A radioactive nuclide is produced at a constant rate x nuclei per second. During each decay, `E_0` energy is released ,50% of this energy is utilised in melting ice at `0^@`C. Find mass of ice that will melt in one mean life. (`lambda`=decay consant, `L_f`=Latent heat of fusion )

Answer

To solve the problem, we need to find the mass of ice that will melt in one mean life of a radioactive nuclide that decays and releases energy. Let's break this down step by step.

### Step 1: Understanding the Production and Decay of Nuclei
A radioactive nuclide is produced at a constant rate of \( x \) nuclei per second. The decay of these nuclei is characterized by a decay constant \( \lambda \).

### Step 2: Mean Life Calculation
The mean life \( \tau \) of a radioactive substance is given by:
\[ \tau = \frac{1}{\lambda} \]
This means that in one mean life, the time \( t \) is equal to \( \tau \).

### Step 3: Number of Nuclei Decayed in One Mean Life
The number of nuclei that decay in one mean life can be calculated using the formula:
\[ N_d = x \cdot \tau \]
Substituting \( \tau \):
\[ N_d = x \cdot \frac{1}{\lambda} \]

### Step 4: Total Energy Released
Each decay releases an energy \( E_0 \). Therefore, the total energy \( E \) released during the decays in one mean life is:
\[ E = N_d \cdot E_0 = \left( x \cdot \frac{1}{\lambda} \right) \cdot E_0 = \frac{x \cdot E_0}{\lambda} \]

### Step 5: Energy Utilized for Melting Ice
It is given that 50% of the released energy is utilized in melting ice. Therefore, the energy used \( E_u \) for melting ice is:
\[ E_u = \frac{1}{2} E = \frac{1}{2} \cdot \frac{x \cdot E_0}{\lambda} = \frac{x \cdot E_0}{2\lambda} \]

### Step 6: Relating Energy to Mass of Ice
The energy used to melt ice can be expressed in terms of the mass of ice \( m \) and the latent heat of fusion \( L_f \):
\[ E_u = m \cdot L_f \]
Equating the two expressions for \( E_u \):
\[ m \cdot L_f = \frac{x \cdot E_0}{2\lambda} \]

### Step 7: Solving for Mass of Ice
Now, we can solve for the mass of ice \( m \):
\[ m = \frac{x \cdot E_0}{2\lambda \cdot L_f} \]

### Final Answer
Thus, the mass of ice that will melt in one mean life is:
\[ m = \frac{x \cdot E_0}{2\lambda \cdot L_f} \]