Introduction

The concepts of heat and temperature are fundamental to our understanding of the physical world. While we intuitively understand temperature as a measure of the 'hotness' or 'coldness' of an object, physics requires a more precise definition. This chapter delves into the thermal properties of matter, defining heat and temperature, exploring how they are measured, and examining the processes by which heat is transferred.

We will study why materials expand when heated, a principle used by blacksmiths to fit iron rings onto wooden wheels. We will also investigate the phenomena of phase changes, such as water boiling or freezing, where temperature remains constant despite the continuous flow of heat. Understanding these concepts is essential for numerous applications, from engineering design to explaining everyday phenomena like the changing direction of sea breezes.

Temperature and Heat

Temperature is a physical quantity that provides a relative measure of how hot or cold an object is. An object with a higher temperature is considered hotter than an object with a lower temperature. Our sense of touch gives us a qualitative idea of temperature, but it is unreliable and limited for scientific purposes.

When two systems at different temperatures are brought into contact, energy is transferred from the hotter system to the colder one. This transfer of energy continues until both systems reach the same temperature, a state known as thermal equilibrium.

Heat (Q) is defined as the form of energy that is transferred between two systems (or a system and its surroundings) by virtue of a temperature difference. Heat always flows from a region of higher temperature to a region of lower temperature.

• The SI unit of heat is the joule (J).
• The SI unit of temperature is the kelvin (K). The degree Celsius (°C) is also a commonly used unit.

When a body is heated, it can undergo several changes, such as a rise in temperature, thermal expansion, or a change of state (e.g., melting or boiling).

Measurement of Temperature

Temperature is measured using a device called a thermometer. Thermometers work based on a physical property of a substance that changes predictably with temperature. This property is called the thermometric property.

The most common thermometric property is the thermal expansion of a liquid. Liquid-in-glass thermometers (using mercury or alcohol) rely on the principle that the volume of the liquid expands or contracts linearly with temperature over a wide range.

Temperature Scales

To define a temperature scale, two fixed reference points are needed. The most common reference points are:

• Ice Point: The temperature at which pure water freezes at standard atmospheric pressure.
• Steam Point: The temperature at which pure water boils at standard atmospheric pressure.

Celsius Scale (°C)

On the Celsius scale, the ice point is defined as 0 °C and the steam point as 100 °C. The interval between these points is divided into 100 equal parts.

Fahrenheit Scale (°F)

On the Fahrenheit scale, the ice point is 32 °F and the steam point is 212 °F. The interval is divided into 180 equal parts.

Conversion Formula

The relationship between the Celsius temperature ($t_C$) and Fahrenheit temperature ($t_F$) is linear and given by the formula:

$\frac{t_F - 32}{180} = \frac{t_C}{100}$

This simplifies to:

$t_F = \frac{9}{5}t_C + 32$

Ideal-Gas Equation and Absolute Temperature

Thermometers using different liquids can give slightly different readings between the fixed points due to non-uniform expansion. A gas thermometer, especially using a low-density gas, provides a more fundamental and consistent measure of temperature.

The state of a gas is described by its pressure (P), volume (V), and temperature (T). For low-density gases, these variables are related by the ideal-gas law:

$PV = \mu RT$

where $\mu$ is the number of moles of the gas and $R$ is the universal gas constant ($R \approx 8.31 \text{ J mol}^{-1} \text{ K}^{-1}$).

Absolute Temperature Scale (Kelvin Scale)

In a constant-volume gas thermometer, the volume of the gas is kept fixed, so the ideal gas law simplifies to $P \propto T$. This means the pressure of the gas can be used as a direct measure of temperature.

If we plot the pressure of a low-density gas versus its temperature in Celsius and extrapolate the line to zero pressure, all gases are found to intersect the temperature axis at the same point: -273.15 °C. This temperature is called absolute zero. It is the theoretical temperature at which a gas would exert no pressure.

The Kelvin temperature scale (K) is based on this absolute zero. On this scale, 0 K is defined as -273.15 °C.

