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Description

CALORIMETRY & HEAT TRANSFER

Definition of Heat : Heat is a form of energy which is transferred between a system and its surrounding as a result of temperature difference only

Thermal Expansion : Expansion due to increase in temperature

Type of thermal expansion

Coefficient of expansion

- (i) Linear
- Lim
- (ii) Superficial
- Lim
- (iii) Volume
- Lim
- (a) (b) 2
- t 0
- t 0
- t 0
- 1 l l0 Δt
- 1 A A0 Δt
- 1 V V0 Δt

For temperature change t change in length l = l0 t Area A = A0t volume V = V0t

2 and 3 are coefficient of linear expansion in X ,

Variation in density : With increase of temperature volume increases so density decreases and vice-versa

- (1 t ) For solids values of are generally small so we can write d'= d0 (1–t) (using bimomial expansion)

Note : (i) for liquids are in order of 10–3 (ii) For water density increases from 0 to 4°C so is –ve (0 to 4° C) and for 4° C to higher temperature is +ve

Thermal Stress : A rod of length l0 is clamped between two fixed walls with distance l0

If temperature is changed by amount t then F stress = (area assumed to be constant) A strain =

- l l0

FA Fl 0 F Y = l l'= = At Al 0 F = Y A t

( varies with distance) Let = ax+b

( varies with tempearture) Let = f (T) l =

- l0 dT

similarly if is in K then put T1 and T2 in K

use units according to the given units of C) T2 T1

Heat transfer in phase change Q = mL L'= latent heat of substance in cal/ gm/ °C or in Kcal/ kg/ °C Lice = 80 cal/ gm for ice L'steam = 540 cal/ gm (A) (i)

- TRANSFER

Steady State : In this state heat absorption stops and temperature gradient throughout the rod dT becomes constant i

- = constant

dx Before steady state : Temp of rod at any point changes

it can be considered always in steady state

Ohm’s law for Thermal Conduction in Steady State : Let the two ends of rod of length l'is maintained at temp T1 and T2( T1 > T2) dQ T1 T2 Thermal current = R dT Th Where thermal resistance RTh =

dT = temperature gradient dx

Stefan’s Law : Rate of heat emitted by a body at temp T K from per unit area E =T4 J/sec/m2 dQ Radiation power = P = AT T4 watt dT If a body is placed in a surrounding of temperature TS dQ = A (T4 – Ts4) dT valid only for black body

Radiation: Every body radiates electromagnetic radiation of all possible wavelength at all temp>0 K

heat from general body Emissivity or emmisive power e = heat from black body If temp of body falls by dT in time dt dT eA 4 (T Ts4 ) (dT/dt = rate of cooling) dt mS

T1 T2 K T1 T2 Ts (used generally in objective questions) t mS 2 dT K (T Ts ) dt mS

(for better results use this generally in subjective)

Wein’s black body radiation At every temperature (>0K) a body radiates energy radiations of all wavelengths

then mT = b where b = is a constant (Wein’s constant) T = temperature of body

Calorimetry & Heat Transfer

- (i) (ii) (iii) (iv) Q
- 2 (a) (b) Q

one of steel and one of brass,

- as shown in figure

00 m long

- 60 mm and the length of the bar AB is 0

? Calculate the extension of the steel wire and the energy stored in it

Calculate the diameter of the brass wire

where should the mass be suspended so that AB would remain horizontal

? The Young modulus for steel = 2

- 0 × 1011 Pa,

the Young modulus for brass = 1

- 0 × 1011 Pa

- area of cross-section A,

[Density = d] It is pulled on a horizontal frictionless floor with a constant horizontal force F = [dALg]/2 applied at one end

An aluminium container of mass 100 gm contains 200 gm of ice at – 20°C

Heat is added to the system at the rate of 100 cal/s

Find the temperature of the system after 4 minutes (specific heat of ice = 0

- 5 and L'= 80 cal/gm,
- specific heat of Al = 0
- 2 cal/gm/°C)

The liquid in the left vertical limb is maintained at a temperature = 0°C while the liquid in the right limb is maintained at a temperature = 100°C

A thin walled metal tank of surface area 5m2 is filled with water tank and contains an immersion heater dissipating 1 kW

The tank is covered with 4 cm thick layer of insulation whose thermal conductivity is 0

