Question

The half-life of radium-226 is 1,600 years and the half-life of carbon-14 is 5,730 years. In 1-2 sentences, mathematically prove that the decay constant for radium-226 is larger than the decay constant for carbon-14.

Answer 1

The decay constant for radium-226 is larger than the decay constant for carbon-14 due to its obtained value.

What is the rate of decay?

The reduction in the number of radionuclides per unit time is the rate of decay.

The decay constant of the radium-226 is;

$\lambda_1=\frac{ 16}{100}\times 266 \\\\ \lambda_1= -36.16$

The decay constant of the carbon-14 is;

$\lambda_2=\frac{ -57.3}{100}\times 14 \\\\ \lambda_1= -802.2$

Hence, the decay constant for radium-226 is larger than the decay constant for carbon-14.

To learn more about the rate of decay refer to the link;

https://brainly.com/question/14486631

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Answer 2

Answer:

  0.693/1600 > 0.693/5730

Explanation:

The decay constant is the reciprocal of the time it takes for the radioactivity to decline to about 36.8% of its original value. It is the value of k in the multiplier e^(-kt), which tells the fraction remaining after time t.

When compared to the half-life, h, the value of k is ...

  k = 0.693/h . . . . . decay constant for half-life h

Effectively, the decay constant is proportional to the inverse of the half-life. The constant of proportionality is ln(2) ≈ 0.693.

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decay constant for radium-226:

  0.693/1600 ≈ 4.33×10^-4

decay constant for carbon-14:

  0.693/5730 ≈ 1.21×10^-4

Since 4.33×10^-4 is greater than 1.21×10^-4, the decay constant of radium-226 is larger than that of carbon-14.

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Additional comment

Taking the reciprocal of a positive number reverses the ordering:

  1600 < 5730

  1/1600 > 1/5730

The half-life of radium-226 is less than that of carbon-14, so its decay constant will be larger.