1. Concept of Temperature:

(i) Temperature is the degree of hotness or coldness of a body. It is a flow of heat from higher temperature to lower temperature.

(ii) When two bodies are at the same temperature, no heat flows from one body to the other (thermal equilibrium).

(iii) Temperature is a fundamental scalar physical quantity. S.I. unit $Kelvin$ . Dimension $= K$ .

(iv) $NTP$ or $STP$ implies $273.15 K$ or $(0 ° C)$

2. Temperature Scales:

Name of the scale	Symbol	Lower fixed point (LFP)	Upper fixed point (UFP)	Number of divisions on the scale
Celsius	$° C$	$0 ° C$	$100 ° C$	$100$
Fahrenheit	$° F$	$32 ° F$	$212 ° F$	$180$
Kelvin	$K$	$273.15 K$	$373 .15 K$	$100$

$\frac{C - 0}{100 - 0} = \frac{F - 32}{212 - 32} = \frac{K - 273.15}{373.15 - 273.15} = \frac{X - LFP}{UFP - LFP}$

$\Rightarrow \frac{Δ C}{100} = \frac{Δ F}{180} = \frac{Δ K}{100} = \frac{Δ X}{UFP - LFP}$

3. Thermometry:

In some substances, few physical properties varies linearly with the change of temperature and it is used for temperature measurement.

(i) Length of liquid in the tube

(ii) Pressure of gas (at constant volume)

(ii) Volume of gas (at constant pressure)

4. Types of Thermometers:

(i) Mercury thermometers: Mercury has the large thermal coefficient of expansion and change is uniform. It also has high thermal conductivity and low specific heat, two desirable properties for good thermometers,

$T = \frac{l - l_{0}}{l_{100} - l_{0}} \times 100 ° C$ ,

where $l =$ length of the mercury column at a given temperature, $l_{0} =$ length of the mercury column at the freezing point of water and $l_{100} =$ length of the mercury column at the boiling point of water

(ii) Constant pressure gas thermometers:

If pressure is constant, then $V \propto T$ ,

$T = \frac{V - V_{0}}{V_{100} - V_{0}} \times 100 ° C$ ,

where $V =$ volume at a given temperature, $V_{0} =$ volume at $0 ° C$ and $V_{100} =$ volume at $100 ° C$

(iii) Constant volume gas thermometers:

If volume is constant, then $P \propto T$ ,

$T = \frac{P - P_{0}}{P_{100} - P_{0}} \times 100 ° C$ ,

where $P =$ pressure of the gas at a given temperature, $P_{0} =$ pressure at $0 ° C$ and $P_{100} =$ pressure at $100 ° C$

5. Thermal Expansion:

Increase in temperature results in an increase in the amplitude of vibration and energy of atoms which leads to an increase in the distance between the atoms.

(i) Linear expansion (in solids):

(a) $l = l_{0} (1 + α Δ T)$ , where $l =$ length of the rod at any given temperature, $l_{0} =$ length of the rod at $0 ° C$ , $α =$ coefficient of linear expansion and $Δ T =$ change in temperature

(b) Unit of $α$ : $° C^{- 1}$ or $K^{- 1}$ .

(ii) Areal expansion (in solids):

(a) $A = A_{0} (1 + β Δ T)$ , where $A =$ area at a given temperature, $A_{0} =$ area at temperature $0 ° C$ and $β =$ coefficient of aerial expansion

(b) Unit of $β$ : $° C^{- 1} or K^{- 1}$ .

(iii) Volume expansion (in solids, liquids and gases):

(a) $V = V_{0} (1 + γ Δ T)$ , where $V =$ volume at a given temperature, $V_{0} =$ volume at $0 ° C$ and $γ =$ coefficient of volume expansion

(b) Unit of $γ$ : $° C^{- 1} or K^{- 1}$ .

(iv) For isotropic solids,

$α : β : γ = 1 : 2 : 3$

(v) Thermal expansion of an isotropic object may be imagined as a photographic enlargement.

(vi) For anisotropic materials, $β_{xy} = α_{x} + α_{y}$ and $γ = α_{x} + α_{y} + α_{z}$

(vii) If $α$ is variable: $Δ l = \int_{T_{1}}^{T_{2}} l_{0} α d T$

6. Thermal Expansion Applications

(i) Bi-metallic strip: Different materials have different coefficient of linear expansion and if joined together, the strip will bend with metal of greater $α$ on outer side (convex side). It is used to make or break electrical contact.

