Thermal Properties of Matter

Thermal Properties of Matter" focuses on how different types of materials respond when exposed to heat. It covers key ideas like heat and temperature, specific heat capacity, and changes in the state of matter, such as melting or boiling. It also looks at latent heat, which is the energy absorbed or released during these changes without a change in temperature. Another important part of the topic is thermal expansion—how materials expand when heated. This can happen in different ways: linear (length), superficial (area), or cubical (volume).The topic also explores how heat is transferred through materials using conduction (through solids), convection (through liquids and gases), and radiation (without any medium). It includes important laws like Newton’s Law of Cooling, which explains how objects lose heat to their surroundings, and Stefan’s Law, which relates to the heat radiated by a body.

1.0Heat and Temperature

Thermal equilibrium: Two bodies are said to be in thermal equilibrium, when no (net) heat flows from one body to the other i.e., when both the bodies are at the same temperature.

Zeroth law of thermodynamics: If two objects, A and B, are each in thermal equilibrium with a third object C (such as a thermometer), then A and B are also in thermal equilibrium with each other. This is the essence of the Zeroth Law of Thermodynamics, which establishes the concept of temperature. According to this law, two systems are considered to be in thermal equilibrium if they share the same temperature.                      

2.0Temperature Scale

Relation between different scale 

$\frac{C}{5}=\frac{F-32}{9}=\frac{K-273}{5}=\frac{A-LFP}{UFP-LFP}$

Absolute Zero: At this temperature kinetic energy of gas molecules becomes zero.Negative temperature is not possible in kelvin scale. Absolute Zero=-273.15°C or 0 Kelvin

3.0Specific Heat Capacity

The amount of heat required to change the temperature of a unit mass of a substance by 1°C or 1K is called specific heat.

$S=\frac{Q}{m\Delta T}$

Q=heat, S=Specific Heat Capacity, M=mass of Substance

$\Delta T$=Change In Temperature

Unit SI-Joule/kg-K   ,CGS-cal/g-°C

1. Molar Specific Heat($S_{mol}$): The amount of energy needed to raise the temperature of one mole of a substance by 1°C (or 1K) is called molar heat capacity. The molar heat capacity is the product of molecular weight and specific heat i.e.,If the molecular mass of the substance is M and the mass of the substance is m then number of moles of the substance

$S_{\text{mol}}=\frac{M}{w}\left(\frac{Q}{m\Delta T}\right)$ $\left[\because \mu =\frac{m}{w}\right]$

$S_{\text{mol}}=\frac{Q}{\mu \Delta T}\Rightarrow S=\frac{Q}{\mu \Delta T}=Q\left[\because \mu =1\text{ mole and }\Delta T=1K\right]$

$\text{Unit:}SI\to \frac{J}{\text{mol}\cdot K}or\frac{J}{{\text{mol}}^{\circ }C};CGS\to \frac{\text{cal}}{{\text{mol}}^{\circ }C}$

2. Gram Specific Heat $(S_{gram})$:The amount of energy needed to raise the temperature of one gram of a substance by 1°C (or 1K) is called gram heat capacity.

$S_{\text{gram}}=\frac{Q}{m\Delta T}$

If m =1gram and $\Delta$T =1 K

$S_{gram}=Q$

$\text{Unit:}SI\to \frac{J}{\text{gram}\cdot K}or\frac{J}{{\text{gram}}^{\circ }C};CGS\to \frac{\text{cal}}{{\text{gram}}^{\circ }C}$

Relation between Gram and Molar Specific heat

$Q=m{S}_{\text{gram}},\Delta T=\mu S_{\text{mol}}\Delta T$

$m{S}_{\text{gram}}=\mu S_{\text{mol}}$

$m{S}_{\text{gram}}=\frac{m}{M}{S}_{\text{mol}}$

$S_{\text{mol}}=\frac{M}{w}{S}_{\text{gram}}$

4.0Change of State and Latent Heat

The heat required for a phase or state change,

$Q=mL\Rightarrow L=\frac{Q}{m}=,L$ = latent heat.

