Heat is a form of energy which causes the sensation of hotness or coldness. Heat is the energy of random motion of molecules constituting the body. It flows from a hot body to a cold body.

`text(E.g. :)` If we dip our finger in hot water we have a sensation of hotness. Similarly, if we touch a block of ice the sensation is that of coldness. In the former case the heat energy has moved into the finger, while in the latter case the heat energy has moved our of the finger. Thus, hotness or coldness basically indicates whether heat energy is flowing into our body or out of our body.

The amount of heat energy present in a body is determined by the total sum of the kinetic energy and potential energy of its molecules.

• Temperature is the effect of heat energy which determines the thermal state of a given substance.
• In other words, it determines the degree of hotness or coldness of a substance.

• If a body is at a higher temperature than its surroundings, it means that heat energy will flow out of the body. Similarly, if a body is at a lower temperature than its surroundings, it means that heat energy will flow into the body.

`text(Role of Temperature in Transfer of Heat)`
When two bodies at different temperature are brought in contact with each other, the heat energy always flows from a body at higher temperature to a body at lower temperature, till the temperature equalizes. Thus, it is the temperature of a body which determines the direction of flow of heat energy.

• The instrument used for the measurement of temperature is called thermometer.
• All thermometers are based on the fact that matter expands on hearing. "Thus, we have solid thermometers, liquid thermometers and gas thermometers.
• The solid thermometers are less sensitive and the gas thermometers are most sensitive, because solid expand far less as compared to gas. However, for general purposes, we use liquid thermometers, using mercury or alcohol as thermometric liquids.

`text(Properties of Thermometric liquid)`
(i) It should have low specific heat capacity, so that it rapidly attains the temperature of a given substance, without absorbing any appreciable amount of hear energy from it.
(ii) It should have a uniform rate of expansion, such that a linear scale can be easily marked.
(iii) It should have large expansion for a unit degree rise in temperature, so that its expansion is visible to the unaided eye.
(iv) It should have a high boiling point and low freezing point, so that a wide range of temperature changes could be recorded by a single thermometer.
(v) It should be shiny and opaque so that it is clearly visible in glass.
(vi) It should not stick to the sides of the glass tube.
(vii) It should exert low vapor pressure.
(viii) It should be a good conductor of heat.
(ix) It should be easily available in pure state.

`text(Reasons for using Mercury as a Thermometric liquid)`
(i) It has low specific heat capacity.
(ii) Its expansion is uniform.
(iii) It has a high Boiling Point (`357^o C`) and low Freezing Point (`- 39^o C`).
(iv) It is opaque and shiny.
(v) It does not stick to the side of the glass.
(vi) It exerts very low vapor pressure.
(vii) It is a good conductor of heat.
(viii) It is easily available in pure state.

`text(Disadvantages of Water as Thermometric Liquid)`
(i) It has the highest specific heat capacity (4.2 J/gK).
(ii) Its expansion is not uniform.
(iii) Its expansion per degree rise in temperature is very small.
(iv) Its Freezing Point is `0^o C` and Boiling Point is `100^o C`. Thus, the temperature less than `0^o C` and
more that `100^o C` cannot be measured.
(v) It is transparent.
(vi) It sticks to the sides of glass.
(vii) It evaporates under vacuum conditions.
(viii) It is a bad conductor of beat.
(ix) It cannot be obtained in cent percent pure form easily.

To measure temperature, two fixed points are taken; one of them is freezing point of water, known as `text(ice-point)` and other point is boiling point of water, known as `text(steam point)`.
Some temperature scales are given below

(i) `text(Celsius Scale) (°C)` In this scale of temperature, the melting point of ice is taken as 0°C and the boiling point of water as 100°C and the space between these two points is divided into 100 equal parts. This scale was designed by Anders Celsius in 1710.

(ii) `text(Fahrenheit Scale) (°F)` In this scale, the melting point of ice is taken as `32^@F` and the boiling point of water as 212°F and the space between these two points is divided into 180 equal parts. This scale was designed by Gabriel Fahrenheit in 1717.

(iii) `text(Kelvin Scale) (K)` In this scale, the ice point and the steam point (boiling point) are taken as 273 K and 373 K, respectively and the space between these two points is divided into 100 equal parts. It was designed by Kelvin.