2 W/m/K

Find the temperature of the tank in the steady state

It is found that at different temperatures the volume of air inside the flask remains the same

If the volume of mercury in the flask is 300 cm3,

then find volume of the flask (given that coefficient of volume expansion of mercury and coefficient of linear expansion of glass are 1

- 8 × 10–4 (°C)–1 and 9 × 10–6 (°C)–1 respectively)

A clock pendulum made of invar has a period of 0

- 5 sec at 20°C

- aporoximately

How much fast or slow will the clock run in 106 sec

(invar=1×10–6/°C) A pan filled with hot food cools from 50

- 1 °C to 49
- 9 °C in 5 sec

How long will it take to cool from 40

- 1 °C to 39

9°C if room temperature is 30°C

A composite rod made of three rods of equal length and cross-section as shown in the fig

The end A and end B are at constant temperatures

All heat entering the face A goes out of the end B there being no loss of heat from the sides of the bar

Find the effective thermal conductivity of the bar A

An iron bar (Young’s modulus = 1011 N/m2 ,

= 10–6 /°C) 1 m long and 10–3 m2 in area is heated from

0°C to 100°C without being allowed to bend or expand

both of same mass & emissivity are heated to same initial temperature and kept under identical conditions

A cylindrical rod with one end in a stream chamber and other end in ice cause melting of 0

- 1 gm of ice/sec

If the rod is replaced with another rod of half the length and double the radius of first and thermal conductivity of second rod is 1/4 that of first,

find the rate of ice melting in gm/sec Three aluminium rods of equal length form an equilateral triangle ABC

and al = 4 3 10 6 / C

D and C are maintained at 20°C,

- 90°C and 0°C

If two rods of length L'and 2 L'having coefficients of linear expansion and 2 respectively are connected so that total length becomes 3 L,

determine the average coefficient of linear expansion of the composite rod

A volume of 120 ml of drink (half alcohol + half water by mass) originally at a temperature of 25°C is cooled by adding 20 gm ice at 0°C

find the final temperature of the drink

- (density of drink = 0

833 gm/cc,

- specific heat of alcohol = 0
- 6 cal/gm/°C)

A solid receives heat by radiation over its surface at the rate of 4 kW

and heat is generated at a rate of 1

- 7 kW over the volume of the solid

The rate of change of the average temperature of the solid is 0

5°Cs–1

Find the heat capacity of the solid

The figure shows the face and interface temperature of a composite slab containing of four layers of two materials having identical thickness

Under steady state condition,

find the value of temperature

- 2 cal g–1 (C°)–1 is dropped into A and a 5 gm piece of metal Y into B

The equilibrium temperature in A is 22°C and in B 23°C

Find the specific heat of metal Y in cal g–1 (C°)–1

Two spheres of same radius R have their densities in the ratio 8 : 1 and the ratio of their specific heats are 1 : 4

If by radiation their rates of fall of temperature are same,

then find the ratio of their rates of losing heat

the corners A and C are maintained at T1 and T2 respectively

The rate of heat flow from A to C is

find the total rate of heat flow

A hot liquid contained in a container of negligible heat capacity loses temperature at rate 3 K/min,

just before it begins to solidify

The temperature remains constant for 30 min

Find the ratio of specific heat capacity of liquid to specific latent heat of fusion is in K–1 (given that rate of losing heat is constant)

Bansal Classes

Calorimetry & Heat Transfer

- (i) (ii) (iii)

A thermostatted chamber at small height h above earth's surface maintained at 30°C has a clock fitted in it with an uncompensated pendulum

The clock designer correctly designs it for height h,

- but for temperature of 20°C

If this chamber is taken to earth's surface,

the clock in it would click correct time

Find the coefficient of linear expansion of material of pendulum

(earth's radius is R) The coefficient of volume expansion of mercury is 20 times the coefficient of linear expansion of glass

Find the volume of mercury that must be poured into a glass vessel of volume V so that the volume above mercury may remain constant at all temperature

Two 50 gm ice cubes are dropped into 250 gm of water into a glass

- (specific heat of ice = 0
- 5 cal/gm/°C and L'= 80 cal/gm)

Water is heated from 10°C to 90°C in a residential hot water heater at a rate of 70 litre per minute

