Physics 11th Thermal Properties of Matter

THERMAL PROPERTIES OF MATTER

1. Concept of Heat and Temperature

• Familiar sensations of hotness and coldness are described with adjectives such as hot , warm, cold, cool etc.
• When we touch an object, we use our temperature sence to ascribe to object a property called temperature.
• Temperature of a body determines whether it feels hot or cold to the touch. The hotter you feel on touching the body higher is its temperature.
• So we can say thet Temperature is relative measurement of hotness or coldness of a body.
• An observation about hot and cold bodies in contact is that, when they both are in contact temperature of cold body increases and that of hot body decreases. This happens because energy is transferred from hot body to cold body when they are in contact and this is a nonmechanical process.
• This energy which is transferred from one body to another without any mechanical work involved is known as HEAT.
• Heat is a form of energy and heat transfer from one body to another takes place by virtue of temperature difference only also heat transfer takes place from body at higher temperature to body at lower temperature.
• S.I. unit of heat is Joule(J) and that of temperature is Kelvin(K).

2. Measurement of temperature

• Measurement of temperature can be obtained using a thermometer.
• Construction of thermometers generally require a measurable property of a substance which monotonically changes with temperature.
• Examples of some common type of thermometers
  (1) Mercury in a glass thermometer.The height of mercury in the tube is taken as thermometric parameter.
  (2) Constant Volume gas thermometer-Gas in bulb is maintaned at constant volume.The mean pressure of gas is taken as thermomtric parameter.
  (3) Constant Pressure gas thermometer-Gas in bulb is maintaned at constant pressure.Volume of gas is taken as thermomtric parameter.
  (4) Resistance therometer-Electric resistance of a metal wire increases monitonically with the temperature and may be used to define temperature scale. Such thermometers are resistance theromometers.
• Thermometers are calibrated to asign a numerical value to any given temperature.
• Defination of any standard scale needs two fixed refrence points and these points can be corelated to physical phenomenon reproducible at the same temperature.
• Two such standard points are freezing and boiling points of water at same pressure.
• Two such familiar scales used for measurement of temperature are Celsius and Fahrenhite scale.
• Temperature in celsius is measured in degree.
• Fahrenhite scale has a smaller degree then celsius scale and a diffrent zero of temperature.
• Relation between Celsius and fahrenhite scale is
  T_F=9/5 T_C + 32°.
  where,
  T_F - Fahrenhite Temperature.
  T_C - Celsius Temperature.
  Letters C & F are used to distinguish measurements on two scale thus
       0° C= 32° F
  this means that 0° C on celsius scale measures the same temperature as 32° F on the fahrenhite scale.
• On fahrenhite scale melting point of ice and boiling point of water have values 32° F and 212° F and that on celsius scale are 0° C and 100° C.
• If we now talk of Kelvin scale ,the melting point of ice and boiling point of water in the scale are 273.15 K and 373.15 K respectively.
• Size of a degree in celsius and kelvin scale are same.
• Relation between Celsius and kelvin scale is
       T_C = T_K - 273.15 K
  where,
  T_K - Temperature in Kelvin
  T_C - Temperature in celsius
  
• Another modern fixed point of temperature is the triple point of water.
• Three phases of water i.e, ice,water,water vapour co-exist at this value of temperature and pressure where
  T_tr=273.16 K
  P_tr= 0.46 cm of HG.
  where,
  T_tr & P_tr are triple point temperature and pressure.
• Now if P_tr is pressure of an ideal gas thermometer at triple point temperature T_tr and if P is pressure at some other temperature T then corresponding temperature is
       T= P( T_tr / P_tr)
  provided T_tr & P_tr are low.
• Electric resitance of metal wire increase monitonically with temperature and may be used to define the temperature scale
• If R₀ & R₁₀₀ are resistance of metal wire at ice and steam point respectively then temperature t can be defined corresponding to resistance R_T as follows