(iv) `text(Reaumur Scale) (R)` In this scale, ice point and, boiling point are taken as `0^@R` and `80^@R` respectively. `1^@R` is equal to the 80th part of difference between two points. This scale was designed by R A Reaumur in 1730.

(v) `text(Rankine Scale) (Ra)` In this scale, ice point and steam point are taken as 460° Ra and 672° Ra, respectively. `1^@Ra` is equal to the 212th part of difference between two points.

(vi) `text(Clinical Thermometer)` It is a specially adapted Fahrenheit thermometer used by doctors to record the temperature of human
body. The marking are from `95^oF` to `110^oF`, because the temperature of human body does not fall below `95^oF` or rise above `11 0°F`, as in either case death occurs.

`text(Relations between various temperature scales)`
`C/5=(F-32)/9 =(K-273)/5 = R/4 = (Ra- 460)/10.6`

The device which measures the temperature of the body, is called thermometer.
Some different types of thermometers are given below

`text(Constant Volume Gas Thermometer)`
If `p_0, p_(100) , p_(tr) ` and `p_t` are the pressures of gas at temperatures `0^@C,100^@C, ` triple point of water and unknown temperature (`t^@C`) respective! keeping the volume constant, then
`t=((p-p_0)/(p_(100) -p_0))text()^@C` or `T=(273.16 p/p_(tr)) K`

`text(Platinum Resistance Thermometer)`
If `R_0, R_(100) , R_(tr) ` and `R_t` are the resistances of a platinum wire at temperatures `0^@C,100^@C, ` triple point of water and unknown temperature (`t^@C`) respective! keeping the volume constant, then
`t=((R-R_0)/(R_(100) -R_0))text()^@C`
or `T=(R_T/R_(tr) xxT_(tr))K=(R_t/R_tr xx 276.16)K`

`text(Mercury Thermometer)`
In this thermometer, the length of a mercury column from some fixed point is taken as thermometric property. Thus,
`t=((l_t-l_0)/(l_(100)-l_0))xx100^@C` or `T=(l_t/l_(tr) xx273.16)K`

A change in temperature of a body causes change in its dimensions. When the body's temperature is increased, body expands in dimensions. It is called thermal expansion.

`text(Linear Expansion)` (Expansion in Length of a Solid)
Consider a rod of length `l_1` at a temperature `theta_1`. Let it he heated to a temperature `theta_2` and tb increased length of the rod be `l_2`, then

`l_2 = l_1 ( 1 + alpha Delta theta)`

where , `alpha =` coefficient of linear expansion and `theta = theta_2 - theta_1`

`text(Areal Expansion)` (Expansion in Surface Area)
It `A_1` is the area of solid at `theta_1 °C` and `A_2` is the area at `theta_2 °C`, then

`A_2 = A_1 ( 1 + beta Delta theta)`

where, `beta =` coefficient of areal ( superficial ) expansion and `Delta theta = theta_2 - theta_1`

`text(Volume Expansion)` ( Expansion in Volume)
If `V_1` is volume of solid at `theta_1 °C` and `V_2` is the volume at `theta_2 °C`, then

`V_2 = V_1 ( 1 + gamma Delta theta)`

where `gamma =` coefficient of cubical (volume) expansion and `Delta theta = theta_2 - Delta theta_1`

`text(Relationship Between)` `alpha, beta` `text(and)` `gamma`
`beta=2 alpha` and `gamma=3alpha`
i.e., `gamma/3=beta/2=alpha` or `alpha:beta:gamma=1:2:3`

`text(The Case of Glass Tumbler)`
This is due to the fact that glass is a poor conductor of heat, so when boiling water is put in the glass tumbler, the inside portion of the glass walls becomes hot and expands while the outside portion of the glass walls remains cold and does not expand quickly. Due to the rapid expansion of the inside glass wall but slow expansion of the outside portion, a strain is set up in the glass and it cracks.

`text(Railway Tracks)`
The rails of railway tracks are made up of steel. While laying the railway tracks, a small gap is left between the successive lengths of rail. This is because rails expands in summer due to considerable rise in atmospheric temperature. The gap is provided to allow this expansion. If no gap is left, the expansion will cause the rails to bend sideways which may cause derailment.