Unit:SI=J/kg or CGS=cal/g

1. Latent heat (L): The heat supplied to a substance of unit mass that causes a change in its state at a constant temperature.      
2. Latent heat of Fusion ($L_f$): The heat supplied to a substance that changes it from solid to liquid at its melting point and 1 atm pressure is called the latent heat of fusion. For ice, the latent heat of fusion is 80 kcal/kg.       
3. Latent heat of vaporization ($L_v$):The heat supplied to a substance that changes it from liquid to vapor at its boiling point and 1 atm pressure is called the latent heat of vaporization. For water, the latent heat of vaporization is 540 kcal/kg.  

5.0Heating Curve

6.0Thermal Expansion

• Thermal Expansion occurs when matter is heated without a change in state, leading to an increase in size.
• According to the atomic theory of matter, thermal expansion is due to the asymmetry of the potential energy curve between atoms.
• As temperature increases (from $T_{1}to{T}_2$ ), atomic vibration amplitude and energy (from $E_{1}to{E}_2$) increase, causing the average interatomic distance to grow (from $r_{1}to{r}_2$).
• This increase in distance leads to the overall expansion of matter. If the potential energy curve were symmetric, no expansion would occur upon heating.
• Solids exhibit minimal thermal expansion due to strong intermolecular forces, while gases expand the most due to weak intermolecular forces.
• Solids can expand:
  In one dimension → Linear expansion
  In two dimensions → Superficial expansion
  In three dimensions → Volumetric expansion
• Liquids and gases primarily undergo volume expansion.

7.0Potential Energy v/s Distance Curve

• Heating matter usually causes it to expand.
• According to atomic theory, thermal expansion is due to asymmetry in the potential energy curve.
• As temperature rises  the amplitude of atomic vibrations and their energy increases
• This results in an increase in the average distance between atoms .
• If the potential energy curve were symmetrical, no thermal expansion would occur.

                   $T_2>T_2,E_2>E_1,r_{2avg.}>r_{1avg.}$

8.0Linear Expansion

When the rod is heated, its increase in length ΔL is proportional to its original length L0 and change in temperature ΔT where ΔT is in °C or K.

$L=L_0(1+\alpha \Delta T)$ where L is the length after heating the rod.

Note: Thermal expansion is typically a 3-D phenomenon. However, when the other two dimensions of an object are negligible compared to one, the expansion is significant only in that dimension, and this is referred to as linear expansion.

9.0Superficial (Areal) Expansion

When the temperature of a 2D object is changed, its area changes, then the expansion is called superficial expansion. 

$A=A_0(1+\beta \Delta T)$

$\beta =2\alpha$

10.0Volume Expansion

When a solid is heated and its volume increases, then the expansion is called volume or cubical expansion.

$V=V_0(1+\gamma \Delta T)$

$\gamma =3\alpha$

$\alpha :\beta :\gamma =1:2:3$

Unit of $\alpha :\beta :\gamma$ is 1/°C or 1/K or K-1

11.0Variation of Time Period of Pendulum Clocks                 

• A pendulum clock's time is based on the pendulum's oscillations.
• The second hand advances by one second when the pendulum reaches its extreme position.
• The second hand moves by two seconds for each complete oscillation.
• A pendulum with a 2-second time period is called a "seconds pendulum."

Therefore change (loss or gain) in time per unit time lapsed is $\frac{T^{\prime }-T}{T}=\frac{1}{2}\alpha \Delta \theta$

Gain or loss in time in duration of 't' in $\Delta T=\frac{1}{2}\alpha \Delta \theta T$ if T is the correct time then

1. $\theta <{\theta }_0,T<T^{\prime }\Rightarrow$clock becomes fast and gain time
2. $\theta >{\theta }_0,T^{\prime }>T\Rightarrow$clock becomes slow and loose time

12.0Measurement of Length by Metallic Scale

Case (i) : When object is expanded only; No expansion of scale

$l_2=l_1[1+{\alpha }_0({\theta }_2-{\theta }_1)]$

Case (ii) : When only the measurement instrument is expanded, the actual length of the object will not change but the measured value (MV) decreases.          

