Question

An important application of exponential functions is working with half-life of radioactive isotopes in chemistry. These isotopes emit particles and decay into stable forms, in doing so they lose mass over time. Half-life of an isotope is the time it takes for the amount to decay by half. For example, the half life of Bromine-85 is 3 minutes. This means if you start with 60g of Br-85, 3 minutes later 30g will remain. How much Br-85 will remain after 20 minutes?

Answer 1

Approximately 0.94 grams of Bromine-85 will remain after 20 minutes, calculated by using the exponential decay formula for half-lives.

Step-by-step explanation:

To calculate the remaining amount of Bromine-85 after any number of half-lives, you can use the formula:

N = No(1/2)ⁿ

Where:

• N is the remaining quantity
• No is the original quantity
• n is the number of half-lives elapsed

Since we know that the half-life of Bromine-85 is 3 minutes and we are looking at a time span of 20 minutes, we divide the total time by the half-life:

20 minutes / 3 minutes per half-life = 6.67 half-lives

We can round down to 6 half-lives as we can't have a fraction of a half-life. Now using the original quantity of 60g:

N = 60g (1/2)⁶

N = 60g (1/64)

N = 0.9375g

So, approximately 0.94 grams of Bromine-85 will remain after 20 minutes.

Answer 2

Answer : The amount left after 20 minutes is, 0.592 grams.

Explanation :

Half-life of Bromine-85 = 3 min

First we have to calculate the rate constant, we use the formula :

`k=\frac{0.693}{3\text{ min}}`

`k=0.231\text{ min}^(-1)`

Now we have to calculate the amount left after decay.

Expression for rate law for first order kinetics is given by:

`t=(2.303)/(k)\log(a)/(a-x)`

where,

k = rate constant

t = time taken by sample = 20 min

a = initial amount of the reactant = 60 g

a - x = amount left after decay process = ?

Now put all the given values in above equation, we get

`20=(2.303)/(0.231)\log(60)/(a-x)`

`a-x=0.592g`

Therefore, the amount left after 20 minutes is, 0.592 grams.