  T =	(RT-R0)x100
  	R100-R0

• A platinum wire is oftently used to construct a thermometer which is known as platinum resistance thermometer.
• Gas thermometer can also define Celsius scale
• If P₀ is pressure of gas at ice point and P₁₀₀₀ is pressure of gas at steam point then temperature T corresponding to a pressure P of gas is defined by

• T =	(P-P0)x100
  	P100-P0

  
  

  
  

  
  

  
  

  3. Absolute Temperature

• In gas thermometer diffrent gases are used for measuring high temperatures and temperature reading are found to be independent of the nature of the gas used
• Graphs between pressure and temperature for diffrent gases are plotted below
   
• It is found that when graph were extrapolated for temperature below 0° C,pressure comes out to be zero at -273° C and this is the lowest temperature.
• Lord kelvin suggested that instead of 0° C which is the melting point of ice,-273° C should be regarded as the zero of the temperature scale
• Such a scale of temperature is absolute scale of temperature and -273.15° C as absolute zero of this new scale and is denoted as 0K and steam point in this scale correspond to 373.15 K

4. Ideal Gas Equation

• Pressure of all the gases changes with the temperature in a similar fashion for low temperature.Also many properties of gases are common at low pressure
• The pressure,volume and temperature in kelvin of such gases obey the equation
       PV=nRT                               (1)
  where n is amount of gas in number of moles and is given as ,
  n = [Total no of molecules in given mass of gas]/[Avagadro Number(N_A)]
  N_A=6.023 × 10²³
  R is universal gas constant and its value is R=8.316 J/mol-K 
• Equation (1) is known as ideal gas equation and a gas obeying this equation is known as ideal gas.

5. Thermal Expansion

• Most of the solid material expand when heated.
• Increase in dimension of a body due to increase in its temperature is called thermal expansion.
• For small change in temperature ΔT of a rod of length L, the fractional change in length ΔL/L is directly propertional to ΔT(Fig 3)
  
  
  
       ΔL/L=α_LΔT                (2)
  or ,
       ΔL=α_LLΔT                (3)
• Constant α_L characterizes the thermal expansion properties of a particulaqr material and it is known as coefficient of linear expansion.
• For materials having no prefential direction,every linear dimension changes according to equation (3) and L could equally well represent the thickness of the rod,side lenght of the square sheet etc.
• Normally metals expand more and have high value of α.
• Again consider the intial surface area A of any surface and A' is the area of the solid when the temperature of the body changes by ΔT then increase in surface area is given by
       ΔA=α_AAΔT                (4)
  where α_A is the coefficient of area expansion.
• Similary we can define coefficient of volume expansion as fractional change in volume ΔV/V of a substance for a temperature change ΔT as
  

  αV =	ΔV
  	VΔT

• K^-1 is the unit of these coefficents of expansions.
• These three coefficent are not strictly constant for a substance and there value is depends on temperaturerange in which they are being measured.
• As an example, fig below shows that coefficient of volume expansion increase with temperrature and takes a constant value above 500K
  
   

Relation between volume and linear coefficient of expansion for solid material:

• Consider a solid parallopide with dimension L₁,L₂ and L₃ then its volume is
       V= L₁L₂L₃
• When temperature increase by a amount ΔT then each linear dimension changes and then new volume is
       V+ΔV=L₁L₂L₃(1+α_LΔT)³
          =V(1+α_LΔT)³
          =V(1+3α_LΔT+3α_L²(ΔT)²+α_L³(ΔT)³)
  if ΔT is small then higher order of ΔT can be neglected.Thus we find
       V+ΔV=V(1+3α_LΔT)
  or,
       ΔV=3α_LV₀ΔT
• Comparing this with equation (5) we find
       α_V=3α_L

6. Thermal stress

• If we fix the ends of a rod rigidly so that it can not expand or contract then with change in temperature, tensile or compressive stress known as thermal stresses will set up in the rod.
• To compute the thermal stress,consider a rod of length L and crossectional area A with its both end rigidly fixed and temperature is then reduced by an amount ΔT.
• The fractional change in rod if the rod is free to contract would be
       ΔL/L=αΔT                          (6)
  where ΔL and ΔT both are negative.
• As we have fixed the ends of the rod, so it is not free to contract and this causes an increase in tension so as to produce an equal and opposite fractional change in length.
• From the defination of Young Modulus
  
