A researcher made160mg of radioactive sodium(Na24) and found that there was only 20 mg left after 45 hours. a). What is the half-life of Na24?
Since the amount of NA24 is decreasing, let's apply the formula of exponential decay, which is:
`y = ae^(kt)`
where y- amount of substance at time t, a - amount of substance at the start and k - rate of decay.
To solve for the half-life of NA24, we have to determine k first. So, substitute y=20, a=160 and t=45 to the formula.
To simplify the equation, divide both sides by 160.
To bring down k, take the natural logarithm of both sides.
`ln0.125 = lne^(45k)`
Then at the right side of the equation, apply the power property of logarithm which is ` ln x^m= m*ln x ` .
Note that ln e= 1.
And divide both sides by 45.
Now that we know the value of k, let's solve proceed to solve for half-life of NA24.
Note that half-life means the time when y = a/2. So to determine t, substitute y = a/2 and k=-0.046 to the formula of exponential decay and follow the steps above.
Divide both sides by a.
Take the natural logarithm of both sides.
Then, apply the power property of logarithm.
And, divide both sides by -0.046.
Hence, the half-life of NA24 is 15.068 hours.