Thermodynamics: Heat, Temperature, and Thermal Phenomena

9.1 Molecular Concept of Thermal Energy, Heat, and Temperature

At a microscopic level, thermal energy is the internal energy of a system due to the random motion of its atoms and molecules. This includes translational, rotational, and vibrational kinetic energies. The more vigorously these particles move, the higher the thermal energy.

• Thermal Energy: The total internal energy of a system associated with the microscopic motion of its constituent particles.
• Temperature: A measure of the average translational kinetic energy of the particles in a substance. It is an intensive property, meaning it does not depend on the amount of substance. A higher temperature indicates faster average particle motion.
• Heat: The transfer of thermal energy from a region of higher temperature to a region of lower temperature. Heat is energy in transit, not a property of a system. It's the process by which thermal energy moves.

The cause of heat flow is a temperature difference. Heat always flows spontaneously from a hotter body to a colder body, in the direction of decreasing temperature, until thermal equilibrium is reached.

9.2 Meaning of Thermal Equilibrium and Zeroth Law of Thermodynamics

Thermal equilibrium is a state where two or more systems in thermal contact have no net heat flow between them. This occurs when all systems reach the same temperature. At thermal equilibrium, the average kinetic energy of the particles in each system is the same.

The Zeroth Law of Thermodynamics states that if two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. This law is fundamental because it provides the basis for the definition of temperature and the operation of thermometers.

9.3 Thermal Equilibrium as a Working Principle of Mercury Thermometer

A mercury thermometer works on the principle of thermal equilibrium and thermal expansion. When the thermometer is placed in contact with an object, heat flows between the object and the mercury in the bulb until they reach thermal equilibrium. As the mercury absorbs heat, it expands, and its volume increases, causing it to rise in the capillary tube. The height of the mercury column then corresponds to the temperature of the object. The scale on the thermometer is calibrated to reflect these temperature readings.

10.1 Linear Expansion and its Measurement

Linear expansion refers to the change in length of a solid material when its temperature changes. For a small temperature change, the change in length (ΔL) is directly proportional to the original length (L₀) and the change in temperature (ΔT).

The formula for linear expansion is:

ΔL = α L₀ ΔT

Where:

• ΔL = Change in length
• L₀ = Original length
• ΔT = Change in temperature
• α (alpha) = Coefficient of linear expansion (a material-specific constant)

The coefficient of linear expansion (α) is measured in units of per degree Celsius (ºC⁻¹) or per Kelvin (K⁻¹).

10.2 Cubical Expansion, Superficial Expansion and its Relation with Linear Expansion

Superficial expansion (Area expansion) refers to the change in surface area of a solid when its temperature changes. Similarly, cubical expansion (Volume expansion) refers to the change in volume of a solid, liquid, or gas.

Formulas:

• Superficial Expansion: ΔA = β A₀ ΔT, where β (beta) is the coefficient of superficial expansion.
• Cubical Expansion: ΔV = γ V₀ ΔT, where γ (gamma) is the coefficient of cubical expansion.

For isotropic materials (materials that expand equally in all directions), there's a simple relationship between these coefficients:

• β ≈ 2α
• γ ≈ 3α

10.3 Liquid Expansion: Absolute and Apparent

Liquids also expand when heated, but their expansion is more complex to measure than solids because they are always contained within a vessel that also expands. We distinguish between absolute and apparent expansion.

• Absolute Expansion: The actual increase in the volume of the liquid. This is the expansion the liquid would undergo if the container did not expand.
• Apparent Expansion: The observed increase in the volume of the liquid relative to the container. This is what we typically measure directly.

The relationship is: γ_absolute = γ_apparent + γ_container

Where γ_container is the coefficient of cubical expansion of the material of the container.

10.4 Dulong and Petit Method of Determining Expansivity of Liquid

The Dulong and Petit method is an experimental technique used to determine the coefficient of absolute expansion of a liquid. It typically involves using a U-tube or two communicating tubes containing the liquid, with one arm heated and the other kept at a lower temperature. By measuring the difference in height of the liquid columns and knowing the temperatures, the absolute expansivity can be calculated, accounting for the expansion of the container.

