Photons absorbed in matter are converted to heat. A source emitting n photons/ sec of frequency `nu` is used to convert 1kg of ice at `0^(@)C` to water at `0^(@)C`. Then, the time T taken for the conversion

Options

A. decreases with increasing n, with v fixed.
B. decreases with n fixed, v increasing.
C. remains constant with n and v changing such that nv=constant.
D. All of these.

Explanation

To solve the problem of determining the time \( T \) taken for converting 1 kg of ice at \( 0^\circ C \) to water at \( 0^\circ C \) using a source emitting \( n \) photons per second of frequency \( \nu \), we can follow these steps:

### Step 1: Calculate the heat required for the conversion
The heat required \( Q \) to convert ice to water at the same temperature can be calculated using the formula:
\[ Q = m \cdot L \]
where:
- \( m \) is the mass of the ice (1 kg = 1000 g),
- \( L \) is the latent heat of fusion of ice (approximately \( 80 \, \text{cal/g} \)).

Substituting the values:
\[ Q = 1000 \, \text{g} \cdot 80 \, \text{cal/g} = 80000 \, \text{cal} \]

### Step 2: Relate the energy from photons to heat
The energy \( E \) of one photon can be expressed using Planck's equation:
\[ E = h \cdot \nu \]
where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( \nu \) is the frequency of the photons.

The total energy emitted by the source in \( T \) seconds, emitting \( n \) photons per second, is:
\[ E_{\text{total}} = n \cdot E \cdot T = n \cdot (h \cdot \nu) \cdot T \]

### Step 3: Set the total energy equal to the heat required
Since the total energy from the photons must equal the heat required to convert the ice to water, we can set up the equation:
\[ n \cdot (h \cdot \nu) \cdot T = Q \]
Substituting \( Q \) from Step 1:
\[ n \cdot (h \cdot \nu) \cdot T = 80000 \, \text{cal} \]

### Step 4: Solve for time \( T \)
Rearranging the equation to solve for \( T \):
\[ T = \frac{80000 \, \text{cal}}{n \cdot (h \cdot \nu)} \]

### Step 5: Convert calories to joules (if necessary)
If you need the answer in joules, remember that \( 1 \, \text{cal} = 4.184 \, \text{J} \):
\[ 80000 \, \text{cal} = 80000 \cdot 4.184 \, \text{J} = 334720 \, \text{J} \]
Thus, the equation becomes:
\[ T = \frac{334720 \, \text{J}}{n \cdot (h \cdot \nu)} \]

### Final Result
The time \( T \) taken for the conversion is:
\[ T = \frac{334720}{n \cdot (h \cdot \nu)} \]