$MV=l_1[1+{\alpha }_s({\theta }_2-{\theta }_1)]$

Case (iii): If both expanded simultaneously                        

$MV=l_1[1+({\alpha }_0-{\alpha }_s)({\theta }_2-{\theta }_1)]$

13.0Thermal Stress of a Material

When the temperature of the rod is decreased or increased under constrained conditions, compressive or tensile stresses are developed in the rod. These stresses are known as thermal stresses.

$Strain=\frac{\Delta L}{L_0}=\frac{FinalLength-OriginalLength}{OriginalLength}=\alpha \Delta T$

original length refers to the natural length of rod at new temperature.

14.0The Bimetallic Strip

• A bimetallic strip is made of two metals with different coefficients of linear expansion (e.g., brass and steel).
• Brass (α = 19 × 10⁻⁶ /°C) has a higher expansion coefficient than steel (α = 12 × 10⁻⁶ /°C).
• The metals are joined together by welding or riveting.
• When heated up , the brass expands more than the steel, causing the strip to bend.
• The brass side of the strip has a larger radius than the steel side.
• Upon cooling, the strip bends in the opposite direction.

Thermal Expansion in Liquids

• Liquids experience volume expansion instead of linear or superficial expansion.
• Initially, as both the liquid and vessel are heated, the liquid level falls due to the vessel expanding more than the liquid.
• Later, the liquid level rises as the liquid expands faster than the vessel.
• The actual increase in the liquid's volume is the sum of the apparent volume increase and the vessel's volume increase.

Co–Efficient of Real Expansion $({\gamma }_r)$

${\gamma }_r=\frac{Realincreaseinvolume}{InitialVolume\times \Delta \theta }=\frac{(\Delta V)}{V\times \Delta \theta }$

Co-Efficient of expansion of flask

${\gamma }_{Vessel}=\frac{(\Delta V{)}_{Vessel}}{V\times \Delta \theta }$

${\gamma }_{real}={\gamma }_{Apparent}+{\gamma }_{Vessel}$

15.0Variation of Density with Temperature

• Mass of substance does not change with change in temperature so with increase of temperature, volume increases so density decreases and vice-versa. $d=\frac{d_0}{(1+\gamma \Delta T)}$
• For solids values of  are generally small $d=d_0(1-\gamma \Delta T)$

Anomalous expansion of water

• Water's density increases from 0°C to 4°C, making negative; from 4°C onward, is positive.
• At 4°C, water's density is maximum.
• This causes ice to form at the surface of lakes in cold weather.
• As winter approaches, surface water cools, becomes denser, and sinks, leading to the formation of ice at 0°C.
• Ice on the surface slows further freezing, as ice is a poor heat conductor.
• Aquatic life survives winter because the lake bottom remains at around 4°C, unfrozen.

Variation of Force of Buoyancy with Temperature       

$F_B^{\prime }=F_B=[1+({\gamma }_s-{\gamma }_L)\Delta \theta ]$

16.0Calorimetry

When two bodies at different temperatures are brought into contact, heat flows from the body at a higher temperature to the one at a lower temperature until thermal equilibrium is reached. The hotter body loses heat, while the cooler body gains it, resulting in both reaching the same final temperature.

Heat Lost = Heat Gained (Principle of calorimetry)

The principle of calorimetry represents the law of conservation of heat energy.

$m_1{s}_1\Delta T_1=m_2{s}_2\Delta T_2$

$m_1{s}_1(T_1-T_{mix})=m_2{s}_2(T_{mix}-T_2)$ (s = specific heat)

$(T_1>T_{mix}>T_2)$

Water Equivalent

The water equivalent of a body is the mass of water (W in grams) that would absorb or release the same amount of heat as the body does when its temperature changes by the same amount.