11.1 Newton’s Law of Cooling

Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in temperatures between the body and its surroundings, provided that the temperature difference is small and the heat loss mechanism is primarily convection and radiation.

dQ/dt = -k (T - T_s)

Where:

• dQ/dt = Rate of heat loss
• k = Positive constant depending on the surface area, nature of the surface, and surrounding conditions
• T = Temperature of the body
• T_s = Temperature of the surroundings

11.2 Measurement of Specific Heat Capacity of Solids and Liquids

Specific heat capacity (c) is the amount of heat energy required to raise the temperature of 1 kilogram of a substance by 1 degree Celsius (or 1 Kelvin). The formula for heat absorbed or released is:

Q = mcΔT

Where:

• Q = Heat energy (Joules)
• m = Mass of the substance (kg)
• c = Specific heat capacity (J kg⁻¹ ºC⁻¹ or J kg⁻¹ K⁻¹)
• ΔT = Change in temperature (ºC or K)

Specific heat capacity is typically measured using calorimetry, where heat transfer between substances in an insulated container (calorimeter) is analyzed based on the principle of conservation of energy (heat lost by hot body = heat gained by cold body).

11.3 Change of Phases: Latent Heat

A phase change (e.g., melting, freezing, boiling, condensation) occurs when a substance changes its physical state without a change in temperature. During a phase change, the absorbed or released heat energy is used to break or form intermolecular bonds, not to increase the kinetic energy of the particles.

Latent heat is the heat energy absorbed or released per unit mass during a phase change at constant temperature. It is 'latent' because it doesn't cause a temperature change.

11.4 Specific Latent Heat of Fusion and Vaporization

• Specific Latent Heat of Fusion (L_f): The amount of heat energy required to change 1 kg of a substance from solid to liquid (melting) or liquid to solid (freezing) at its melting point without a change in temperature.
• Specific Latent Heat of Vaporization (L_v): The amount of heat energy required to change 1 kg of a substance from liquid to gas (boiling/vaporization) or gas to liquid (condensation) at its boiling point without a change in temperature.

The formula for latent heat is:

Q = mL

Where L is either L_f or L_v.

11.5 Measurement of Specific Latent Heat of Fusion and Vaporization

These values are also typically measured using calorimetric methods. For specific latent heat of fusion, a known mass of ice (at 0ºC) can be added to water in a calorimeter, and the temperature change of the water/calorimeter system is used to calculate the heat absorbed by the ice to melt. For specific latent heat of vaporization, steam (at 100ºC) can be condensed into water in a calorimeter, and the heat released by the steam is measured.

11.6 Triple Point

The triple point of a substance is the unique temperature and pressure at which the three phases (solid, liquid, and gas) of that substance coexist in thermodynamic equilibrium. For water, the triple point is at 0.01 ºC (273.16 K) and a pressure of 611.73 Pascals. The triple point is a very precise and reproducible fixed point, making it useful for calibrating thermometers and defining temperature scales.

12.1 Conduction: Thermal Conductivity and Measurement

Conduction is the transfer of heat through direct contact, primarily in solids. It occurs due to the vibration and collision of particles (atoms/molecules) and, in metals, also by the movement of free electrons. Heat flows from hotter regions to colder regions within the material.

The rate of heat conduction (H or dQ/dt) through a material is given by Fourier's Law of Heat Conduction:

H = -kA (dT/dx)

Where:

• H = Rate of heat flow (Watts or J/s)
• k = Thermal conductivity (W m⁻¹ K⁻¹) - a measure of how well a material conducts heat. Good conductors (metals) have high k values; insulators (wood, air) have low k values.
• A = Cross-sectional area through which heat flows (m²)
• (dT/dx) = Temperature gradient (K m⁻¹) - the change in temperature per unit distance. The negative sign indicates heat flows in the direction of decreasing temperature.

Thermal conductivity is typically measured using methods like the Searle's bar method for good conductors or Lee's disc method for poor conductors.

12.2 Convection

Convection is the transfer of heat through the movement of fluids (liquids or gases). When a fluid is heated, it becomes less dense and rises, while cooler, denser fluid sinks, creating a convection current. This movement carries thermal energy from one place to another.

• Natural Convection: Driven by density differences (e.g., boiling water, sea breezes).
• Forced Convection: Fluid movement is aided by external means like fans or pumps (e.g., air conditioning, car radiator).

12.3 Radiation: Ideal Radiator

Radiation is the transfer of heat through electromagnetic waves. Unlike conduction and convection, radiation does not require a medium and can travel through a vacuum (e.g., heat from the sun reaching Earth). All objects at a temperature above absolute zero emit thermal radiation.

An ideal radiator, also known as a black body, is a hypothetical object that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It also emits thermal radiation at the maximum possible rate for its temperature.

12.4 Black-Body Radiation

Black-body radiation is the thermal electromagnetic radiation emitted by a black body at a given temperature. The spectrum of black-body radiation depends only on the temperature of the object, not its material or surface properties. Studying black-body radiation led to the development of quantum mechanics.

12.5 Stefan – Boltzmann Law

The Stefan-Boltzmann Law quantifies the total energy radiated per unit surface area of a black body across all wavelengths per unit time. It states that this power is directly proportional to the fourth power of the black body's absolute temperature.

