Question

titanium-51 decays with a half life of 6 minutes. What fraction of the titanium would remain after one hour?​

Answer 1

Answer:

$\large \boxed{9.76 \times 10^{-4}}$

Explanation:

Let A₀ = the original amount of titanium-51.

The amount remaining after one half-life is ½A₀.

After two half-lives, the amount remaining is ½ ×½A₀ = (½)²A₀.

After three half-lives, the amount remaining is ½ ×(½)²A₀ = (½)³A₀, and so on.

We can write a general formula for the amount remaining:

A = A₀(½)ⁿ

where n is the number of half-lives

$n = \dfrac{t}{t_{\frac{1}{2}}}$

The fraction remaining is

$\text{Fraction} = \dfrac{A}{A_{0}} = \left (\dfrac{1}{2}\right)^{n}$

Data:

$t_{\frac{1}{2}} = \text{6 min}$

t = 1 h

Calculations:

(a) Convert the time to minutes

$t = \text{1 h} \times \dfrac{\text{60 min}}{\text{1 h}} = \text{60 min}$

(b) Calculate n

$n = \dfrac{60}{6} = 10$

(c) Calculate the fraction remaining

$\dfrac{A}{A_{0}} =  \left (\dfrac{1}{2}\right)^{10} =  \mathbf{9.76 \times 10^{-4}}\\\\\text{The fraction of titanium-51 remaining after 1 h is \large \boxed{\mathbf{9.76 \times 10^{-4}}} }$