$(H_C{)}_{substance}=(H_C{)}_{water}$

$M_s{S}_s=M_w{S}_w$

$M_s{S}_s=W\times 1cal/g°C$

Water Equivalent = W in gram

• Numerically water equivalent and heat capacity are same but their unit is different.

Note : whenever water equivalent or heat capacity of container is given in that case heat consumed by container will also be calculated.

17.0Heat Transfer

• Conduction: Conduction, or thermal Conduction, is a process of heat transfer in which heat is transferred from one particle to another without dislocating the particle from its equilibrium position.
• Transient State : In this state, the temperature of each part of the rod varies with time.
• Steady state : After a long time, when any part absorbs no heat, the temperature of every part is constant and decreases uniformly from the hotter end to the colder end. 

Note: In steady state, each point has a different temperature but remains constant.

18.0Fourier’s Law of Heat Conduction

$i_T=\frac{dQ}{dt}=-KA(\frac{dT}{dx})$

K= Coefficient of Thermal Conductivity

Series Combination

• In Series Combination

$R_{eq}=R_1+R_2$

$\Rightarrow \frac{(L_1+L_2)}{K_{eq}.A}=\frac{L_1}{K_{1}A}+\frac{L_2}{K_{2}A}$

• Equivalent Thermal Conductivity is

$K_{eq}=\frac{L_1+L_2}{\frac{L_1+L_2}{K_1+K_2}}=\frac{\Sigma (L_i)}{\Sigma \frac{L_i}{K_i}}$

If lengths of the rod are the same then (L1=L2=L)

$K_{eq}=\frac{2K_1{K}_2}{K_1+K_2}$

Parallel Combination

• In Parallel Combination

$\Rightarrow \frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}$

$\Rightarrow \frac{K_{eq}.(A_1+A_2)}{L}=\frac{K_1{A}_1}{L}+\frac{K_2{A}_2}{L}$

Equivalent thermal conductivity is

$\Rightarrow K_{eq}=\frac{K_1{A}_1+K_2{A}_2}{A_1+A_2}=\frac{\Sigma (K_i{A}_i)}{\Sigma (A_i)}$

• If Area of both rods are the same then $(A_1=A_2=A)$

$K_{eq}=\frac{K_1+K_2}{2}$

19.0Temperature of Junction($T_0$)

To determine the temperature at the junction, we equate the heat flow rates through both sections.

$\Rightarrow (\frac{dQ}{dt}{)}_1=(\frac{dQ}{dt}{)}_2$

$\Rightarrow \frac{K_{1}A}{L_1}(T_1-T_0)=\frac{K_{2}A}{L_2}(T_0-T_2)$

$\Rightarrow T_0=\frac{K_1{L}_2{T}_1+K_2{L}_1{T}_2}{K_1{L}_2+K_2{L}_1}$

Growth of Ice on Lake

$t\propto (x_2^2-x_1^2)$

Time taken to double and triple the thickness ratio 

• t₁ : t₂ : t₃ :: 1² : 2² : 3²
• t₁ : t₂ : t₃ :: 1 : 4 : 9

Ratio of time taken to form thickness

$(0\to x):(x\to 2x):(2x\to 3x)$

$\Delta t_1:\Delta t_2:\Delta t_3::(x^2-0^2):(2x^2-x^2):(3x^2-2x^2)$

$\Delta t_1:\Delta t_2:\Delta t_3::1:3:5$

1. Convection

• Convection is the transfer of heat through the actual movement of a heated fluid and requires a medium.
• Natural convection occurs due to density differences (e.g., heating a fluid from below).
• Forced convection happens when fluid motion is driven by external devices like fans or pumps.

2. Thermal Radiation

• Radiation is the transfer of heat without requiring a material medium.
• It allows heat to travel through a vacuum, unlike conduction and convection, which need a medium.
• A heated object in a vacuum can still lose heat through radiation.
• Radiation is the only mode of heat transfer that occurs without heating the intervening space or matter.