(ii) Effect on the time period of a simple pendulum:

As the temperature is increased, length of the pendulum increases due to the change in thermal expansion and hence, time period will increase. It becomes slower in summer and loses time.

Fractional change in time-period, $\frac{Δ T}{T} = \frac{1}{2} α Δ θ$ , where $Δ T =$ change in time period, $T =$ time period and $Δ θ =$ change in temperature

(iii) Thermal stress: If rod ends are fixed to prevent expansion or contraction, then due to thermal expansion or contraction, compressive or tensile stress is developed. Hence, the rod will apply a large force on the wall.

(a) Thermal strain $= \frac{Δ L}{L} = α Δ θ$

(b) Thermal stress $= Y α Δ θ$ , where $Y =$ Young's modulus

(iv) Error in scale reading:

If scale gives correct reading at temperature $θ$ , then at temperature $θ' (> θ)$ , it will expand and the reading will be lesser than the true value.

True value $=$ Scale reading $[1 + α (θ' - θ)]$

7. Thermal Expansion of Gases:

(i) Coefficient of volume expansion, $γ_{V} = \frac{Δ V}{V_{0} Δ T} = \frac{1}{T}$

$[P V = n R T$ at constant pressure, $V \propto T \Rightarrow \frac{Δ V}{V} = \frac{Δ T}{T}]$

(ii) Coefficient of pressure expansion, $γ_{P} = \frac{Δ P}{P_{0} Δ T} = \frac{1}{T}$

8. Thermal Expansion in Liquids:

On heating, both liquid and vessel expands, therefore increase in the volume of the liquid $=$ The apparent increase in the volume of liquid $+$ the increase in the volume of the vessel.

(i) Co-efficient of apparent expansion $(γ_{a})$ : It is due to the apparent(that appears to be, but is not) increase in the volume of liquid if expansion of vessel containing the liquid is not taken into account,

$γ_{a} = \frac{Apparent expansion in volume}{Initial volume \times Δθ} = \frac{{(Δ V)}_{a}}{V \times Δ θ}$

(ii) Co-efficient of real expansion $(γ_{r})$ : It is due to the actual increase in the volume of liquid due to heating,

$γ_{r} = \frac{Real increase in volume}{Initial volume \times Δθ} = \frac{{(Δ V)}_{r}}{V \times Δ θ}$

(iii) $γ_{Real} = γ_{Apparent} + γ_{Vessel}$

(iv) Anomalous expansion of water:

It expands on heating if its temperature is greater than $4 ° C$ . In the range $0 ° C$ – $4 ° C$ , water contracts on heating and expands on cooling, i.e., $γ$ is negative. This behaviour of water in the range from $0 ° C$ to $4 ° C$ is called anomalous expansion.

9. Heat:

A form of energy that is transferred between objects with different temperatures. It is a scalar quantity.

Unit: $Joule$

(i) Thermal capacity of a body $= \frac{Q}{Δ T}$ , where $Q =$ amount of heat given and $Δ T =$ change in temperature

It is the amount of heat required to raise the temperature of a given body by $1 ° C$ or $1 K$ .

(iii) Specific heat capacity $= \frac{Q}{m Δ T}$ ( $m =$ mass)

It is the amount of heat required to raise the temperature of unit mass of a body through $1 ° C$ or $1 K$ .

(iv) Molar heat $= \frac{Q}{n Δ T}$ ( $n =$ number of moles)

(v) Water equivalent: Mass of water which when given the same amount of heat as to the body, it changes.

The temperature of water through the same range as that of the body.

Water equivalent of the body,

$W = (mass of body) \times (\frac{specific heat of body}{specific heat of water})$

Unit: $g$ or $kg$ .

(vi) Latent heat (hidden heat): The amount of heat that has to be supplied to (or removed from) a body for its complete change of state (from solid to liquid, liquid to gas, etc.) is called latent heat of the body. Remember that phase transformation is an isothermal (i.e., temperature $=$ constant) change.

10. Principle of Calorimetry:

Heat lost $=$ heat gained.

