One kilowatt electric heater is to be used with 220V DC supply. It converts `(N xx10)/(3)g` of water at `100^(0)C` into steam at `100^(0)C` in one minute. find the value N.

Answer

To solve the problem, we need to find the value of \( N \) in the equation given for the mass of water converted to steam by a 1 kW electric heater in one minute. Let's break this down step by step.

### Step 1: Understand the power of the heater
The power rating of the heater is given as 1 kW, which is equivalent to:
\[ P = 1 \text{ kW} = 1000 \text{ W} = 1000 \text{ J/s} \]

### Step 2: Calculate the energy supplied in one minute
Since we want to find out how much energy is supplied in one minute, we convert the time from minutes to seconds:
\[ T = 1 \text{ minute} = 60 \text{ seconds} \]
Now, we can calculate the total energy supplied by the heater in one minute:
\[ E = P \times T = 1000 \text{ J/s} \times 60 \text{ s} = 60000 \text{ J} \]

### Step 3: Use the latent heat of vaporization
The heater converts \( N \times \frac{10}{3} \) grams of water at \( 100^\circ C \) into steam at \( 100^\circ C \). The latent heat of vaporization for water is approximately \( 540 \text{ cal/g} \). We need to convert this to joules:
\[ 1 \text{ cal} = 4.2 \text{ J} \implies 540 \text{ cal/g} = 540 \times 4.2 \text{ J/g} = 2268 \text{ J/g} \]

### Step 4: Calculate the total heat required for the conversion
The mass of water being converted is:
\[ m = N \times \frac{10}{3} \text{ grams} \]
The total heat \( Q \) required to convert this mass of water to steam is given by:
\[ Q = m \times L = \left(N \times \frac{10}{3}\right) \times 2268 \text{ J} \]

### Step 5: Set up the equation
We know that the energy supplied by the heater in one minute must equal the heat required to convert the water to steam:
\[ 60000 \text{ J} = \left(N \times \frac{10}{3}\right) \times 2268 \]

### Step 6: Solve for \( N \)
Rearranging the equation gives:
\[ N \times \frac{10}{3} = \frac{60000}{2268} \]
Calculating the right side:
\[ \frac{60000}{2268} \approx 26.4 \]
Now, substituting this back:
\[ N \times \frac{10}{3} = 26.4 \implies N = 26.4 \times \frac{3}{10} = 7.92 \]

### Step 7: Round to the nearest whole number
Since \( N \) must be a whole number, we round \( 7.92 \) to \( 8 \).

Thus, the value of \( N \) is:
\[ \boxed{8} \]