Question

A radioactive sample contains 3.00 g of an isotope with a half-life of 3.8 days.

How much of the isotope in grams will remain after 19.8 days?

Answer 1

The isotope in grams will remain after 19.8 days would be 0.081 grams.

The formula to calculate the left mass of a radioactive element can be deduced as -

`\qquad\star\longrightarrow \underline{\boxed{\sf{m =m_(o) * { \bigg((1)/(2) \bigg)}^{ (t)/(T½)} }}} \\`

Where-

• 
  `\sf m_(o)`is the initial mass of a radioactive element
• T½ is the half life time
• t is the time period
• m = Left mass of a radioactive element.

According to the given specific parameters -

• Initial mass,
  `\sf m_(o)` = 3 g
• Half life time, T½= 3.8 days
• Time period, t =19.8 days

Now that we have all the required values, so we can plug them into the formula and solve for the left mass of a radioactive element-

`\qquad \longrightarrow \sf \underline{m =m_(o) * { \bigg((1)/(2) \bigg)}^{ (t)/(T½) }} \\`

`\qquad\longrightarrow \sf m =3 * { \bigg((1)/(2) \bigg)}^{ (19.8)/(3.8) } \\`

`\qquad \longrightarrow \sf m =3 * { \bigg((1)/(2) \bigg)}^{ \frac{\cancel{19.8}}{\cancel{3.8}} } \\`

`\qquad\longrightarrow \sf m =3 * { \bigg((1)/(2) \bigg)}^( 5.21052..... ) \\`

`\qquad\longrightarrow \sf m =3 * 0.02700... \\`

`\qquad\longrightarrow \sf m =0.081020....\;g \\`

`\qquad\longrightarrow \sf \underline{m =\boxed{\sf{0.081\;g}}} \\`

• Henceforth,about 0.081 g of the isotope in grams will remain after 19.8 days.

Answer 2

Answer:So, about 0.093 g of the isotope will remain after 19.8 days.

Step-by-step explanation:

The first step is to find the number of half-lives that have passed during 19.8 days:

Number of half-lives = time elapsed / half-life

Number of half-lives = 19.8 days / 3.8 days per half-life

Number of half-lives ≈ 5.21

This means that the initial amount of the isotope has been halved 5.21 times. The remaining fraction of the original amount can be calculated using the following formula:

Remaining fraction = (1/2)^(number of half-lives)

Substituting the values, we get:

Remaining fraction = (1/2)^5.21

Remaining fraction ≈ 0.031

Therefore, the amount of the isotope remaining after 19.8 days is:

Remaining amount = Remaining fraction x Initial amount

Remaining amount = 0.031 x 3.00 g

Remaining amount ≈ 0.093 g

So, about 0.093 g of the isotope will remain after 19.8 days.