For temperature change $Q = m s Δ T$ and for phase change $Q = m L$ , where $s =$ specific heat of substance and $L =$ latent heat of substance

11. Heating Curve:

If a given mass $(m)$ of a solid, heat $(Q)$ is supplied at a constant rate and a graph is plotted between temperature and time, the graph is called the heating curve.

where $M . P . =$ melting point and $B . P . =$ boiling point

(i) Specific heat (or thermal capacity) $\propto \frac{1}{slope of curve}$

(ii) Latent heat $\propto$ length of horizontal line.

12. General Points:

(i) Specific heat of a body may be greater than its thermal capacity as mass of the body may be less than unity.

(ii) The steam at $100 ° C$ causes severe burn to human body than the water at $100 ° C$ , because steam has greater internal energy than water due to latent heat of vaporization.

(iii) Heat is energy in transit which is transferred from hot body to cold body.

(iv) One calorie is the amount of heat required to raise the temperature of $1 g$ of water through $1 ° C$ (more precisely from $14. 5 ° C$ to $15.5 ° C$ ).

13. Heat Transfer:

There are three modes by which heat energy transfers from a body at a higher temperature to a body at a lower temperature.

(i) In conduction, heat is transferred from one point to another without the actual motion of heated particles.

(ii) In the process of convection, the heated particles of matter actually move.

(iii) In radiation, the intervening medium is not affected and heat is transferred without any material medium.

Conduction	Convection	Radiation
Heat transfer due to temperature difference	Heat transfer due to density difference	Heat transfer without any medium
Due to free-electron or vibration motion of molecules	Actual motion of particles	Electromagnetic radiation
Heat transfer in solid body (in mercury also)	Heat transfer in fluids (liquid $+$ gas)	All
Slow process	Slow process	Fast process $(3 \times 10^{8} m s^{- 1})$
Irregular path	Irregular path	Straight line (like light)

14. Thermal Conduction:

(i) Variable state: In this state, temperature of each part of the rod increases.

(ii) Steady state: At this stage, temperature of each part of the rod becomes constant.

(iii) Temperature gradient: Rate of change of temperature with the distance between any two points along the length of the rod.

Temperature gradient $= \frac{- Δ T}{Δ x}$

Unit: $K m^{- 1}$ or $° C m^{- 1}$

(iv) Rate of heat flow (in steady state), $\frac{d Q}{d t} = - K A \frac{d T}{d x}$ or $\frac{Q}{t} = \frac{K A (T_{1} - T_{2})}{l}$ , where $d T =$ change in temperature on moving from one point to another, $d x =$ change in length in moving from one point to another along the flow of heat, $K =$ coefficient of thermal conductivity and $A =$ area of cross-section of the rod.

(v) Thermal resistance, $R_{H} = \frac{l}{K A}$

Equivalent thermal resistance in series and parallel connection can be written as shown in the figure.

where $K_{1}, K_{2} =$ thermal conductivity of the given rods made of different materials.

(vi) Growth of ice on ponds:

Time taken by ice to grow a thickness from $x_{1}$ to $x_{2}, t = \frac{ρ L}{2 K θ} (x_{2}^{2} - x_{1}^{2})$ , where $K =$ thermal conductivity of ice, $ρ =$ density of ice, $θ =$ magnitude of the temperature of the surrounding and $L =$ latent heat of ice,

15. Radiation:

Energy transfer in the form of electromagnetic waves through any medium.

Due to incident radiations on the surface of a body, following phenomena occur by which the radiation is divided into three parts.

(i) Reflection

(ii) Absorption

(iii) Transmission

From energy conservation,

(a) $Q = Q_{r} + Q_{a} + Q_{t}$ , where $Q =$ amount of radiation incident, $Q_{r} =$ amount of reflected off radiation, $Q_{a} =$ amount of radiation absorbed by a surface and $Q_{t} =$ amount of radiation transmitted through body,

$\Rightarrow \frac{Q_{r}}{Q} + \frac{Q_{a}}{Q} + \frac{Q_{t}}{Q} = 1$

$\Rightarrow r + a + t = 1$

(b) Reflectance or reflecting power: $r = \frac{Q_{r}}{Q}$

(d) Transmittance or transmitting power: $t = \frac{Q_{t}}{Q}$

(e) $r = 1$ and $a = 0, t = 0 \Rightarrow$ Perfect reflector

(f) $a = 1$ and $r = 0, t = 0 \Rightarrow$ Ideal absorber (ideal black body)

(g) $t = 1$ and $a = 0, r = 0 \Rightarrow$ Perfect transmitter (Daithermanons)

(h) Reflection power: $(r) = [\frac{Q_{r}}{Q} \times 100] %$

(i) Absorption power: $(a) = [\frac{Q_{a}}{Q} \times 100] %$

(j) Transmission power: $(t) = [\frac{Q_{t}}{Q} \times 100] %$

(k) A hot body emits radiation of all wavelengths.

