Two radioactive nuclei X & Y initially contain an equal number of atoms. Their half life is 1 hour and 2 hour respectively. calculte the ratio of their rates of disintegration after two hours

To find the ratio of the rates of disintegration of two radioactive nuclei, X and Y, after two hours, we need to understand how radioactive decay works, particularly focusing on the concept of half-life and the decay constant.

Understanding Half-Life and Decay Constant

The half-life of a radioactive substance is the time it takes for half of the radioactive atoms in a sample to decay. For nuclei X and Y, the half-lives are given as follows:

• Nucleus X: 1 hour
• Nucleus Y: 2 hours

The decay constant (λ) is a measure of the probability of decay per unit time and is related to the half-life (T_1/2) by the formula:

λ = ln(2) / T_1/2

Calculating the Decay Constants

Let's calculate the decay constants for both nuclei:

• For nucleus X:

λ_X = ln(2) / 1 hour ≈ 0.693 per hour

• For nucleus Y:

λ_Y = ln(2) / 2 hours ≈ 0.3465 per hour

Determining the Number of Atoms Remaining

Next, we need to find out how many atoms remain after two hours for both nuclei. The formula to calculate the remaining quantity of a radioactive substance is:

N(t) = N₀ * e^-λt

Where:

• N(t) = number of atoms remaining at time t
• N₀ = initial number of atoms
• λ = decay constant
• t = time elapsed

Calculating Remaining Atoms After Two Hours

Assuming both nuclei start with the same number of atoms, let's denote this initial number as N₀.

For nucleus X after 2 hours:

N_X(2) = N₀ * e^{-0.693 * 2} = N₀ * e^-1.386 ≈ N₀ * 0.25

For nucleus Y after 2 hours:

N_Y(2) = N₀ * e^{-0.3465 * 2} = N₀ * e^-0.693 ≈ N₀ * 0.5

Calculating the Rates of Disintegration

The rate of disintegration (R) can be calculated using the formula:

R = λ * N

Finding the Rates for X and Y

Now, we can calculate the rates of disintegration for both nuclei after two hours:

For nucleus X:

R_X = λ_X * N_X(2) = 0.693 * (N₀ * 0.25) = 0.17325 N₀

For nucleus Y:

R_Y = λ_Y * N_Y(2) = 0.3465 * (N₀ * 0.5) = 0.17325 N₀

Calculating the Ratio of Rates

Finally, we can find the ratio of the rates of disintegration:

Ratio R_X : R_Y = 0.17325 N₀ : 0.17325 N₀ = 1 : 1

Thus, the ratio of the rates of disintegration of nuclei X and Y after two hours is 1:1. This means that, despite their different half-lives, the rates of disintegration are equal at this specific time point due to the remaining quantities of each nucleus.