Understanding the Half-Life of Cadmium

Radioactive decay is a significant concept in nuclear science and chemistry, with cadmium providing an interesting case study. Let's explore a specific scenario involving cadmium and how to determine its half-life.

The Radioactive Decay Formula and Its Application to Cadmium

The formula for radioactive decay can be expressed as follows:

[ A A_0 times 0.5^{t/h} ]

Where ( A ) is the final amount of substance remaining, ( A_0 ) is the initial amount, ( t ) is the time elapsed, and ( h ) is the half-life.

Given Data for Cadmium

Consider that 100 grams of cadmium remain until only 12.5 grams are present after 17,190 years. To find the half-life of cadmium, we need to determine ( h ).

Deriving the Half-Life

Start with the formula: Solve for the half-life ( h ): Take the logarithm of both sides: Perform the logarithmic calculation: Finally, solve for ( h ):

Let's apply this formula step by step:

Given: ( A 12.5 , text{g} ), ( A_0 100 , text{g} ), and ( t 17,190 , text{years} ).

Rearrange the formula:

[ 12.5 , text{g} 100 , text{g} times 0.5^{17190/h} ]

Divide both sides by 100 grams:

[ 0.125 0.5^{17190/h} ]

Take the logarithm (base 10) of both sides:

[ log(0.125) left(frac{17190}{h}right) log(0.5) ]

Substitute the logarithmic values:

[ -0.9031 left(frac{17190}{h}right) times (-0.3010) ]

Simplify and solve for ( h ):

[ frac{17190}{h} frac{0.9031}{0.3010} 3 ]

Therefore:

[ h frac{17190}{3} 5730 , text{years} ]

Hence, the half-life of cadmium is 5730 years.

Visual Representation of Half-Life Periods

To illustrate this concept more clearly, let's consider the mass reduction over half-lives:

100 grams → 50 grams → 25 grams → 12.5 grams This sequence represents 3 half-lives.

Therefore, 17,190 years divided by 3 half-lives equals 5,730 years per half-life:

[ frac{17190}{3} 5730 , text{years} ]

Conclusion

The half-life of cadmium is a crucial parameter in understanding its decay behavior. This value allows for accurate predictions and calculations in fields such as radiometric dating and the safe handling of radioactive materials.

FAQ

How many grams of cadmium remain after the first half-life?

After the first half-life, 50 grams of cadmium remain.

How many grams after the second half-life?

After the second half-life, 25 grams of cadmium remain.

How many grams after the third half-life?

After the third half-life, 12.5 grams of cadmium remain.

Divide 17,190 by the number of half-life periods?

A half-life period is 5,730 years, so 17,190 divided by 3 equals 5,730 years.

Is cadmium with such a long half-life common in the universe?

Given the context of known elements and their half-lives, cadmium with a half-life of 5,730 years is unusual and not representative of typical elements in our universe.