`text(Construction of Bridge)`
In construction of bridge, steel girders are used. One end of girder is kept fixed, but the other end is not fixed into the thick concrete (or brick) pillar. It is supported on rollers as shown in Figure so that if there is any rise or fall in temperature during summer or winter, girders may expand or contract without affecting the pillar.

`text(Telephone wires sag more in summer due to Expansion)`
If we look up at the telephone wires (or electric wires) fixed to the poles in summer, we find that they are quite loose. On the other hand, the same wires appear to be tightly fixed in winter. We say that the telephone wires sag more in summer. This is due to the fact that in summer, the temperature is high due to which the wires expand (increase in length) and sag more. In winter the temperature falls down due to which the wires contract (decrease in length) and become tight.

`text(The Pendulum of Clock)`
The pendulum of a clock is generally made of a metal like steel or brass. In summer, due to increase in atmospheric temperature, the length of pendulum increases and therefore the time taken for each oscillation increases and the dock loses time (i.e., it goes slow). In winter, the length of pendulum decreases and the clock gains time. (i.e., it becomes fast). To avoid it, the pendulum must be made of invar for which coefficient of linear expansion is very low.

`"Loosening a Glass Stopper or a Metal Screw Cap"`
A glass stopper or a metal screw cap on a bottle is loosened by warming the neck of bottle. The reason is that on warming the neck of bottle, it expands and so the glass stopper or the metal screw cap in it gets loosened.

`"Fitting the Steel Rim on a Horse-cart Wheel"`
The wheel of a horse cart is fitted with a steel rim. To ensure a tight fit, the rim is made slightly smaller in diameter than the wooden wheel. The steel rim is first heated uniformly till its diameter becomes slightly more than that of the wooden wheel. The rim is then slipped over the wooden wheel and it is then cooled. On cooling the rim contracts and make a tight fit on the wooden wheel.

`"Rivetting"`
For joining the two steel plates, they are placed one above the other and holes are drilled in them. The rivets (small steel rods) are made red hot and they are inserted in the holes of the plates. The ends of the rivet are hammered into the head. The hammering becomes easy because heating of rivets softens them. Now the rivets are allowed to cool. Since they cannot regain their original size due to contraction, they force the plates to come closer and firmly grip them together. Thus the joints become water-proof or steam-proof. Such rivetting is used in joining the steel girders and in joining the boiler plates etc.

`"Fusion the Platinum Wire in Glass Rod"`
The platinum (or nickel-iron alloy) wire is fused in glass rod. The reason is that both the platinum and glass have nearly the same coefficient of linear expansion so they contract equally on being cooled. If the platinum wire is fused in the rod of any other material or if the wire of any other metal is fused in the glass rod, on being cooled the rod will crack because of unequal contractions of the wire and the rod.

Liquids have only cubical expansion because they have no shape and size when liquids are heated, then the vessel, in which liquid is kept also heated. So, first vessel expands and then liquid.

Thermal expansion of liquids are of two types-
`"(i) Apparent: Expansion"`
The expansion of liquids neglecting expansion of vessel is called apparent expansion.
`"Apparent expansion coefficient"` (`gamma_a`) `="Apparent Increase in Volume"/"Initial Volume x Rise in Temperature" = (DeltaV_a)/(V xx Deltat)`

`"(ii) Real Expansion"`
It is the actual increase in volume of liquid.
`"Real expansion coefficient"` (`gamma_r`) `= "Real expansion in volume"/"Initial Volume x Rise in Temperature" = (DeltaV_r)/(V xx Deltat)`

`"Real expansion" = "Expansion of vessel" + "Apparent expansion"`

`"Coefficient of Real Expansion" (gamma_r)`
The true increase in volume of the liquid per unit original volume per degree rise in temperature is called the coefficient of real expansion. Its unit is `"per" ""^oC`

`"Coefficient of Apparent Expansion" (gamma_a)`
The apparent increase in volume of the liquid (relative to the container) per unit original volume per degree rise in temperature is called the apparent coefficient of expansion. Its unit is `"per" ""^oC`

`"Relation between " gamma_a " and " gamma_r`
`gamma_r = gamma_a + gamma_g`

where, `gamma_r =` real expansion of liquid.
`gamma_a =` apparent expansion of liquid.
`gamma_g =` volume expansion coefficient of material vessel.