(l) The wavelength corresponding to maximum emission of radiations shifts from longer wavelength to shorter wavelength as the temperature increases. Due to this, the color of a body appears to be changing.

16. Prevost's Theory of Heat Exchange:

A body is simultaneously emitting radiations to its surroundings and absorbing radiations from the surroundings.

17. Stefan's Boltzmann Law:

Radiated energy emitted by a perfect black body per unit area per sec is $E = σ T^{4}$ , where $T =$ temperature of the black body.

(i) For a general body, $e = ε E = ε σ T^{4}$ , where $0 \leq ε \leq 1$ , $ε =$ emissivity, $σ =$ Stefan's constant and $e =$ total emittance or total emissive power.

(ii) If surrounding has temperature $T_{0}$ , then $e = ε σ (T^{4} - T_{0}^{4})$ .

(iii) Emissivity is defined as the ratio of the energy radiated from a material's surface to that radiated from a perfect emitter known as blackbody, at the same temperature and wavelength and under the same viewing conditions. It is a dimensionless number between $0$ (for a perfect reflector) and $1$ (for a perfect emitter).

18. Newton's Law of Cooling:

If temperature difference is small.

Rate of cooling $\frac{d θ}{d t} \propto (θ - θ_{0})$ , where $\frac{d θ}{d t} =$ rate of change of temperature of an object, $θ =$ temperature of the object at any general time and $θ_{0} =$ temperature of the surrounding.

$\Rightarrow θ = θ_{0} + (θ_{1} - θ_{0}) e^{- k t}$ , where $k =$ constant and $θ_{1} =$ initial temperature of the object

when a body cools from $θ_{1}$ to $θ_{2}$ in time $t$ in a surrounding of temperature $θ_{0}$ then,

$\frac{θ_{1} - θ_{2}}{t} = k [\frac{θ_{1} + θ_{2}}{2} - θ_{0}]$ [where $k =$ constant]

19. Kirchhoff's Law:

The ratio of emissive power to absorptive power is the same for all surfaces at the same temperature and is equal to the emissive power of a perfectly black body at that temperature, $\frac{e_{1}}{a_{1}} = \frac{e_{2}}{a_{2}} = {(\frac{E}{A})}_{Perfectly black body}$ but for a perfectly blackbody $A = 1$ , i.e., $\frac{e}{a} = E$ .

Therefore, a good absorber is a good emitter.

(i) Perfectly black body: A body that absorbs all the radiation incident on it is called a perfectly black body.

(ii) Absorptive power $(a)$ : Absorptive power of a surface is defined as the ratio of the radiant energy absorbed by it in a given time to the total radiant energy incident on it in the same time.

For an ideal black body, absorptive power $= 1$

(iii) Emissive power $(ε)$ : For a given surface, it is defined as the radiant energy emitted per second per unit area of the surface.

20. Wien's Displacement Law:

Product of the wavelength $λ_{m}$ of the most intense radiation emitted by a black body and absolute temperature of the black body is a constant. $λ_{m} T = b = 2.89 \times 10^{- 3} mK =$ Wein's constant

Area under $e_{λ} - λ$ graph $= \int_{0}^{\infty} e_{λ} d λ = e = σ T^{4}$

21. Solar Constant:

The energy emitted by the sun per second (where $R$ is the radius of the sun and $T$ its temperature),

This energy will spread over a sphere of radius $r (=$ average distance between sun and earth) while reaching earth, so the intensity of the solar radiation at the surface of the earth (called solar constant $S$ ) will be given by,

$S = \frac{P}{4 π r^{2}} = \frac{4 π R^{2} σ T^{4}}{4 π r^{2}} = 5.67 \times 1 0^{- 8} W m^{- 2} K^{- 4}$

Variation of molar specific heat of a metal with temperature is best depicted by

Important Questions on Thermal Properties of Matter

Learn and Prepare for any exam you want !