The number of quanta of radiation of frequncy `4.98 xx 10^(14) s^(-1)` required to melt 100 g of ice are ( latent heat of melting of ice is 33 joule per g)

Options

A. `10^(20)`
B. `10^(22)`
C. `10^(14)`
D. ` 6.023 xx 10^(21)`

Answer

To solve the problem of determining the number of quanta of radiation required to melt 100 g of ice, we will follow these steps:

### Step 1: Calculate the total energy required to melt the ice
The latent heat of melting of ice is given as 33 joules per gram. Therefore, for 100 grams of ice, the total energy (Q) required can be calculated as:

\[ Q = \text{mass} \times \text{latent heat} = 100 \, \text{g} \times 33 \, \text{J/g} = 3300 \, \text{J} \]

### Step 2: Use the formula for energy of a quantum of radiation
The energy (E) of a single quantum of radiation can be calculated using the formula:

\[ E = h \nu \]

where:
- \( h \) (Planck's constant) = \( 6.626 \times 10^{-34} \, \text{J s} \)
- \( \nu \) (frequency) = \( 4.98 \times 10^{14} \, \text{s}^{-1} \)

### Step 3: Calculate the energy of one quantum
Substituting the values into the formula:

\[ E = (6.626 \times 10^{-34} \, \text{J s}) \times (4.98 \times 10^{14} \, \text{s}^{-1}) \]

Calculating this gives:

\[ E \approx 3.30 \times 10^{-19} \, \text{J} \]

### Step 4: Calculate the number of quanta required
To find the number of quanta (n) required to provide the total energy (Q), we use the formula:

\[ n = \frac{Q}{E} \]

Substituting the values we calculated:

\[ n = \frac{3300 \, \text{J}}{3.30 \times 10^{-19} \, \text{J}} \]

Calculating this gives:

\[ n \approx 1.00 \times 10^{22} \]

### Conclusion
Thus, the number of quanta of radiation required to melt 100 g of ice is approximately \( 10^{22} \).

### Final Answer
The correct answer is \( 10^{22} \).

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