**Introductory Exercise 19.1**

**Take c**_{ice }= 0.53 cal/g-Â°C, c_{water} = 1.0 cal/g-Â°C,(L_{f})_{water} = 80cal/g and(L_{v})_{water} = 529 cal/g unless given in the question,

**Ques 1: In a container of negligible mass 140 g of ice initially at -15Â° C is added to 200 g of water that has a temperature of 40Â°C. If no heat is lost to the surroundings, what is the final temperature of the system and masses of water and ice in mixture?**

**Sol: **Let mixture is water at Î¸Â°C (where 0Â°C< Î¸ < 40Â°C)

Heat given by water = Heat taken by ice

âˆ´ (200) (1) (40 - Î¸) = (140) (0.53) (15) + (140)(80)+(140)(l)(Î¸ - 0)

Solving we get,

Î¸ = -12.7Â°C

Since, Î¸ < 0Â° C and we have assumed the mixture to be water whose temperature can't be less than 0Â°C.

Hence, mixture temperature Î¸ = 0Â° C. Heat given by water in reaching upto 0Â° C is,

Î¸ = (200) (1) (40 - 0) = 8000 cal.

Let m mass of ice melts by this heat, then 8000 = (140) (0.53) (15)+(m) (80)

Solving we get m = 86 g

âˆ´ Mass of water = 200 + 86 = 286 g

Mass of ice = 140 - 86 = 54 g

**Ques 2: The temperatures of equal masses of three different liquids A, Sand Care 12Â°C, 19Â°C and 28Â°C respectively. The temperature when A and B are mixed is 16Â°C and when S and C are mixed is 23Â°C. What would be the temperature when A and C are mixed?**

Sol: A + B ms_{A} (16 - 12) = ms_{B} (19' - 16)

4s_{A} = 3s_{B}

B + C ms_{B} (23 - 19) = ms_{C} (28 - 23)

4s_{B }= 5s_{C} ...(ii)

Solving these two equations, we get

A + C

Solving, we get

Î¸ = 20.25Â° C

**Ques 3: Equal masses of ice (at 0Â°C) and water are in contact. Find the temperature of water needed to just melt the complete ice.**

Sol: mL = ms (Î¸ - 0Â°)

âˆ´

**Ques 4: A nuclear power plant generates 500 MW of waste heat that must be carried away by water pumped from a lake. If the water temperature is to rise by 10Â°C, what is the required flow rate in kg/s?**

Sol:

** **

**Introductory Exercise 19.2**

**Ques 1: Suppose a liquid in a container is heated at the top rather than at the bottom. What is the main process by which the rest of the liquid becomes hot?**

Sol: In convection, liquid is heated from the bottom.

**Ques 2: The inner and outer surfaces of a hollow spherical shell o f inner radius 'a' and outer radius 'b' are maintained at temperatures T**_{1} and T_{2} (< T_{1}). The thermal conductivity of material of the shell is k. Find the rate of heat flow from inner to outer surface.

Sol: Let us take an element at distance r from centre of thickness dr. Applying the formula of or thermal resistance.

= thermal resistance of this element

Rate of heat flow,

**Ques 3: Show that the SI units of thermal conductivity are W/m-K.**

Sol:

âˆ´

**Ques 4: A carpenter builds an outer house wall wit h a layer of wood 2.0 cm thick on the outside and a layer of an insulation 3.5 cm thick as the inside wall surface. The wood has k = 0.08 W/m -K and the insulation has k = 0.01 W/m -K. The interior surface temperature is 19Â° C and the exterior surface temperature is -10Â° C.**

Sol: (a) H_{1} = H_{2}

âˆ´

or

âˆ´

âˆ´

Solving, we get

Î¸= -8.1Â°C

= 7.7 W/m^{2}

**Ques 5: A pot with a steel bottom 1.2 cm thick rests on a hot stove. The area of the bottom of the pot is 0.150 m**^{2}. The water inside the pot is at 100Â°C and 0.440 kg are evaporated every 5.0 minute. Find the temperature of the lower surface of the pot, which is in contact with the stove. Take L_{v} = 2.256 Ã— 10^{6 }J/kg and k_{steel} = 50.2 W/m-K.

Sol:

âˆ´

= 105Â°C

**Ques 6: A layer of ice o f thickness y is on the surface of a lake. The air is at a constant temperature -Î¸Â°C and the ice water interface is at 0Â°C. Show that the rate at which the thickness increases is given by,**

where k is the thermal conductivity of the ice, L the latent heat of fusion and p is the density of the ice.

Sol: See the extra points just before solved examples. Growth of ice on ponds. We have already derived that,

âˆ´

âˆ´

**Ques 7: The emissivity of tungsten is 0.4. A tungsten sphere with a radius of 4.0 cm is suspended within a large evacuated enclosure whose walls are at 300 K. What power input is required to maintain the sphere at a temperature of 3000 K if heat conduction along supports is neglected?**

Take Ïƒ= 5.67 Ã— 10^{-8} W/m^{2} -K^{4}.

Sol:

= (0.4)(5.67 Ã— 10^{-8})(4Ï€)(4 Ã— 10^{-2})2[3000)^{4} -(300)^{4}]

= 3.7 Ã— 10^{4}W

**Ques 8: Find SI units of thermal resistance.**

Sol: