Question

A radioactive sample has ${\mathrm{n}}_0$ active atom at $\mathrm{t}=\mathrm{o}$, at the rate of disintegration at any time is $\mathrm{R}$ and the number of atom is $\mathrm{N}$, then ratio. $(\mathrm{R}/\mathrm{N})$ varies with time $(\mathrm{t})$ as.

Short answer

The ratio $\frac{R}{N}$ is equal to the decay constant λ and remains constant with time. Therefore, the ratio does not vary with time.

Step-by-step solution

Radioactive Decay Law

The radioactive decay law states that the rate of decay of a radioactive substance is proportional to the number of active atoms present at any given time. Mathematically, this can be represented as: $R=\lambda N$ where R is the rate of decay at time t, N is the number of active atoms at time t, and λ is the decay constant.

Exponential Decay Equation

We can also write an expression for N, the number of active atoms at any time t, in terms of the initial number of active atoms n₀: $N=n_0{e}^{-\lambda t}$ where N is the number of active atoms at time t, n₀ is the initial number of active atoms, λ is the decay constant, and t is time.

Express R in terms of N

We can write the rate of decay R in terms of the number of active atoms N and the initial number of active atoms n₀ using the radioactive decay law: $R=\lambda N$ Substitute the expression for N from Step 2: $R=\lambda (n_0{e}^{-\lambda t})$

Calculate R/N as a function of t

From the expression for R in Step 3, divide by N to find the ratio R/N as a function of time t: $\frac{R}{N}=\frac{\lambda (n_0{e}^{-\lambda t})}{(n_0{e}^{-\lambda t})}$ Simplify, by canceling the common terms (n₀ and e⁽⁻ᴸᵗ⁾): $\frac{R}{N}=\lambda$

Conclusion

The ratio R/N is equal to the decay constant λ and remains constant with time. Therefore, the ratio does not vary with time.

Key concepts

Decay Constant

In the world of radioactive decay, the decay constant, commonly denoted by the Greek letter $\lambda$, holds significant importance. It's a key factor in understanding how quickly or slowly a radioactive substance undergoes disintegration. The decay constant is a fixed value specific to each radioactive substance. It essentially represents the probability per unit time that an individual atom will decay.
Some points to remember about the decay constant include:

• It is a characteristic of the material: each radioactive isotope has its own decay constant.
• The larger the decay constant, the faster the substance decays.
• It remains constant over time for a given isotope.

Knowing the decay constant allows scientists and researchers to predict the behavior of radioactive materials over time. This predictability makes it easier to calculate the remaining active atoms and the amount of time it takes for a particular sample to decrease to half its initial quantity, known as the half-life.

Exponential Decay

Exponential decay is a process that describes how the quantity of radioactive atoms decreases over time. This term reflects a mathematical model, wherein the number of undecayed atoms decreases at a rate proportional to the current number.The equation that governs radioactive material under exponential decay is formulated as follows:$$N=n_0{e}^{-\lambda t}$$Here, $N$ represents the number of atoms at time $t$, $n_0$ is the initial number of atoms, $\lambda$ is the decay constant, and $t$ is time.
A few additional insights regarding this model:

• This provides a smooth curve when graphed, indicating a decrease over time without fluctuations.
• Such behavior is not unique only to radioactive decay but also applies to other natural phenomena, including biological processes and population declines.
• The model demonstrates that no matter what the initial quantity is, it will ultimately reduce to zero.

Understanding exponential decay enables effective management and prediction of various systems where decay is a consideration, including nuclear waste management and even financial models with decay-like characteristics.

Rate of Disintegration

The rate of disintegration, often referred to as the decay rate $R$, signifies the number of particles or atoms decaying per unit time. This rate is a critical measure because it quantifies how rapidly a sample loses its radioactivity.
According to the radioactive decay law, the rate at any given time $t$ is expressed as:$$R=\lambda N$$Where $\lambda$ is the decay constant and $N$ is the number of active atoms remaining at time $t$.
Here are some important facts about the rate of disintegration:

• It is directly proportional to both the decay constant and the number of remaining atoms.
• As more atoms decay, the rate of disintegration decreases, reflecting the diminishing quantity of radioactive material.
• The initial rate is highest when $N$ is equal to the initial number of atoms $n_0$.

This concept helps determine the safety and timeframe for handling radioactive materials, ensuring they are kept under control before substantial decay lowers the rate significantly.