Question

The half life of radium-226 is 1600 years. If you have 200 grams of radium today how many grams would be present in 8000 years?

Answer 1

`\boxed {\boxed {\sf 6.25 \ grams}}`

Step-by-step explanation:

We are asked to find the mass of a sample of radium-226 after half-life decay. We will use the following formula:

`A= A_o *(1)/(2)^{(t)/(h)}`

In this formula,
`A_o` is the initial amount, t is the time, and h is the half-life.

For this problem, the initial amount is 200 grams of radium-226, the time is 8,000 years, and the half-life is 1,600 years.

`\bullet \ A_o= 200 \ g \\\\bullet \ t= 8,000 \ \\\bullet \ h= 1,600`

Substitute the values into the formula.

`A= 200 \ g * (1)/(2) ^{(8.000)/(1,600)`

Solve the fraction in the exponent.

`A= 200 \ g * (1)/(2)^(5)`

Solve the exponent.

`A= 200 \ g *0.03125`

`A= 6.25 \ g`

In addition, we can solve this another way. First, we find the number of half-lives by dividing the total time by the half-life.

• 8,000/1,600= 5 half-lives

Every half-life, 1/2 of the mass decays. Divide the initial mass in half, then that result in half, and so on 5 times.

• 1. 200 g/2= 100 g
• 2. 100 g / 2 = 50 g
• 3. 50 g / 2 = 25 g
• 4. 25 g / 2 = 12.5 g
• 5. 12.5 g / 6.25 g

After 8,000 years, 6.25 grams of radium-226 remains.

Answer 2

Half life is the time taken by a radio active isotope to reduce by half of its original amount. Radium-226 has a half life of 1602 years meaning that it would take 1602 years for a mass of radium to reduce by half.

Number of half lives in 9612 years = 9612/1602 = 6 half lives

New mass = Original mass x (1/2)n where n is the number of half lives.

Therefore, New mass= 500 x (1/2)∧6

= 500 x 0.015625

= 7.8125 g

Hence the mass of radium after 9612 years will be 7.8125 grams.

Step-by-step explanation: