Chapter 11 - THERMAL PROPERTIES OF MATTER

According to Stefan’s law of radiation, a black body radiates
energy σT 4 from its unit surface area every second where T is the
surface temperature of the black body and σ = 5.67 × 10–8 W/
m2K4 is known as Stefan’s constant. A nuclear weapon may be
thought of as a ball of radius 0.5 m. When detonated, it reaches
temperature of 106K and can be treated as a black body.
(a) Estimate the power it radiates.
(b) If surrounding has water at 30 C ° , how much water can 10%
of the energy produced evaporate in 1 s?

(c) If all this energy U is in the form of radiation, corresponding
momentum is p = U/c. How much momentum per unit time
does it impart on unit area at a distance of 1 km?

According to Stefan’s law of radiation, a black body radiates
energy σT 4 from its unit surface area every second where T is the
surface temperature of the black body and σ = 5.67 × 10–8 W/
m2K4 is known as Stefan’s constant. A nuclear weapon may be
thought of as a ball of radius 0.5 m. When detonated, it reaches
temperature of 106K and can be treated as a black body.
(a) Estimate the power it radiates.
(b) If surrounding has water at 30 C ° , how much water can 10%
of the energy produced evaporate in 1 s?

A thin rod having length L0 at 0°C and coefficient of linear
expansion α has its two ends maintained at temperatures θ 1 and
θ2, respectively. Find its new length.

One day in the morning, Ramesh filled up 1/3 bucket of
hot water from geyser, to take bath. Remaining 2/3 was to
be filled by cold water (at room temperature) to bring
mixture to a comfortable temperature. Suddenly Ramesh
had to attend to something which would take some times,
say 5-10 minutes before he could take bath. Now he had
two options: (i) fill the remaining bucket completely by cold
water and then attend to the work, (ii) first attend to the
work and fill the remaining bucket just before taking bath.
Which option do you think would have kept water warmer ?
Explain.

100 g of water is supercooled to –10°C. At this point, due to
some disturbance mechanised or otherwise some of it suddenly
freezes to ice. What will be the temperature of the resultant
mixture and how much mass would freeze?

During summers in India, one of the common practice to keep
cool is to make ice balls of crushed ice, dip it in flavoured sugar
syrup and sip it. For this a stick is inserted into crushed ice and
is squeezed in the palm to make it into the ball. Equivalently in
winter, in those areas where it snows, people make snow balls
and throw around. Explain the formation of ball out of crushed
ice or snow in the light of P–T diagram of water.

Find out the increase in moment of inertia I of a uniform rod
(coefficient of linear expansion α ) about its perpendicular
bisector when its temperature is slightly increased by ΔT.

These days people use steel utensils with copper bottom. This is
supposed to be good for uniform heating of food. Explain this
effect using the fact that copper is the better conductor.

Calculate the temperature which has same numeral value on
celsius and Fahrenheit scale.

Why does a metal bar appear hotter than a wooden bar at the
same temperature? Equivalently it also appears cooler than
wooden bar if they are both colder than room temperature.

Is the bulb of a thermometer made of diathermic or adiabatic wall?

Discuss briefly energy distribution of a black body radiation. Hence deduce
wien’s displacement law?

Discuss briefly energy distribution of a black body radiation. Hence deduce
wien’s displacement law?

Define the terms reflectance, absorptance and transmittance. How are they
related?

How do you derive Newton’s law of cooling from Stefan’s law?

Calculate the temperature in Kelvin at which a perfectly black body radiates at

the rate of 5.67 w/cm²?

A metal rod of length 20cm and diameter 2cm is covered with a non-conducting

substance. One of it ends is maintained at 100⁰c while the other is at 0⁰c. It is

found that 25g of ice melts in 5 min calculate coefficient of thermal conductivity

of metal?

Why is it hotter at the same distance over the top of the fire than in front of it?

Animals curl into a ball, when they feel very cold?

A body with large reflectivity is a poor emitter why?

If air is a bad conductor of heat, why do we not feel warm without clothes?

A bar of copper of length 75cm and a bar of length 125cm are joined end to end.

Both are of circular cross – section with diameters 2cm. The free ends of copper

and steel are maintained at 100⁰c and 00c respectively. The surfaces of the bars

are thermally insulated. What is the temperature of copper – steel junction?

Thermal conductivity of copper = 9.2x10^-2k cal |m|0c|s and that of steel is 1.1x10^-2

k cal|m|⁰c|s?

A room has a 4m x 4m x10cm concrete roof (K₁ = 1.26w|m|0C). At some instant,

the temperature outside is 46⁰c and radius 32⁰c.

1) Calculate amount of heat flowing per second into the room through the roof.

2) If bricks (K₂ – 0.56w |m|0c) of thickness 7.5cm are laid down on roof, calculate

the new rate of heat flow under the same temperature conditions?

A cylinder of radius R made of a material of thermal conductivity K₁ is surrounded

by a cylindrical shell of inner radius R and other radius 2R made of material of

thermal conductivity K₂. The 2 ends of combined system are maintained at two

different temperatures. There is no loss of heat across the clinical surface and the

system is in steady state calculate the effective thermal conductivity of the

system?

A spherical black body of radius 12cm radiates 450w power at 500k. If the radius
were halved and the temperature doubled, what would be the power radiated?

The tile floor feels colder than the wooden floor even though both floor materials
are at same temperature. Why?

Differentiate between conduction, convection and radiation?

If the radiation from the moon gives maxima at l = 4700 A⁰ and l = 14x10^-6m.

What conclusion can be drawn from the above information?

How do you explain the emission of long - wavelength by the object at low
temperature?

If the temperature of the sun is doubled, the rate of energy received on each will
increases by what factor?

A Sphere is at a temperature of 6oo k. Its cooling rate is R in an external

environment of 200k. If temperature falls to 400k. What is the cooling rate R₁ in

terms of R?

Two rods A and B are of equal length. Each rod has its ends at temperature T₁ and

T₂ (T₁ > T₂). What is the condition that will ensure equal rates of flow through the

rods A and B?

Why is the energy of thermal radiation less than that of visible light?

How do you explain heating of rooms based on principle of convection?

A brass boiler has a base area of 0.15m² and thickness 1.0cm. It boils water at

the rate of 6kg| min when placed on a gas stove, Estimate the temperature of the

part of flame in contact with the boiler. Thermal conductivity of brass =

109J|s|m|⁰C, heat of vaporization of water = 2256J|g?

Animals in the forest find shelter from cold in holes in the snow. Why?

An indirectly heated filament is radiating maximum energy of wavelength 2.16x

10^-7m. Find the net amount of heat energy lost per second per unit area, the

temperature of surrounding air is 13⁰C. Given b = 2.88x10^-3 mk, σ = 5.77x10^-8

J|s|m²|k⁴?

What are thermal radiation? Write its properties of thermal radiation?

The ends of the two rods of different materials with their thermal
conductivities, radii of cross – section and length in the ratio 1:2 are maintained
at the same temperature difference. If the rate of flow of heat through the larger
rod is 4 cal |s, what is the rate of flow through the shorter rod?

A layer of ice 10cm thick is formed on a pond. The temperature of air is – 10⁰C.

Calculate how long it will take for the thickness of ice to increase by 1mm.

Density of ice = 1g|cm³; Thermal conductivity of ice = 0.005Cal|s|cm^|0C; Latent

heat of ice = 80Cal|g?

Two rods A and B are of equal length. Each rod has its ends at temperatures T₁

and T₂. What is the condition that will ensure equal rates of flow of heat through

the rods A and B?

If a body is heated from 27⁰ C to 927⁰C then what will be the ratio of energies of

radiation emitted?

On a winter night, you feel warmer when clouds cover the sky than when sky is
clear. Why?

