The radioactive of a sample is `R_(1)` at a time `T_(1)` and `R_(2)` at a time `T_(2)`. If the half-life of the specimen is `T`, the number of atoms that have disintegrated in the time `(T_(2)-T_(1))` is equal to`(n(R_(1)-R_(2))T)/(ln4) `. Here n is some integral number. What is the value of n?

Explanation

To solve the problem, we need to analyze the decay of a radioactive sample and relate the given quantities. Let's break it down step by step.

### Step 1: Understand the relationship between decay rate and number of atoms
The decay rate \( R \) of a radioactive sample is given by:
\[ R = \lambda N \]
where \( \lambda \) is the decay constant and \( N \) is the number of radioactive atoms present at that time.

### Step 2: Express the number of atoms at times \( T_1 \) and \( T_2 \)
At time \( T_1 \), the rate is \( R_1 \):
\[ R_1 = \lambda N_1 \]
At time \( T_2 \), the rate is \( R_2 \):
\[ R_2 = \lambda N_2 \]
From these equations, we can express the number of atoms:
\[ N_1 = \frac{R_1}{\lambda}, \quad N_2 = \frac{R_2}{\lambda} \]

### Step 3: Determine the number of disintegrated atoms
The number of atoms that have disintegrated between times \( T_1 \) and \( T_2 \) is given by:
\[ \Delta N = N_1 - N_2 = \frac{R_1}{\lambda} - \frac{R_2}{\lambda} = \frac{R_1 - R_2}{\lambda} \]

### Step 4: Relate decay constant \( \lambda \) to half-life \( T \)
The decay constant \( \lambda \) is related to the half-life \( T \) by:
\[ \lambda = \frac{\ln 2}{T} \]

### Step 5: Substitute \( \lambda \) into the expression for \( \Delta N \)
Substituting \( \lambda \) into the expression for the number of disintegrated atoms:
\[ \Delta N = \frac{R_1 - R_2}{\frac{\ln 2}{T}} = \frac{(R_1 - R_2) T}{\ln 2} \]

### Step 6: Relate \( \Delta N \) to the given expression
According to the problem, the number of atoms that have disintegrated in the time \( (T_2 - T_1) \) is given by:
\[ \Delta N = \frac{n(R_1 - R_2) T}{\ln 4} \]

### Step 7: Equate the two expressions for \( \Delta N \)
Setting the two expressions for \( \Delta N \) equal to each other:
\[ \frac{(R_1 - R_2) T}{\ln 2} = \frac{n(R_1 - R_2) T}{\ln 4} \]

### Step 8: Simplify the equation
Assuming \( R_1 \neq R_2 \) and \( T \neq 0 \), we can divide both sides by \( (R_1 - R_2) T \):
\[ \frac{1}{\ln 2} = \frac{n}{\ln 4} \]

### Step 9: Solve for \( n \)
Rearranging gives:
\[ n = \frac{\ln 4}{\ln 2} \]
Using the property of logarithms, we know:
\[ \ln 4 = \ln(2^2) = 2 \ln 2 \]
Thus:
\[ n = \frac{2 \ln 2}{\ln 2} = 2 \]

### Final Answer
The value of \( n \) is:
\[ \boxed{2} \]