20.0Newton’s Law of Cooling

Newton's Law of Cooling asserts that the speed at which an object's temperature changes is directly linked to the disparity between its Temperature. The surrounding ambient Temperature relationship dictates how quickly an object cools or heats as it interacts thermally with its environment.

$T=T_0+e^{-KT}+C$

This equation calculates the time of cooling of a body through a particular range of temperature.

Note-For Numerical Problems (Newton’s Law of Cooling)  

$\frac{T_2-T_1}{t}=-K[\frac{T_1+T_2}{2}-T_0]$

21.0Black Body Radiation and Stefan’s Law

Ideal Black Body

A body that absorbs all thermal radiation incident on it at low temperatures and emits all absorbed radiation at high temperatures, regardless of wavelength. 

Characteristics:

• Emission depends only on temperature, not material or surface properties.
• Emitted radiation is called full or white radiation.
• The spectral energy distribution is continuous across all wavelengths.
• If a heat source has a continuous spectrum (e.g., kerosene lamp, filament), it's considered a black body.
• For an ideal black body:
  Absorptivity (a) = 1
  Reflectivity (r) = 0
• Transmissivity (t) = 0
  Emissivity (e) = 1
• Acts as a perfect absorber at low temperatures and a perfect emitter at high temperatures.
• Color doesn't define a black body (e.g., the Sun is not black).

Examples:

• Wien’s black body
• Ferry’s black body

22.0Kirchhoff's Law

At a given temperature, the ratio of a body's spectral emissive power  to its spectral absorptive power is constant. This constant is equal to the spectral emissive power of an ideal black body at the same temperature:

$\frac{e_{\lambda }}{a_{\lambda }}=\text{Constant}=E_{\lambda }$

Thus, good absorbers are also good emitters, and bad absorbers are bad emitters. According to Kirchhoff's law, the absorptivity of a surface is equal to its emissivity (e=a).

Practical confirmation of Kirchhoff's law was carried out using the Rishi apparatus, with the Leslie container as the primary base.

23.0Stefan's Law

The amount of radiation emitted per second per unit area by a black body is directly proportional to the fourth power of its absolute temperature.

$E=\sigma T^4$

$\sigma$=Stefen's constant = $5.67\times 10^{-8}Watt/m^2{K}^4$

Stefan’s Radiation Law

• An ideal blackbody absorbs all radiation that falls on it, including visible, infrared, ultraviolet, and all other wavelengths.
• A blackbody is both a perfect absorber and a perfect emitter of radiation.
• It emits more radiant power per unit area than any real object at the same temperature.
• The rate of radiation emission per unit area is directly proportional to the fourth power of the absolute temperature.

$P=\sigma A{T}^4$

Rate of Emission of Radiation

$\frac{dQ}{dt}=eA\sigma (T_b^4-T_s^4)=-ms\frac{d{T}_b}{dt}$

24.0Spectral Emissive Power

${\int }_0^{\infty }{E}_{\lambda }d\lambda =\text{total emissive power}$

Total area of graph = Total emissive power = $e\sigma T^4$ (here e = emissivity of surface)

Spectral Energy Distribution Curve of Black Body Radiations

Spectral energy distribution curves are continuous, indicating that radiation is emitted at all wavelengths (from 0 to ∞) at any given temperature, though the intensity varies across different wavelengths.

• As the wavelength increases, the amount of emitted radiation initially increases, reaches a maximum, and then decreases.
• The area under the spectral energy curve at a specific temperature represents the body's spectral emissive power.

25.0Wein's Displacement Law

The wavelength corresponding to maximum emission of radiation decreases with increasing temperature.

$[{\lambda }_m\propto \frac{1}{T}]$

${\lambda }_m.T=b\Rightarrow {\lambda }_{m1}{T}_1={\lambda }_{m2}{T}_2=b$

$\text{Wein’s Constant}=2.89\times 10^{-3}mK.$