A ball is dropped on a floor from a height of 2cm. After the collision, it rises up

to a height of 1.5m. Assuming that 40% of mechanical energy lost goes to

thermal energy into the ball. Calculate the rise in temperature of the ball in the

collision. Specific heat capacity of the ball is 800J/k. Take g = 10m/s²

Calculate the amount of heat required to convert 1.00kg of ice at – 10⁰c into

steam at 100⁰c at normal pressure. Specific heat of ice = 2100J|kg|k. Latent heat

of fusion of ice = 3.36x10⁵J|kg, specific heat of water = 4200J|kg|k. Latent heat of

vaporization of water = 2.25 x10⁶J|kg?

Develop a relation between the co-efficient of linear expansion, co-efficient
superficial expansion and coefficient of cubical expansion of a solid?

Calculate C_p for air, given that Cv =0.162 cal g^-1 k^-1 and density air at N.T. P is

0.001293 g|cm³?

The resistance of certain platinum resistance thermometer is found to be 2.56 Ω

at 0⁰c and 3.56 Ω at 100⁰c. When the thermometer is immersed in a given liquid,

its resistance is observed to 5.06 Ω. Determine the temperature of liquid?

Two bodies at different temperatures T₁, and T₂ are brought in thermal contact

do not necessarily settle down to the mean temperature of T₁ and T₂?

Why does the density of solid | liquid decreases with rise in temperature?

Why the clock pendulums are made of invar, a material of low value of
coefficient of linear expansion?

State Dulong and petit law?

Define triple point of water?

If for hydrogen CP – CV = a and for oxygen CP – CV = b where CP & CV refer to
specific heat at constant pressure and volume then what is the relation between
a and b?

What is critical temperature? How will you differentiate between a gas and a
vapour depending on critical temperature?

Calculate the amount of heat necessary to raise the temperature of 2 moles of HE

gas from 20⁰C to 50⁰C using:-

1) Constant – Volume Process 2) Constant Pressure Process

Here for, He; C_V = 1.5 R and C_P = 2.49R

How is heat loss reduced in Calorimeter?

Specific heat of Argon at constant Pressure is o.125 cal | g | K and at constant
volume is 0.075 cal | g | K. Calculate the density of argon at N.T.P. Given that J =
4.2J | cal?

Why is there a difference in the specific heat curve as given by Delong’s petit law
and the experimental result at low temperatures?

A 0.20 Kg aluminum block at 800C is dropped in a copper calorimeter of mass

0.05 Kg containing 200 cm3 of ethyl alcohol at 20⁰C. What is the final

temperature of the mixture? Given Density of ethyl alcohol = 0.81 g | cm³ ;

specific heat of ethyl alcohol = 0.6 cal | g | ⁰C ; specific heat of copper = 0.094 cal |

g | ⁰C, specific heat of Al = 0.22 cal | g | ⁰C?

A gas can have any value of specific heat depending upon how heating is carried
out. Explain?

If a body has infinite heat capacity? What does it signify?

Why are Calorimeters made up of metal only?

Why is temperature gradient required for flow of heat from one body to another?

A one liter flask contains some mercury. IT is found that at different

temperatures, then volume of air inside the flask remains the same. What is the

volume of mercury in the flask? Given the co-efficient of linear expansion of glass

= 9 × 10^-6 / ⁰C and co-efficient of volume expansion of mercury = 1.8 × 10-4 / ⁰C

Why does heat flow from a body at higher temperature to a body at lower
temperature?

Calculate the difference between two principal specific heats of 1g of helium gas
at N. T. P. Given Molecular weight of Helium = 4 and J = 4.186 J/cal and Universal
Gas constant, R = 8.314J / mole / K?

If one Mole of a monatomic gas is mixed with 3 moles of a diatomic gas. What is
the molecular specific heat of the mixture at constant volume?

The density of a solid at0⁰C and 500⁰C is in the ratio 1.027 : 1. Find the coefficient

of linear expansion of the solid?

If the volume of block of metal changes by 0.12% when it is heated through 200C.
What is the co-efficient of linear expansion of the metal?

Write the advantages and disadvantages of platinum resistance thermometer?

A thermometer has wrong calibration. It reads the melting point of ice as – 10⁰C.

