Q.Photons absorbed in matter are converted to heat. A source emitting $n$ photon/sec of frequency $\nu$ is used to convert $1\ \text{kg}$ of ice at $0\,^\circ\text{C}$ to water at $0\,^\circ\text{C}$. Then, the time $T$ taken for the conversion

The key idea is that each photon carries energy $h\nu$, so the total power delivered is $P = n h \nu$. The heat required to melt 1 kg of ice at $0\,^\circ\text{C}$ is $Q = mL_f$, where $L_f = 3.36 \times 10^5\ \text{J/kg}$.

Step 1: Energy delivered in time $T$:

$E = P T = n h \nu T$.

Step 2: This must equal the melting heat:

$n h \nu T = m L_f$.

Step 3: Solve for $T$:

$T = \dfrac{m L_f}{h} \cdot \dfrac{1}{n \nu}$.

So $T$ is inversely proportional to the product $n\nu$.

• If $n$ increases ($\nu$ fixed), $T$ decreases → (A) is correct. …

The core idea here is simple: each photon carries a fixed packet of energy $E = h\nu$. When $n$ photons strike the ice per second, the power (energy per second) delivered is $P = n h \nu$. Melting ice at $0^\circ\text{C}$ requires a fixed amount of heat — the latent heat of fusion. So the time taken is just the total energy needed divided by the power.

Let’s walk through it.

1. Energy needed to melt the ice The ice is at $0^\circ\text{C}$ and must become water at $0^\circ\text{C}$. No temperature change occurs — only the phase change. The heat required is

where $m = 1\ \text{kg}$ and $L_f = 3.34 \times 10^5\ \text{J/kg}$ (the latent heat of fusion of ice). So $Q$ is a fixed number.

2. Power delivered by the photon source Each photon has energy $h\nu$. With $n$ photons emitted per second, the power absorbed (assuming all photons are absorbed and converted to heat) is

3. Time to deliver the required heat Since power is energy per unit time,

All quantities in the numerator are constants. So $T$ is inversely proportional to $n\nu$.

4. Now examine each option

   (A) “decreases with increasing $n$, with $\nu$ fixed.”

   If $\nu$ is fixed, $T \propto 1/n$. So yes, $T$ decreases as $n$ increases. Correct.

   (B) “decreases with $n$ fixed, $\nu$ increasing.”

   If $n$ is fixed, $T \propto 1/\nu$. So $T$ decreases as $\nu$ increases. Correct. …

Method: Analyzing Proportionality Relations from a Physical Formula

When a question gives several qualitative statements about how a time, rate, or other quantity changes as different variables are adjusted, the reliable technique is to first derive the exact formula relating them, then read the proportionality directly off that formula — never guess the trend intuitively.

Steps

Step 1: Identify the fixed physical requirement

Here, converting a fixed mass of ice at its melting point to water requires a fixed amount of heat energy, $Q=mL_f$, regardless of how that energy is delivered.

Step 2: Write the rate at which energy is supplied, in terms of the given variables

where $n$ is the photon rate and $\nu$ is the frequency — this is the same "power = photon flux × photon energy" relation used throughout photon-counting problems.

Step 3: Combine into a single equation for the quantity being asked about

$T = \frac{Q}{P} = \frac{m L_f}{n h \nu}$ …