• As the temperature of water increases from `0` to `4^o C`, the density of water increases and as temperature increases beyond `4^o C`, the density decreases.

• The variation in the density in the water with temperature is shown in the figure.

When a given mass of a solid or a liquid is heated, its volume increases. Accordingly, the density of a solid or a liquid decreases on heating.

Let `V` and `V'` be volumes of a solid (or a liquid) at temperature `T` and (`T + DeltaT`) respectively. If `gamma` is its coefficient of cubical expansion, then `V^' =V(1+gammaDeltaT)`

Let `rho` and `rho^'` be densities of the solid (or the liquid) at temperature `T` and (`T + DeltaT`).

`rho=M/V` and `rho^' =M/V^'`

In expansion of gases, there are two coefficients
`"(i) Volume Coefficient"` (`gamma_V`)
The change in volume of gas per unit volume per unit degree celsius at constant pressure is known as coefficient of volume expansion.

`"(ii) Pressure Coefficient"` (`gamma_p`)
The change in pressure of gas per unit degree celsius at constant volume is known as pressure coefficient.

• Heat energy is measured in calories.

• The quantity of heat energy required to raise the temperature of 1 g of pure water through 1 oc is called one calorie.

• The calorie is a very small unit of heat energy for practical purposes. Thus, a bigger unit called kilocalorie is used.

• The quantity of heat energy required to raise the temperature of 1 kg of pure water through `1^oC` is called one kilocalorie.
1 kilocalorie = 1000 calories.

• Kilocalorie is sometimes called big calorie or Doctor's calorie or Calorie (with capital C). The energy of the foods and the fuel is measured in kilocalories.

• We know energy is measured in Joules. As heat energy is a form of energy, therefore, it should also be measured in Joules, rather than calories or kilocalories, for strict scientific purpose. However, doctors still continue with kilocalories. Following are the equivalent of calorie and kilocalorie in joules.
(i) 1 calorie = 4.186 J = 4.2 J (approx.)
(ii) 1 kilocalorie = 4186 J = 4200 J (approx.)

• Heat energy absorbed and given out by a cold body is directly proportional to the mass of the body. If H is the heat energy absorbed or given out and m is the mass of the body, then `H prop m`.

• Heat energy absorbed and given out by a cold body is directly proportional to the temperature difference between the body and surrounding. If H is the heat energy absorbed or given out and `Deltat` is the temperature difference between the body and surrounding, then `H prop Deltat`.

• `H=mcDeltat`, where c is the constant of proportionality and is commonly called specific heat capacity.

• The amount of heat energy required to raise the temperature of unit mass of a substance through `1^oC` or `1 K` is called specific heat capacity.

• In the expression `H = mcDeltat` if there is a unit mass of substance (1 g), such that it is heated through `1^oC` by supplying `H` J of heat energy, then
Heat absorbed `H J=1 g xx c xx 1^oC`
`:.` Specific heat capacity, `c=H/(1xx1) (J/(g^oC))`

`"Units of Specific Heat Capacity"`
• In C.G.S. system unit of specific heat capacity is `"erg " g^(-1) ""^oC^(-1)`.

• In S.I. system unit of specific heat capacity is `J kg^(-1) K^(-1)`.

• Molar specific heat of a substance is defined as the amount of heat required to raise the temperature of one gram mole of the substance through a unit degree.

• It is represented by (capital) C, and is measured in `"cal " "mol"^(- 1) K^(-1)` or `J "mole"^(-1) K^(- 1)`.

• By definition, one mole of any substance is a quantity of the substance, whose mass in gram is numerically equal to the molecular mass M. Therefore, `C=Mc`

There are two types of molar specific heat
`"Molar specific heat at constant volume"` (`C_V`)
It is defined as the amount of heat required to raise the temperature of 1 mole of the gas through `1°C` (or 1 K). When its volume is kept constant.