It reads 60⁰C in place of 50⁰C. What is the temperature of boiling point of water

on the scale?

Two thermometers are constructed in the same way except that one has a
spherical bulb and the other an elongated cylindrical bulb. Which one will
response quickly to temperature change?

Show the variation of specific heat at constant pressure with temperature?

How would a thermometer be different if glass expanded more with increasing
temperature than mercury?

Why is mercury used in making thermometers?

A body cools from 80 °C to 50 °C in 5 minutes. Calculate the time it takes to cool from 60 °C to 30 °C. The temperature of the surroundings is 20 °C.

Explain why:
(a) a body with large reflectivity is a poor emitter
(b) a brass tumbler feels much colder than a wooden tray on a chilly day
(c) an optical pyrometer (for measuring high temperatures) calibrated for an ideal black body radiation gives too low a value for the temperature of a red hot iron piece in the open, but gives a correct value for the temperature when the same piece is in the furnace
(d) the earth without its atmosphere would be inhospitably cold
(e) heating systems based on circulation of steam are more efficient in warming a building than those based on circulation of hot water

A brass boiler has a base area of 0.15 m² and thickness 1.0 cm. It boils water at the rate of 6.0 kg/min when placed on a gas stove. Estimate the temperature of the part of the flame in contact with the boiler. Thermal conductivity of brass = 109 J s ^–1 m^–1 K^–1; Heat of vaporisation of water = 2256 × 10³ J kg^–1.

 A ‘thermacole’ icebox is a cheap and efficient method for storing small quantities of cooked food in summer in particular. A cubical icebox of side 30 cm has a thickness of 5.0 cm. If 4.0 kg of ice is put in the box, estimate the amount of ice remaining after 6 h. The outside temperature is 45 °C, and co-efficient of thermal conductivity of thermacole is 0.01 J s^–1 m^–1 K^–1. [Heat of fusion of water = 335 × 10³ J kg^–1]

A child running a temperature of 101°F is given an antipyrin (i.e. a medicine that lowers fever) which causes an increase in the rate of evaporation of sweat from his body. If the fever is brought down to 98 °F in 20 min, what is the average rate of extra evaporation caused, by the drug? Assume the evaporation mechanism to be the only way by which heat is lost. The mass of the child is 30 kg. The specific heat of human body is approximately the same as that of water, and latent heat of evaporation of water at that temperature is about 580 cal g^–1.

Answer the following questions based on the P–T phase diagram of CO₂:
(a) CO₂ at 1 atm pressure and temperature – 60 °C is compressed isothermally. Does it go through a liquid phase?
(b) What happens when CO₂ at 4 atm pressure is cooled from room temperature at constant pressure?
(c) Describe qualitatively the changes in a given mass of solid CO₂ at 10 atm pressure and temperature –65 °C as it is heated up to room temperature at constant pressure.
(d) CO₂ is heated to a temperature 70° C and compressed isothermally. What changes in its properties do you expect to observe?

Answer the following questions based on the P-T phase diagram of carbon dioxide:
(a) At what temperature and pressure can the solid, liquid and vapour phases of CO₂ co-exist in equilibrium?
(b) What is the effect of decrease of pressure on the fusion and boiling point of CO₂?
(c) What are the critical temperature and pressure for CO₂? What is their significance?
(d) Is CO₂ solid, liquid or gas at (a) –70 °C under 1 atm, (b) –60° C under 10 atm, (c) 15 °C under 56 atm?

. Given below are observations on molar specific heats at room temperature of some common gases.

In an experiment on the specific heat of a metal, a 0.20 kg block of the metal at 150 °C is dropped in a copper calorimeter (of water equivalent 0.025 kg) containing 150 cm³ of water at 27 °C. The final temperature is 40 °C. Compute the specific heat of the metal. If heat losses to the surroundings are not negligible, is your answer greater or smaller than the actual value for specific heat of the metal?

A copper block of mass 2.5 kg is heated in a furnace to a temperature of 500 °C and then placed on a large ice block. What is the maximum amount of ice that can melt? (Specific heat of copper = 0.39 J g^–1 K^–1; heat of fusion of water = 335 J g^–1).

