Calculate the rate of flow of heat through a metal sheet `0.02` m thick and area `50 xx 10^(-4) "m"^(2)` with its two sides at 273 K and 293 K respectively. Given `K=0.2` cal cm`""^(-1) "s"^(-1) "c"^(-1)`.

Explanation

To calculate the rate of flow of heat through a metal sheet, we can use the formula for thermal conduction:

\[ \frac{Q}{t} = \frac{k \cdot A \cdot (T_2 - T_1)}{x} \]

Where:
- \( \frac{Q}{t} \) is the rate of heat flow (in Joules per second),
- \( k \) is the thermal conductivity (in Joules per meter per second per Kelvin),
- \( A \) is the area of the sheet (in square meters),
- \( T_2 \) and \( T_1 \) are the temperatures on either side of the sheet (in Kelvin),
- \( x \) is the thickness of the sheet (in meters).

### Step 1: Convert the thermal conductivity from calories to Joules
Given:
- \( k = 0.2 \, \text{cal cm}^{-1} \text{s}^{-1} \text{°C}^{-1} \)

We know:
- \( 1 \, \text{cal} = 4.184 \, \text{J} \)
- \( 1 \, \text{cm} = 0.01 \, \text{m} \)

Thus, we convert \( k \):
\[ k = 0.2 \, \text{cal cm}^{-1} \text{s}^{-1} \text{°C}^{-1} = 0.2 \times 4.184 \, \text{J} \cdot (100) \, \text{m}^{-1} \text{s}^{-1} \text{K}^{-1} \]
\[ k = 0.2 \times 4.184 \times 100 = 836.8 \, \text{J m}^{-1} \text{s}^{-1} \text{K}^{-1} \]

### Step 2: Identify the values for the calculation
- Area \( A = 50 \times 10^{-4} \, \text{m}^2 = 0.005 \, \text{m}^2 \)
- Thickness \( x = 0.02 \, \text{m} \)
- Temperature difference \( T_2 - T_1 = 293 \, \text{K} - 273 \, \text{K} = 20 \, \text{K} \)

### Step 3: Substitute the values into the formula
Now we can substitute all the values into the formula:
\[ \frac{Q}{t} = \frac{836.8 \, \text{J m}^{-1} \text{s}^{-1} \text{K}^{-1} \cdot 0.005 \, \text{m}^2 \cdot 20 \, \text{K}}{0.02 \, \text{m}} \]

### Step 4: Calculate the rate of heat flow
Calculating the numerator:
\[ = 836.8 \cdot 0.005 \cdot 20 = 83.68 \, \text{J} \]

Now divide by the thickness:
\[ \frac{Q}{t} = \frac{83.68}{0.02} = 4184 \, \text{J/s} \]

### Final Result
The rate of flow of heat through the metal sheet is:
\[ \frac{Q}{t} = 4184 \, \text{J/s} \]