`"Molar specific heat at constant pressure"` (`C_P`)
It is defined as the amount of heat required to raise the temperature of 1 mole of the gas through `1°C` (or 1 K), when its pressure is kept constant.

`"Mayer's Formula"`
If specific heat at constant pressure (`C_P`) is greater the specific heat at constant volume (`C_V`), then molar specific heat,
`C_P - C_V =R`
where R = gas constant

• Amount of heat energy required to raise the temperature of one gram of water through `1^oC` is one calories (by definition), therefore
specific heat of water `= 1 cal//g ""^oC = 1 cal//g K`

• Specific heat of ice `= 0.5 cal//g ""^oC`

• Specific heat of steam `= 0.47 cal//g ""^oC`

• Thermal capacity of a body is defined as the amount of heat required to raise the temperature of the (whole) body through `1^oC` or `1 K`. We know that amount of heat energy required (`DeltaQ`) to raise the temperature of mass `m` of a body through temperature range (`DeltaT`) is

`DeltaQ=cm DeltaT`, where c is specific heat of the body

• When `DeltaT=1`, Thermal Capacity `S=c xx m`

• It is measured in `JK^(-1)` or `"cal " ""^oC^(-1)`.

• The water equivalent of a body is defined as the mass of water which requires the same amount of heat as is required by the given body for the same rise of temperature.

• Water equivalent = Mass x Specific heat
`w=mc`

• The SI unit of water equivalent is kg.

• The amount of heat required to change the state of unit mass of a substance at constant temperature is called latent heat of the substance. If mass m of a substance undergoes a change from one state to another, then the amount of heat required for the process is
`Q=mL`
Where, L is the latent heat of the substance.

• The SI unit of latent heat is J/Kg and the CGS unit of latent heat is cal/g.

• There are two types of latent heat
`"Latent heat of Fusion"`
* The amount of heat required to change the state of unit mass of a substance from solid to liquid at its melting point is called latent heat of fusion.
* In case of ice the latent heat of fusion of ice is 80 cal/gm.

`"Latent heat of vaporisation"`
* The amount of heat required to change the state of unit mass of a substance from liquid to vapour at its boiling point is called latent heat of vaporisation.
* In case of water the latent heat of vaporisation is 536 cal/gm.

`"Note"`
Latent heat of vaporisation is more than the latent heat of fusion. This is because when a substance gets converted from liquid to vapour, there is a large increase in volume. Hence, more amount of heat is required. But when a solid gets converted to a liquid, then the increase in volume is negligible. Hence, very less amount of heat is required.

`"Melting and Melting Point"`
The process of change of state from solid to liquid is called melting. The temperature at which solid starts to liquefy is known as the melting point of that solid.
The melting point of a substance at atmospheric pressure is called normal melting point.

`"Fusion and Freezing Point"`
The process of change of state from liquid to solid is called fusion. The temperature at which liquid starts to freeze is known as the freezing point of the liquid.

`"Vaporisation and Boiling Point"`
The process of change of state from liquid to vapour (or gas) is called vaporisation. During the change of state (completely), the temperature remains constant which implies both liquid and vapour are at the thermal equilibrium. The temperature at which the liquid starts to evaporate is called the boiling point of the liquid.

`"Sublimation"`
The process of change of state directly from solid to vapour (or gas) is known as sublimation. There is no matter of liquid state of substance. The reverse process of sublimation is not possible e.g., camphor, nepthalene balls etc.

• The branch of physics that deals with the measurement of heat is called calorimetry.

• The principle of calorimetry states that the heat gained by the cold body must be equal to the heat lost by hot body, provided there is no exchange of heat with the surroundings.

Heat gained = Heat lost

• This principle is a consequence of the law conservation of energy.

• This principle is a consequence of the law of conservation of energy.

• All matter is trade up of molecules. The molecules of a gas are in state of rapid and continuous motion. Their ·velocity depends on temperature. Using this molecular motion, various properties of a gas like temperature, pressure, energy etc can be explained. Hence, this theory is called kinetic theory of gases.

• Kinetic theory of gases was developed by Claussius and Maxwell.