- 2 kg/m3 is used in the heater,

which has a transfer efficiency of 32%

Find the gas consumption rate in cubic meters per hour

(heat combustion for natural gas is 8400 kcal/kg) A metal rod A of 25cm lengths expands by 0

- 040 cm for the same rise in temperature

- 03 cm on heating from 0°C to 50°C

A substance is in the solid form at 0°C

The amount of heat added to this substance and its temperature are plotted in the following graph

find from the graph the mass of the substance

the specific latent heat of the melting process,

and the specific heat of the substance in the liquid state

One end of copper rod of uniform cross-section and of length 1

- 5 meters is in contact with melting ice and the other end with boiling water

At what point along its length should a temperature of 200°C be maintained,

- so that in steady state,

the mass of ice melting is equal to that of steam produced in the same interval of time

? Assume that the whole system is insulated from the surroundings

A vessel containing 100 gm water at 0°C is suspended in the middle of a room

- it melts in 10 hours

The maximum in the energy distribution spectrum of the sun is at 4753 Å and its temperature is 6050K

What will be the temperature of the star whose energy distribution shows a maximum at 9506 Å

List of recommended questions from I

247 to 2

254 to 2

Calorimetry & Heat Transfer

EXERCISE – II

A wire of cross-secitonal area 4 × 10–4 m2 modulus of elasticity 2 × 1011 N/m2 and length 1 m is stretched between two vertical rigid poles

A mass of 1 kg is suspended at its middle

Calculate the angle it makes with the horizontal

A metal spherical shell of mean radius 20 cm and wall thickness 1 mm is completely filled with a liquid of bulk modulus = 10 GPa,

and volumetric thermal expansion coefficient = 2 × 10–4 /K

with the liquid at atmospheric pressure

The Young’s modulus of the metal shell is 100 GPa and its linear thermal expansion coefficient is 1 × 10–5 /K

at what value of T will it rupture

A copper calorimeter of mass 100 gm contains 200 gm of a mixture of ice and water

Steam at 100°C under normal pressure is passed into the calorimeter and the temperature of the mixture is allowed to rise to 50°C

If the mass of the calorimeter and its contents is now 330 gm,

what was the ratio of ice and water in the beginning

- ? Neglect heat losses

- 42 × 103 J kg–1K–1,

- 2 × 103 J kg–1K–1,

- 36 × 105 J kg–1 Latent heat of condensation of steam = 22
- 5 × 105 Jkg–1

An isosceles triangle is formed with a rod of length l1 and coefficient of linear expansion 1 for the base and two thin rods each of length l2 and coefficient of linear expansion 2 for the two pieces,

if the distance between the apex and the midpoint of the base remain unchanged as the temperatures varied show that

- 6 (a) (b) Q
- l1 2 2
- l2 1

A solid substance of mass 10 gm at – 10°C was heated to – 2°C (still in the solid state)

The heat required was 64 calories

Another 880 calories was required to raise the temperature of the substance (now in the liquid state) to 1°C,

while 900 calories was required to raise the temperature from –2°C to 3°C

Calculate the specific heat capacities of the substances in the solid and liquid state in calories per kilogram per kelvin

Show that the latent heat of fusion L'is related to the melting point temperature tm by L'= 85400 + 200 tm

The mass of the steel block and the drill is 180 gm

If the entire mechanical work is used up in producing heat and the rate of raise in temperature of the block and the drill is 0

5 °C/s

and the torque required to drive the drill

- 1 and J = 4

2 J/cal

- 25 kg and a cross sectional area 5 cm2 increases its length by 0
- 3 mm upon heating from 0°C

What amount of heat is spent for heating the rod

? The coefficient of linear expansion for brass is 2×10–5/K,

- its specific heat is 0

39 kJ/kg

K and the density of brass is 8

- 5 × 103 kg/m3

Bansal Classes

A submarine made of steel weighing 109 g has to take 108 g of water in order to submerge when the temperature of the sea is 10°C

How much less water it will have to take in when the sea is at 15°C

? (Coefficient of cubic expansion of sea water = 2 × 10–4/°C,

coefficient of linear expansion of steel = 1

- 2 × 10–5/°C)

Heat is added at a known rate to a stream of the liquid as it passes through the calorimeter at a known rate

Then a measurement of the resulting temperature difference between the inflow and the outflow points of the liquid stream enables us to compute the specific heat of the liquid

