Nuclear Decay of Sodium-22: Understanding Half Life, Exponential Decay, and Calculation of Remaining Mass

Nuclear decay is a fundamental process in radiology and nuclear physics, where unstable atomic nuclei lose energy by emitting radiation. This process is governed by precise mathematical formulas that help predict the behavior of radioactive isotopes.

One of the key isotopes in this context is 22Na, or Sodium-22, which has a half-life of approximately 15 hours. This means that it takes 15 hours for half of the radioactive nuclei to decay into a more stable form, typically 22Mg (Magnesium-22).

Understanding the Half-Life Concept

The half-life of a radioisotope is the time required for one-half of the amount of unstable material to degrade into a more stable material. For Sodium-22, this half-life is 15 hours. This concept is crucial for understanding how radioactive isotopes behave over time.

Calculating the Remaining Mass After 5 Hours

Given a sample of Sodium-22 with an initial mass of 2 grams, we can calculate the remaining mass after 5 hours using the formula for exponential decay:

Exponential Decay Formula

Nt N0 (1/2)t/T1/2

Where:

Nt is the remaining mass after time t. N0 is the initial mass of the sample. T1/2 is the half-life of the isotope. t is the elapsed time.

For our example:

N_0 2 grams T_{1/2} 15 hours t 5 hours

Plugging these values into the formula:

N_5 2(1/2)5/15 2(1/2)1/3

Calculating the exponent span(1/2)1/3

span(1/2)1/3

Now calculating the remaining mass:

N_5 ≈ 2 * 0.7937 1.5874 grams

Calculating Time to Decay to 0.4 Grams

To find the time it takes for the sample to decay to 0.4 grams, we rearrange the decay formula:

0.4 2(1/2)t/15

Dividing both sides by 2:

(1/2) (1/2)t/15

Taking the logarithm of both sides:

log(1/2) t/15 log(1/2)

Now solve for t:

t 15 * [log(1/2) / log(1/2)]

Calculating the logarithms:

log(1/2) ≈ -0.30103

Substitute the values into the equation:

t ≈ 15 * [-0.69897 / -0.30103] ≈ 15 * 2.32193 ≈ 34.82 hours

Summary:

The mass remaining after 5 hours is approximately 1.59 grams. The time to decay to 0.4 grams is approximately 34.82 hours.

Conclusion

Understanding the half-life and exponential decay of isotopes like Sodium-22 is crucial for applications in radiology, nuclear physics, and environmental science. This knowledge helps in predicting the behavior of radioactive materials and ensures safe handling and disposal practices.