 - \text{{percentage decayed}} = 1 - 0.75 = 0.25 \]
Step 2: Calculate the natural logarithm of the inverse of the percentage left:
\[ \log (1/\text{{percentage left}}) = \log (1/0.25) \]
Using a scientific calculator, we find:
\[ \log (1/0.25) \approx 1.386 \]
Step 3: Calculate the natural logarithm of 2:
\[ \log 2 \approx 0.693 \]
Step 4: Substitute the values into the formula and calculate the time:
\[ \text{{Time}} = 80 \times \left( \frac{{1.386}}{{0.693}} \right) = 160 \]
Therefore, it would take approximately 160 hours for 75% of the isotope to decay.
To determine the time it takes for a certain percentage of a radioactive isotope to decay, we can use the concept of half-life. The half-life is the amount of time it takes for half of the substance to decay.
In this case, the half-life of the radioactive isotope is given as 80 hours. We want to find out how long it takes for 75% of the isotope to decay.
Let's break down the problem into steps:
Step 1: Calculate the number of half-lives required for 75% to decay.
Since the half-life is 80 hours, divide the percentage by 50% (half-life) to find the number of half-lives needed.
75% / 50% = 1.5 half-lives
Step 2: Determine the time it takes for the given number of half-lives.
Multiply the number of half-lives by the duration of one half-life.
1.5 half-lives * 80 hours = 120 hours
Therefore, it would take 120 hours for 75% of the radioactive isotope to decay.