- 2 g/cm3 flows through a calorimeter at the rate of 10 cm3/s

Heat is added by means of a 250-W electric heating coil,

and a temperature difference of 25°C is established in steady-state conditions between the inflow and the outflow points

Find the specific heat of the liquid

Toluene liquid of volume 300 cm3 at 0°C is contained in a beaker an another quantity of toluene of volume 110 cm3 at 100°C is in another beaker

(The combined volume is 410 cm3)

Given the coefficient of volume expansion = 0

- 001/C and all forms of heat losses can be ignored

- -20°C is filled upto height h = 10 cm in a uniform cylindrical vessel

water from second vessel is poured into first vessel and it is found that level of upper surface falls through h = 0

- 5 cm 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 of water

w = 1 gm cm–3 Density of ice,

- i = 0
- 9 gm/cm3 Specific heat of water,

sw = 1 cal/gm 0C Specific heat of ice,

- 5 cal/gm0C Specific latent heat of ice,

A composite body consists of two rectangular plates of the same dimensions but different thermal conductivities KA and KB

The composite body can be placed such that flow of heat takes place either parallel to the interface or perpendicular to it

K of the composite body for the parallel and perpendicular orientations

A highly conducting solid cylinder of radius a and length l'is surrounded by a co-axial layer of a material having thermal conductivity K and negligible heat capacity

which is higher than temperature of the cylinder

calculate time required to increase temperature of the cylinder from T1 to T2

The lower end of the duct is maintained at a temperature T1 which is greater than the melting point Tm of cast iron and the upper end at a temperature T2 which is less than the temperature of the melting point of cast iron

It is given that the conductivity of liquid cast iron is equal to k times the conductivity of solid cast iron

Bansal Classes

- (a) (b)

temperature gradient along the rod in steady state

total heat absorbed by the rod to reach steady state

- 4 m an area of cross-section 0
- 04m2 is placed coaxially on a thin metal disc of mass 0
- 4 kg and of the same cross-section

The upper face of the cylinder is maintained at a constant temperature of 400K and the initial temperature of the disc is 300K

If the thermal conductivity of the material of the cylinder is 10 watt/m-K and the specific heat of the material of the disc in 600 J/kgK,

how long will it take for the temperature of the disc to increase to 350K

- ? Assume,
- for purposes of calculation,

the thermal conductivity of the disc to be very high and the system to be thermally insulated except for the upper face of the cylinder

when its temperature is 127°C

Find the rate at which another solid copper sphere of twice the radius lose its temperature at 327°C,

- if in both the cases,

the room temperature is maintained at 27°C

- 5 m and of uniform cross-sectional area is maintained at some constant temperature

The other end B of this rod is radiating energy into vacuum and the wavelength with maximum energy density emitted from this end is 0 = 75000 Å

determine the temperature of the end A

the rod is thermally insulated

The shell of a space station is a blackened sphere in which a temperature T = 500K is maintained due to operation of appliances of the station

Find the temperature of the shell if the station is enveloped by a thin spherical black screen of nearly the same radius as the radius of the shell

? The temperature of surrounding is 20°C

Calculate the mass of the steam required for this purpose

- [JEE '96]

The arrangement is thermally insulated

The coefficients of thermal conductivity of A & B are 300 W/mº C and 200 W/mº C respectively

After steady state is reached the temperature T of the interface will be ______

- [JEE' 96]

A double pane window used for insulating a room thermally from outside consists of two glass sheets each of area 1 m2 and thickness 0

- 01 m separated by a 0
- 05m thick stagnant air space

the room glass interface and the glass outdoor interface are at constant temperatures of 270C and 00C respectively

Given thermal conductivities of glass and air as 0

8 and 0

- 08 W m–1K–1 respectively
- [JEE’97]

The apparatus shown in the figure consists of four glass columns connected by horizontal sections

- 8 cm & 51 cm respectively

- [JEE '97]

If the radius were halved and the temperature doubled,

the power radiated in watt would be : (A) 225 (B) 450 (C) 900 (D) 1800

If all the solar energy falling on a lens of area 0

- 2 m2 is focussed on to a block of ice of mass 280 grams,

the time taken to melt the ice will be ______ minutes

(Latent heat of fusion of ice = 3

- 3 x 105 J/kg) [JEE '97]

