The latent heat of fusion of ice is `330 J//g`. Calculate the number of photons of radiation of frequency `5xx10^(13) s^(-1)` to cause the melting of `1` mole of ice. Take `h=6.6xx10^(-34) J.S.` Express your answer as `X xx 10^(22)`, what is the value of 'X'

Explanation

To solve the problem, we need to calculate the number of photons required to melt 1 mole of ice using the given latent heat of fusion and the frequency of the radiation. Let's break it down step by step.

### Step 1: Calculate the energy required to melt 1 mole of ice.
The latent heat of fusion of ice is given as \( 330 \, \text{J/g} \). The molar mass of ice (water) is approximately \( 18 \, \text{g/mol} \).

\[ \text{Energy required} = \text{mass} \times \text{latent heat} = 18 \, \text{g} \times 330 \, \text{J/g} = 5940 \, \text{J} \]

### Step 2: Calculate the energy of a single photon.
The energy of a photon can be calculated using the formula:

\[ E = h \nu \]

Where:
- \( h = 6.63 \times 10^{-34} \, \text{J.s} \) (Planck's constant)
- \( \nu = 5 \times 10^{13} \, \text{s}^{-1} \) (frequency)

Substituting the values:

\[ E = (6.63 \times 10^{-34} \, \text{J.s}) \times (5 \times 10^{13} \, \text{s}^{-1}) = 3.315 \times 10^{-20} \, \text{J} \]

### Step 3: Calculate the number of photons required.
To find the number of photons (\( n \)) needed to provide the total energy required to melt 1 mole of ice, we can use the formula:

\[ n = \frac{\text{Total Energy}}{\text{Energy per photon}} = \frac{5940 \, \text{J}}{3.315 \times 10^{-20} \, \text{J}} \]

Calculating \( n \):

\[ n = \frac{5940}{3.315 \times 10^{-20}} \approx 1.79 \times 10^{23} \]

### Step 4: Express the answer in the form \( X \times 10^{22} \).
To express \( 1.79 \times 10^{23} \) in the form \( X \times 10^{22} \):

\[ 1.79 \times 10^{23} = 17.9 \times 10^{22} \]

Thus, \( X = 17.9 \).

### Final Answer:
The value of \( X \) is approximately \( 17.9 \).

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