The relation between half`-`life period `(t_(1//2))` and disintegration constant `(lambda)` is expressed as

Options

A. `lambda=(0.693)/(t_(1//2))`
B. `lambda=0.693t_(1//2)`
C. `lambda=(693)/(t_(1//2))`
D. `lambda=693t_(1//2)`

Explanation

To derive the relationship between the half-life period (t₁/₂) and the disintegration constant (λ), we can follow these steps:

### Step 1: Understand the Concept of Half-Life
The half-life period (t₁/₂) is defined as the time required for half of the radioactive nuclei in a sample to decay. This is a crucial concept in nuclear chemistry as it helps in understanding the stability and decay of radioactive isotopes.

**Hint:** Remember that half-life is about the time it takes for half of a substance to decay.

### Step 2: Recognize the Nature of Radioactive Decay
Radioactive decay follows first-order kinetics. This means that the rate of decay of a radioactive substance is directly proportional to the number of undecayed nuclei present at any time.

**Hint:** First-order kinetics implies that the rate of reaction depends on the concentration of the reactant.

### Step 3: Write the First-Order Rate Equation
For a first-order reaction, the rate of decay can be expressed as:
\[ \frac{dN}{dt} = -\lambda N \]
where:
- \( N \) = number of undecayed nuclei
- \( \lambda \) = disintegration constant

**Hint:** The negative sign indicates that the number of undecayed nuclei decreases over time.

### Step 4: Integrate the Rate Equation
Integrating the above equation gives us the relationship between the number of undecayed nuclei and time. The integrated form is:
\[ N(t) = N_0 e^{-\lambda t} \]
where \( N_0 \) is the initial number of nuclei.

**Hint:** The exponential function describes how the quantity decreases over time.

### Step 5: Define Half-Life in Terms of N
At half-life (t = t₁/₂), the number of undecayed nuclei is half of the initial amount:
\[ N(t_{1/2}) = \frac{N_0}{2} \]

**Hint:** Substitute \( N(t_{1/2}) \) into the integrated equation.

### Step 6: Set Up the Equation for Half-Life
Substituting into the integrated equation gives:
\[ \frac{N_0}{2} = N_0 e^{-\lambda t_{1/2}} \]

### Step 7: Simplify the Equation
Dividing both sides by \( N_0 \) (assuming \( N_0 \neq 0 \)):
\[ \frac{1}{2} = e^{-\lambda t_{1/2}} \]

### Step 8: Take the Natural Logarithm
Taking the natural logarithm of both sides:
\[ \ln\left(\frac{1}{2}\right) = -\lambda t_{1/2} \]

### Step 9: Solve for Half-Life
Using the property of logarithms, we know that \( \ln\left(\frac{1}{2}\right) = -\ln(2) \):
\[ -\ln(2) = -\lambda t_{1/2} \]
Thus, we can rearrange to find:
\[ \lambda t_{1/2} = \ln(2) \]

### Step 10: Final Relationship
Rearranging gives us the final relationship:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
Since \( \ln(2) \approx 0.693 \), we can express it as:
\[ \lambda = \frac{0.693}{t_{1/2}} \]

### Conclusion
The relationship between the half-life period (t₁/₂) and the disintegration constant (λ) is:
\[ \lambda = \frac{0.693}{t_{1/2}} \]

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