  

  Y =	F/A
  	ΔL/L

  ΔL/L =	F
  	AY

  where ΔL is positive
• Tensile forces is determined by the requirement that total fractional changes in length,Thermal expansion plus elastic strain must be zero.
       αΔT + F/AY=0
       F=-AYαΔT                         (7)
  Thus tensile stress in rod is
       F/A=-YαΔT
• Hence F/A is positive in case of decrease in temperature since change in temperature becomes negative.
• In case where change in temperature tends to increase in temperature of the body then F and F/A becomes negative, corresponding development of to compressive force and strength. 
• Finaly stress which is is positive with decrease in temperature is Tensile stree and stress which is negative with increase in temperature is Compressive stress.

7. Specific Heat Capacity

• If a system undergoes a change of temperature from T to T+ΔT during the transfer of ΔQ amount of heat then heat capacity c of the system is defined as the ratio of
  

  c =	ΔQ
  	ΔT

• Thus Heat capacity per unit mass of a substance is its specific heat capacity.

  c =	ΔQ
  	mΔT

   where,
  m - mass of the substance
  ΔQ - Heat absorbed or rejected by the substance
  ΔT - Change in Temperature
• Specific heat capacity depends on the nature of substance.
• It is constant characterstics of the substance and is independent of the ammount of substance.
• It also depends on the temperature of the substance 
• Its unit is J Kg^-1 K^-1
• If the amount of substance is specified in terms of no of moles n instead of mass m then the heat capacity per mole of the substance is
  

  C =	ΔQ
  	nΔT

  and is known as Molar Specific Heat capacity.
• It is constant characterstics of the substance and independent of the ammount of substance
• It depends on the nature of the substance ,temperature and amount of heat supplied
• Its unit is J mol^-1 K^-1
• In case of gases , when a gas is heated, ordinarly there is change in volume as well as pressure in addition to change in temperature
• For simiplicity either volume or pressure can be kept constant.Thus gas have two specific heat capacities
  1) Specific heat capacity at constant volume C_V
  2) Specific heat capacity at constant pressure C_P

8. Calorimetery

• Calorimetery means measurement of Heat.
• Calorimeter is the device used to measure heat and it is cylinderical vessel made of copper and provided by a stirrer and a lid.
• This vessel is kept in a wooden block to isolate it thermally from suroundings.A theromometer is used to measure the temperature of the content in the calorimeter.
• When bodies at diffrent temperature are mixed in a calorimeter,they exchange heat with each other.
• Bodies at higher temperature loose heat while bodies at low temperature gain heat.Contents of the calorimeter is continously stirred to keep temperature of contents uniform 
• Thus principle of calorimetery states that the total heat given by hot objects is equal to the total heat received by cold objects.

9.Change of phases:

• There are three phases of matter i.e, Solid, Liquid and Gas.
• Substance for example, H₂O exists in solid phase as Ice,in liquid phase as Water and in gas state as Steam.
• Transtion from one phase to another phase are accompained by absorption or liberation of heat and usually by change in volume even at constant T.
• Change of phase from solid to liquid is called melting , from liquid to solid is called fusion and from liquid to gas is called vaporisation
• Once the temperature for phase change is reached( e,g melting or boiling temperature) no further temperature change occurs until all the substance has undergone phase change.
  
  Melting Point:-Temperature at which solid and liquid phase are in thermal equilibrium with each other
  Boiling point:-Temperature at which liquid and vapour phase are in thermal equilibrium with each other
• Change of phase from solid state to Vapour state without passing through liquid state is called sublimation for e.g Dry Ice and Iodine sublimes.

