An immersion heater is placed in crushed ice at $- 40 ° C$ , contained in a perspex jar, such that water at $50 ° C$ is formed. If the power of the heater is $200 W$ and it is switched on for $3 \min$ and $20 s$ , calculate the initial mass of ice. [S.H.C. of ice $= 2.1 {Jg}^{- 1} C^{- 1}$ and latent heat of ice $= 336 {Jg}^{- 1}$ ]

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Solution

Step 1: Given data
Power of heater $= 200 W$ .
Specific Heat Capacity of ice is $c_{1} = 2.1 {Jg}^{- 1} C^{- 1}$ .
Specific latent heat of ice is $L = 336 {Jg}^{- 1}$ .
Heater is switched on for $3 \min$ and $20 s = 200 s$ .
Step 2: Finding the heat required $(H_{1})$ to convert ice at $- 40 ° C$ to $0 ° C$
The amount of heat required is $m c_{1} θ_{R}$ , where $m$ is the initial mass of ice.
Substituting the known values,
$H_{1} = m \times 2.1 \times [0 - (- 40)] = m \times 2.1 \times 40 = 84 m J$
Step 3: Finding the heat required $(H_{2})$ to convert ice at $0 ° C$ to water at $0 ° C$
The amount of heat required is $m L$ .
Putting the values,
$H_{2} = m \times 336 = 336 m J$
Step 4: Finding the heat required $(H_{3})$ to convert water at $0 ° C$ to water at $50 ° C$
Specific heat capacity of water is $c_{2} = 4.2 J / g ° C$ .
The amount of heat required is $m c_{2} θ_{R}$ .
Putting the values,
$H_{3} = m \times 4.2 \times (50 - 0) = m \times 4.2 \times 50 = 210 m J$
Step 5: Finding the total amount of heat required
Total amount of heat required is $H = H_{1} + H_{2} + H_{3} = 84 m + 336 m + 210 m = 630 m J$ .
Step 6: Finding the total amount of heat produced
For $3 \min$ and $20 s$ , the amount of heat energy produced is given as $H' = P \times T = 200 \times (200 s) = 40000 J$ .
Step 6: Finding the initial mass of ice
We know, heat required $H$ will be equal to the heat produced $H'$ ..
Putting the values,
$630 m = 40000 m = \frac{40000}{630} m = 63.49 g$
Hence, the initial mass of ice is $63.49$ grams.

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An immersion heater is placed in crushed ice at -40°C, contained in a perspex jar, such that water at 50°C is formed. If the power of the heater is 200W and it is switched on for 3min and 20s, calculate the initial mass of ice. [S.H.C. of ice =2.1Jg-1C-1 and latent heat of ice =336Jg-1]