`text(Assumptions of Kinetic Theory of Gases )`
The entire structure of the kinetic theory of gases is based on the following assumptions

• All gases consist of molecules. The molecules are rigid, elastic spheres identical in all respects for a given gas and different for different gases.

• The size of the gas molecules is very small as compared to the distance between them.

• The molecules of a gas are in a state of continuous random motion, moving in all directions with all possible velocities.

• During the random motion, the molecules collide with one another and with the walls of the vessel.

• The collisions are perfectly elastic and there are no forces of attraction or repulsion between the molecules.

• Between two collisions a molecule moves in a straight path with a uniform velocity.

• The collisions are almost instantaneous i.e., the time of collision of two molecules is negligible as compared to time interval between two successive collisions.

• Inspite of the molecular collisions, the density remains uniform throughout the gas.

Mass (m), volume (V ), pressure (p) and temperature (T) of a gas are the measurable properties. The Law's which inter-relate these properties, are called gas Laws. Let's discuss the various gas laws which give the between measurable properties of gases.

`text(Boyle's law)`
It states that for a given mass of an ideal gas at constant temperature (called isothermal process), the volume of a gas is inversely proportional to its pressure,
i.e. `V prop 1/p`

or pV = constant or `p_1V_1 = p_2V_2 = p_3V_3 = ...`

This law can also be shown graphically.

`text(Charles' Law)`
It states that for a given mass of an ideal gas at constant pressure, (called isobaric process) volume of a gas is directly proportional to its absolute temperature
i.e. `V prop T` [ if m and p are constant ]

or `V/T` = constant or ` V_1 /T_1 = V_2/T_2`

This law can also be shown graphically.

`text(Gay-Lussac's Law or Pressure Law)`
It states that for a given mass of an ideal gas at constant volume (called isochoric process), pressure of a gas is directly proportional to its absolute temperature
i.e `p prop T` [ if m and V are constants ]

or `p/T = ` constant or `p_1/T_1 = p_2/T_2`
Here , temperature is in kelvin

This law can also be shown graphically.

An ideal or a perfect gas is that gas which strictly obey the gas law, (such as Boyle's law , Charle's law , Gay lussac's law etc )

Following are the characteristics of the ideal gas

• The size of the molecule of an ideal rs zero, i.e., each molecule of the ideal gas is a point mass with no dimensions.

• There is no force of attraction or repulsion amongst the molecules of an ideal gas.

`text(Equation of State or Ideal Gas Equation)`
The equation which relates the pressure (P), volume (V) and temperature (T) of the given state of an ideal gas is known as ideal gas equation or equation of state.
For 1 mole of gas `(PV)/T =R` (constant ) `=> PV =RT`

Where , `R = ` universal gas constant

`text(Vander Waal's Gas Equations)`
• For 1 mole of gas `( P + a/V^2) (V - b ) = RT`

• For `mu` moles of gas `( P + (a mu^2)/V^2) ( V - mu b ) = mu RT`
Here, a and b are constant i.e. called Vander Waal's constant.

• Real gases obey this equation at high pressure and low temperature.

`"Critical temperature"` (`T_c`)
The maximum temperature below which a gas can be liquefied by pressure done is called critical temperature and it is characteristic of the gas. A gas cannot be liquefied, if its temperature is more than critical temperature.

e.g. `CO_2(31.1°C), O_2 (- 118°C), N_2 (- 147.1°C)` and `H_2O (374.1 °C)`.

`"Critical pressure"` (`P_c`)
The minimum pressure necessary to liquify a gas at critical temperature is defined as critical pressure.

e.g. `CO_2` (73.87 bar) and `O_2` (49.7 atm).

`"Critical volume"` (`V_c`)
The volume of 1 mole of gas at critical pressure and critical temperature is defined as critical volume.

e.g `CO_2 ( 95 xx 10^-6 m^3)`

Molecules of gases collide with each other and also collide with the walls of the vessel. Thus, gas applies a pressure on the walls of the container, this pressure is called gaseous pressure.

`p =1/3 (mn)/V barv^2`

`=> P =1/2 rho barv^2`

where, m = mass of one molecule,
n= number of molecules of the gas,
V = volume of the vessel,
` bar v^2 = ` root mean square velocity
and `rho =` density of gas