Ice at $- 20^{\circ} C$ is filled upto height $h = 10 c m$ in a uniform cylindrical vessel. Water at temperature $θ^{\circ} C$ is filled in another identical vessel up to the same height $h = 10 c m$ . Now, water from second vessel is poured into first vessel and it is found that level of upper surface falls through $Δ h = 0.5 c m$ when thermal equilibrium is reached. Neglecting thermal capacity of vessels, change in density of water due to change in temperature and loss of heat due to radiation, calculate initial temperature $θ$ of water in $^{o} C$ .
Given,
Density of water, $ρ_{w} = 1 g c m^{- 3}$
Density of ice, $ρ_{i} = 0.9 g c m^{- 3}$
Specific heat of water, $s_{w} = 1 c a l / g^{\circ} C$
Specific heat of ice, $s_{i} = 0.5 c a l / g^{\circ} C$
Specific latent heat of ice, $L = 80 c a l / g$

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Solution

Total initial mass is $(0.9) (10 A) + (1) (10 A)$ where A is cross section area of cylinder. Now, let the height of ice in final situation be $x$ . Then the height of water will be $19.5 - x$ .
By conservation of mass,

$(x A) (0.9) + (A (19.5 - x)) (1) = (0.9) (10 A) + (1) (10 A)$

Solving this, we get $x = 5 c m$
Now, heat lost by water in cooling from $θ$ to $0^{o} C$ is consumed in heating of ice from ${- 20}^{o} C$ to $0^{o} C$ and conversion of $5 c m$ of ice into water at $0^{o} C$ .

Hence, $(10 A) (1) (θ - 0) = (0.9) (10 A) (0.5) (0 - (- 20)) + (5 A) (0.9) (80)$

$(4.5) (100) = (10) (θ)$

Ans, $θ = 45^{o} C$

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Q. A vessel contains

110 g

of water. The heat capacity of the vessel is equal to that of

10 g

of water. The initial temperature of water in the vessel is

10^{\circ} C

. While the water and vessel are in thermal equilibrium,

220 g

of hot water at

70^{\circ} C

is poured into the vessel. Find the final temperature of the vessel + water system (neglecting radiation loss).

1 kg

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is mixed with

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30^{\circ} C

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Given specific heat of ice

= 0.5 {cal/g}^{\circ} C

,
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A vessel of negligible heat capacity contains 40 g of ice in it at 0⁰C. A metal of mass 50 g at 50⁰C is placed into the ice to melt it. The final temperature of the contents of the vessel is 10⁰C. Find the specific capacity of the metal.

(Specific latent heat of fusion of ice = 336 J/g and Specific heat capacity of water = 4.2 J/g⁰C)

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