A solid body X of heat capacity C is kept in an atmosphere whose temperature is TA = 300K

the temperature of X is T0 = 400K

the body X is connected to a larger body Y at atmospheric temperature TA,

through a conducting rod of length L,

cross-sectional area A and thermal conductivity K

Find the temperature of X at time t = 3t1

- [JEE’ 98]

The energy of radiation emitted by this object with wavelength between 499 nm and 500 nm is U1,

between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3

- 88 × 106 nm K

Bansal Classes

Calorimetry & Heat Transfer

A bimetallic strip is formed out of two identical strips one of copper and the other of brass

On heating,

the temperature of the strip goes up by T and the strip bends to form an arc of radius of curvature R

Then R is : (A) proportional at T (B) inversely proportional to T [JEE’ 99] (C) proportional to |B – C| (D) inversely proportional to |B – C| A block of ice at – 10°C is slowly heated and converted to steam at 100°C

Which of the following curves represents the phenomenon qualitatively

- ? [JEE (Scr) 2000] (A)

T2 and T3 respectively are as shown

Their temperatures are such that [JEE (Scr) 2000] (A) T1 > T2 > T3 (B) T1 > T3 > T2 (C) T2 > T3 > T1 (C) T3 > T2 > T1

Three rods made of the same material and having the same cross-section have been joined as shown in the figure

The left and right ends are kept at 0°C and 90°C respectively

The temperature of the junction of the three rods will be [JEE(Scr)2001] (A) 45°C (B) 60°C (C) 30°C (D) 20°C

It is observed that (A) initially it is the darkest body and at later times the brightest

(B) it the darkest body at all times (C) it cannot be distinguished at all times

(D) initially it is the darkest body and at later times it cannot be distinguished

- [JEE(Scr)2002]

- 1 kg at 0°C is placed in an isolated container which is at 227°C

The specific heat S of the container varies with temperature T according the empirical relations = A + BT,

where A = 100 cal/kg-K and B = 2 × 10–2 cal/kg-K2

If the final temperature of the container is 27°C,

determine the mass of the container

(Latent heat of fusion for water = 8 × 104 cal/kg

and other steel of length l2 having coefficient of linear expansion s are joined end to end

Then the value of l' l'is 1 2 s (A) a s

- s (B) a s
- a s s

[JEE' (Scr) 2003]

- 2 kg ice at – 20°C is mixed with 5 kg water at 20°C

5cal/g°C,

specific heat of water = 1 cal/g°C,

Latent heat of fusion of ice = 80 cal/g

[JEE' (Scr) 2003] (A) 6 kg (B) 5 kg (C) 4 kg (D) 2 kg If emissivity of bodies X and Y are ex and ey and absorptive power are Ax and Ay then [JEE' (Scr) 2003] (A) ey > ex

Ay > Ax (B) ey < ex

Hot oil is circulated through an insulated container with a wooden lid at the top whose conductivity K = 0

- 149 J/(m-°C-sec),
- thickness t = 5 mm,
- emissivity = 0

Temperature of the top of the lid in steady state is at Tl = 127°

If the ambient temperature Ta = 27°C

- 17 10 8 ) 3
- temperature of the oil
- (Given =

- and C having radii 2 m,
- 4 m and 6 m respectively are coated with carbon black on their outer surfaces

- 400 nm and 500 nm respectively

QB and QC respectively

(a) QA is maximum (B) QB is maximum [JEE' 2004 (Scr

)] (C) QC is maximum (D) QA = QB = QC

Two identical conducting rods are first connected independently to two vessels,

one containing water at 100°C and the other containing ice at 0° C

In the second case,

the rods are joined end to end and connected to the same vessels

)] Liquid oxygen at 50 K is heated to 300 K at constant pressure of 1 atm

[JEE' 2004 (Scr

)] A cube of coefficient of linear expansion s is floating in a bath containing a liquid of coefficient of volume expansion l

the depth upto which the cube is submerged in the liquid remains the same

Find the relation between s and l,

- showing all the steps

[JEE 2004] One end of a rod of length L'and cross-sectional area A is kept in a furnace of temperature T1

The thermal conductivity of the material of the rod is K and emissivity of the rod is e

It is given that T2 = TS + T where T