10. Latent Heat:

• The amount of heat per unit mass that must be transfered as heat when a sample completely undergoes phase change is called latent Heat of the substance for the process.
• Thus when a sample of mass m completely undergoes phase change,the total energy transfered is
  Q=Lm
  where,
  L- Latent heat and is characterstics of a substance
• Its unit is JKg^-1.
• Latent Heat for a solid liquid change is called Latent Heat of Fusion L_f.
• Latent Heat for a liquid gas change is called Latent Heat of Vaporisation L_v.

Solved Examples

Question 1
A circular hole of diameter 2.00 cm is made in an aluminium plate at 0 ⁰ C .what will be the diameter at 100⁰ C?
Linear expansion for aluminium = 2.3 * 10^-3 / ⁰ C

Solution:
Diameter of circular hole in aluminium plate at 0⁰ C=2.0 cm
With increase in temperature from 0⁰ C to 100⁰ C diameter of ring increases 

using
L=L₀(1+αΔT) 
where L0=2.0 cm 
α = 2.3 * 10^-3 / ⁰ C
ΔT=(100 -0)=100 ⁰ C
we can find diameter at 100⁰ C
L=2(1+2.3*10^-3*100)
=2.46 cm

Question 2
The pressure of the gas in constant volume gas thermometer are 80 cm,90cm and 100cm of mercury at the ice point,the steam point and in a heated wax bath resp.Find the temperature of the wax bath

Solution
Given that
Pressure at the ice point P_ice= 80 cm of Hg
Pressure at the steam point P_steam= 90 cm of Hg
Pressure at the wax bath P_wax= 100 cm of Hg
T=(P_wax-P_ice)X100/(P_steam-P_ice)
T = (100 - 80)X100/(90-80)
= 20X100/10
=200 ⁰C

Question 3
A rod of length L having coefficent of Linear expansion a is lying freely on the floor.it is heated so that temperature changes by b .Find the longitidunal strain developed in the rod
a. 0
b. ab
c. -ab
d. none of the above
Solution
There was no restriction for it expansion.So no tensile or compressive force developed.Longitudinal strains happens only when tensile or compressive force developed in the rod.So answer is a

Question 4
.if a is coefficent of Linear expansion,b coefficent of areal expansion,c coefficent of Volume expansion.Which of the following is true
a. b=2a
b. c=3a
c. b=3a
d. a=2b
Solution
Answer is c

Question 5
.which is of them is not used as the measurable properties in thermometer?
a.Resistance of platinum wire
b.Constant volume of gas
c.Contant pressure of gas
d.None of the above
Solution
Answer is d

Question 6
.when a solid metalic sphere is heated.the largest percentage increase occurs in its
a.Diameter
b. Surface area
c. Volume
d. density
Solution
Answer is c

Question 7
.the density of the liquid depends upon
a. Nature of the liquid
b. Temperature of the liquid
c. Volume of the liquid
d. Mass of the liquid
Solution
Answer is a and b
Question 8

A metallic sphere has a cavity of diameter D at its center.If the sphere is heated,the diameter of the . cavity will
a. Decrease
b. Increase
c. Remain unchanged
d. none of the above
Solution
Answer is b

Question 9
.A metallic circular disc having a circular hole at its center rotates about it axis passing through the center and perpendicular to it plane.when the disc is heated
a. Its speed will decrease
b. Diameter will increase
c. Moment of inertia will increase
d. its speed will increase
Solution
(a),(c)
Due to thermal expansion,the diameter of the disc as well of the hole will increase.therefore the moment of inertia will increase resulting in a increase in the angular speed.
Question 10
A resistance thermometer is such that resistance varies with temperature as
R_T=R₀(1+aT+bT⁵)
where T represent Temperature on Celsius scale And a,b,R₀ are constants.R₀ unit is ohm
Based on above data ,Find out the unit of a,b
Solution


As per dimension analysis Unit on both sides should be equal
Now since R & R₀ both unit are same
Quantity 1+aT+bT⁵ should be dimension less
so at should be dimension less
so a unit is C^-1
similarly bT⁵ should be dimensionless
so b unit is C^-5