Final-Answer:
To solve this problem, we can use the concept of radioactive decay and the formula for calculating the remaining amount of a substance based on its half-life.
The formula for the remaining amount of a radioactive substance after a certain time is given by:
[ N(t) = N0 times left(frac{1}{2}\right)^frac{t}{T{1/2}} ]
Where:
- (N(t)) is the amount remaining after time 't'
- (N0) is the initial amount
- (T{1/2}) is the half-life
- (t) is the time that has elapsed
Given that the half-life of carbon-10 is 19.29 seconds, and we have 1.53 kg of carbon-10, we want to find the time it takes until we have less than 10.0 g (which is 0.01 kg).
We'll start by converting the initial mass from kilograms to grams:
[1.53,kg = 1530,g]
Now, we'll set up the equation:
[0.01,kg = 1530,g times left(frac{1}{2}right)^frac{t}{19.29}]
To solve for 't', we can take the natural logarithm of both sides of the equation:
[ ln{0.01} = ln{1530} + frac{t}{19.29} times ln{left(frac{1}{2}right)} ]
[t = 19.29 times frac{ln{(1530)} - ln{(0.01)}}{ln{(0.5)}}]
Using a calculator, we can solve for 't' to find the time it will take until we have less than 10.0 g of carbon-10.
When we substitute the values and solve for 't':
[t approx 60.882s]
Therefore, it will take approximately 60.882 seconds until we have less than 10.0 g of carbon-10 remaining.
