Specific heat of water is `4.2 J//g.^(@)C`. If light of frequency `3 xx 10^(29)Hz` is used to heat 400 g of water from `20^(@)C` to `40^(@)C`, the number of photons needed will be

Options

A. `1.69xx10^9`
B. `1.69xx10^8`
C. `2.80xx10^4`
D. `2.80xx10^5`

Answer

To solve the problem, we need to calculate the number of photons required to heat 400 g of water from 20°C to 40°C using light of a given frequency. Here’s a step-by-step solution:

### Step 1: Calculate the heat required (Q) to raise the temperature of the water.

The formula to calculate the heat required is:
\[ Q = m \cdot s \cdot \Delta T \]
where:
- \( m \) = mass of water = 400 g
- \( s \) = specific heat of water = 4.2 J/g°C
- \( \Delta T \) = change in temperature = \( 40°C - 20°C = 20°C \)

Substituting the values:
\[ Q = 400 \, \text{g} \cdot 4.2 \, \text{J/g°C} \cdot 20 \, \text{°C} \]
\[ Q = 400 \cdot 4.2 \cdot 20 = 33600 \, \text{J} \]

### Step 2: Calculate the energy of one photon.

The energy of a photon can be calculated using the formula:
\[ E = h \cdot f \]
where:
- \( h \) = Planck's constant = \( 6.63 \times 10^{-34} \, \text{J s} \)
- \( f \) = frequency of light = \( 3 \times 10^{29} \, \text{Hz} \)

Substituting the values:
\[ E = 6.63 \times 10^{-34} \, \text{J s} \cdot 3 \times 10^{29} \, \text{Hz} \]
\[ E = 1.989 \times 10^{-4} \, \text{J} \]

### Step 3: Calculate the number of photons required.

The number of photons (\( n \)) required can be calculated using the formula:
\[ n = \frac{Q}{E} \]

Substituting the values:
\[ n = \frac{33600 \, \text{J}}{1.989 \times 10^{-4} \, \text{J}} \]
\[ n \approx 1.69 \times 10^{8} \]

### Final Answer:
The number of photons needed is approximately \( 1.69 \times 10^{8} \).

---