A 10 kW drilling machine is used to drill a bore in a small aluminium block of mass 8.0 kg. How much is the rise in temperature of the block in 2.5 minutes, assuming 50% of power is used up in heating the machine itself or lost to the surroundings. Specific heat of aluminium = 0.91 J g^–1 K^–1.

The coefficient of volume expansion of glycerin is 49 × 10^–5 K^–1. What is the fractional change in its density for a 30 °C rise in temperature?

A brass rod of length 50 cm and diameter 3.0 mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250 °C, if the original lengths are at 40.0 °C? Is there a ‘thermal stress’ developed at the junction? The ends of the rod are free to expand (Co-efficient of linear expansion of brass = 2.0 × 10^–5 K^–1, steel = 1.2 × 10^–5 K^–1).

A brass wire 1.8 m long at 27 °C is held taut with little tension between two rigid supports. If the wire is cooled to a temperature of –39 °C, what is the tension developed in the wire, if its diameter is 2.0 mm? Co-efficient of linear expansion of brass = 2.0 × 10^–5 K^–1; Young’s modulus of brass = 0.91 × 10¹¹ Pa.

A hole is drilled in a copper sheet. The diameter of the hole is 4.24 cm at 27.0 °C. What is the change in the diameter of the hole when the sheet is heated to 227 °C? Coefficient of linear expansion of copper = 1.70 × 10^–5 K^–1.

A large steel wheel is to be fitted on to a shaft of the same material. At 27 °C, the outer diameter of the shaft is 8.70 cm and the diameter of the central hole in the wheel is 8.69 cm. The shaft is cooled using ‘dry ice’. At what temperature of the shaft does the wheel slip on the shaft? Assume coefficient of linear expansion of the steel to be constant over the required temperature range: α_steel = 1.20 × 10^–5 K^–1.

A steel tape 1m long is correctly calibrated for a temperature of 27.0 °C. The length of a steel rod measured by this tape is found to be 63.0 cm on a hot day when the temperature is 45.0 °C. What is the actual length of the steel rod on that day? What is the length of the same steel rod on a day when the temperature is 27.0 °C? Coefficient of linear expansion of steel = 1.20 × 10^–5 K^–1.

Two ideal gas thermometers Aand Buse oxygen and hydrogen respectively. The following observations are made:

(a) What is the absolute temperature of normal melting point of sulphur as read by thermometers Aand B?
(b) What do you think is the reason behind the slight difference in answers of thermometers Aand B? (The thermometers are not faulty). What further procedure is needed in the experiment to reduce the discrepancy between the two readings?

 Answer the following:
(a) The triple-point of water is a standard fixed point in modern thermometry. Why? What is wrong in taking the melting point of ice and the boiling point of water as standard fixed points (as was originally done in the Celsius scale)?
(b) There were two fixed points in the original Celsius scale as mentioned above which were assigned the number 0 °C and 100 °C respectively. On the absolute scale, one of the fixed points is the triple-point of water, which on the Kelvin absolute scale is assigned the number 273.16 K. What is the other fixed point on this (Kelvin) scale?
(c) The absolute temperature (Kelvin scale) T is related to the temperature t_c on the Celsius scale by
t_c = T – 273.15
Why do we have 273.15 in this relation, and not 273.16?
(d) What is the temperature of the triple-point of water on an absolute scale whose unit interval size is equal to that of the Fahrenheit scale?

 The electrical resistance in ohms of a certain thermometer varies with temperature according to the approximate law:
R = R_o [1 + α (T – T_o)]
The resistance is 101.6 Ω at the triple-point of water 273.16 K, and 165.5 Ω at the normal melting point of lead (600.5 K). What is the temperature when the resistance is 123.4 Ω?

Two absolute scales A and B have triple points of water defined to be 200 A and 350 B. What is the relation between T_A and T_B?

 The triple points of neon and carbon dioxide are 24.57 K and 216.55 K respectively. Express these temperatures on the Celsius and Fahrenheit scales.