P/A = σT⁴

Where:

• P/A = Radiated power per unit area (W/m²)
• σ (sigma) = Stefan-Boltzmann constant (5.67 x 10⁻⁸ W m⁻² K⁻⁴)
• T = Absolute temperature of the black body (Kelvin)

For a non-black body (real object), an emissivity factor (e, between 0 and 1) is included: P/A = eσT⁴.

13.1 Ideal Gas Equation

The Ideal Gas Equation (also known as the General Gas Equation) relates the pressure, volume, temperature, and number of moles of an ideal gas:

PV = nRT

Where:

• P = Absolute pressure of the gas (Pascals)
• V = Volume of the gas (m³)
• n = Number of moles of gas
• R = Ideal gas constant (8.314 J mol⁻¹ K⁻¹)
• T = Absolute temperature of the gas (Kelvin)

Alternatively, for a fixed mass of gas, Boyle's Law (PV=constant at constant T), Charles's Law (V/T=constant at constant P), and Gay-Lussac's Law (P/T=constant at constant V) can be derived from the ideal gas equation.

13.2 Molecular Properties of Matter

The behavior of gases is explained by the properties of their constituent molecules:

• Molecules are in constant, random motion.
• There are large distances between molecules compared to their size.
• Molecules exert negligible forces on each other except during collisions.
• Collisions between molecules and with the container walls are perfectly elastic.

13.3 Kinetic-Molecular Model of an Ideal Gas

The Kinetic-Molecular Model (KMT) provides a microscopic explanation for the macroscopic properties of ideal gases. It's based on the following postulates:

• Gases consist of large numbers of identical molecules that are in continuous, random motion.
• The volume occupied by the molecules themselves is negligible compared to the total volume of the container.
• There are no attractive or repulsive forces between gas molecules.
• Collisions between gas molecules and between molecules and the container walls are perfectly elastic (no net loss of kinetic energy).
• The average kinetic energy of the gas molecules is directly proportional to the absolute temperature of the gas.

13.4 Derivation of Pressure Exerted by Gas

Using the KMT, the pressure exerted by an ideal gas on the walls of its container can be derived. The pressure arises from the countless elastic collisions of gas molecules with the container walls. Each collision imparts a tiny impulse to the wall, and the sum of these impulses over time and area results in a measurable pressure.

The derivation often starts by considering a single molecule in a cubical container and then extending it to N molecules, leading to the formula:

P = (1/3) (N/V) m

Where:

• P = Pressure
• N = Total number of molecules
• V = Volume
• m = Mass of one molecule
• = Mean square speed of the molecules

13.5 Average Translational Kinetic Energy of Gas Molecule

From the KMT and the ideal gas law, it can be shown that the average translational kinetic energy (KE_avg) of a gas molecule is directly proportional to the absolute temperature:

KE_avg = (1/2) m = (3/2) kT

Where:

• m = Mass of one molecule
• = Mean square speed
• k = Boltzmann constant
• T = Absolute temperature

This equation is crucial as it directly links the microscopic property (average kinetic energy) to the macroscopic property (temperature).

13.6 Boltzmann Constant, Root Mean Square Speed

• Boltzmann Constant (k): A fundamental physical constant relating the average kinetic energy of particles in a gas to the temperature of the gas. It is the gas constant (R) divided by Avogadro's number (N_A): k = R/N_A ≈ 1.38 x 10⁻²³ J/K.
• Root Mean Square Speed (v_rms): The square root of the average of the squares of the speeds of the gas molecules. It is a measure of the typical speed of the molecules in a gas.

v_rms = √ = √(3kT/m) = √(3RT/M)

Where M is the molar mass of the gas.

13.7 Heat Capacities: Gases and Solids

Heat capacity (C = Q/ΔT) refers to the amount of heat required to change the temperature of an entire object by 1 degree. Specific heat capacity (c = C/m) is for a unit mass. For gases, we also consider molar heat capacity (C_m = C/n).

• Heat Capacities of Gases: Gases have two principal specific heat capacities:
• C_p (at constant pressure): More heat is required because the gas does work against the surroundings as it expands.
• C_v (at constant volume): Less heat is required as no work is done against the surroundings. C_p > C_v. The relationship between them is C_p - C_v = R (Mayer's relation for ideal gases).
• Heat Capacities of Solids: For solids, the distinction between C_p and C_v is usually negligible because solids do not expand significantly when heated. According to the Dulong and Petit Law (at high temperatures), the molar heat capacity of most solid elements is approximately 3R (≈ 25 J mol⁻¹ K⁻¹), implying that each atom has 6 degrees of freedom (3 kinetic, 3 potential for vibration).