The size of a kelvin is the same as the size of a degree Celsius. The relationship between the Kelvin temperature (T) and the Celsius temperature ($t_C$) is:

$T(\text{K}) = t_C(^\circ\text{C}) + 273.15$

Thermal Expansion

Most substances expand when their temperature increases and contract when their temperature decreases. This phenomenon is called thermal expansion. The change in dimensions is a result of the increased kinetic energy of the atoms, which causes them to vibrate with greater amplitude and increase the average interatomic distance.

1. Linear Expansion

This refers to the expansion in one dimension, such as the length of a rod.

The fractional change in length ($\Delta l/l$) is directly proportional to the change in temperature ($\Delta T$).

$\frac{\Delta l}{l} = \alpha_l \Delta T$

where $\alpha_l$ is the coefficient of linear expansion. Its unit is K⁻¹ or °C⁻¹.

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2. Area Expansion

This refers to the expansion in two dimensions, such as the area of a sheet.

$\frac{\Delta A}{A} = \alpha_A \Delta T$

where $\alpha_A$ is the coefficient of area expansion. For an isotropic material, $\alpha_A = 2\alpha_l$.

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3. Volume Expansion

This refers to the expansion in three dimensions, such as the volume of a cube or a liquid.

$\frac{\Delta V}{V} = \alpha_V \Delta T$

where $\alpha_V$ is the coefficient of volume expansion. For an isotropic solid, $\alpha_V = 3\alpha_l$.

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Anomalous Expansion of Water

Water exhibits an unusual behaviour. Instead of expanding continuously on heating, it contracts when heated from 0 °C to 4 °C. This means that water has its maximum density at 4 °C. Above 4 °C, it expands normally.

This anomalous property is crucial for aquatic life in cold climates. As a lake cools, the surface water at 4 °C is the densest and sinks. Water cooler than 4 °C is less dense and stays at the surface, where it eventually freezes. This allows the water at the bottom of the lake to remain at 4 °C, enabling fish and other organisms to survive.

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Specific Heat Capacity

The amount of heat required to change the temperature of a substance depends on its mass, the desired temperature change, and the nature of the substance itself.

The heat capacity (S) of a substance is the amount of heat ($\Delta Q$) required to change its temperature by $\Delta T$.

$S = \frac{\Delta Q}{\Delta T}$ (Unit: J/K)

A more useful property is the specific heat capacity (s), which is the heat capacity per unit mass.

Specific Heat Capacity (s): It is the amount of heat required to raise the temperature of a unit mass of a substance by one degree (1 K or 1 °C).

$s = \frac{1}{m} \frac{\Delta Q}{\Delta T}$

From this, the heat absorbed or released by a mass $m$ during a temperature change $\Delta T$ is:

$\Delta Q = ms \Delta T$

• The SI unit of specific heat capacity is J kg⁻¹ K⁻¹.
• Water has a very high specific heat capacity (4186 J kg⁻¹ K⁻¹) compared to most other common substances. This is why it is an excellent coolant (e.g., in car radiators) and is used in hot water bags for heating. Its high specific heat also moderates coastal climates.

Molar Specific Heat Capacity (C)

If the amount of substance is given in moles ($\mu$), we define the molar specific heat capacity (C).

$C = \frac{1}{\mu} \frac{\Delta Q}{\Delta T}$

The SI unit is J mol⁻¹ K⁻¹.

For gases, the specific heat capacity depends on the process. The two important types are:

• Molar specific heat at constant pressure ($C_p$)
• Molar specific heat at constant volume ($C_v$)

Calorimetry

Calorimetry is the measurement of heat. It is based on the principle of conservation of energy applied to heat transfer.

Principle of Calorimetry: In an isolated system, when a hot body is brought into contact with a cold body, the heat lost by the hot body is equal to the heat gained by the cold body.

Heat Lost = Heat Gained

This principle is used to determine quantities like specific heat capacity or latent heat. The device used for these measurements is a calorimeter. A simple calorimeter consists of an insulated metallic vessel (often copper) with a stirrer and a lid for a thermometer. The insulation minimizes heat exchange with the surroundings.

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Change of State

Matter can exist in three primary states: solid, liquid, and gas. A transition from one state to another is called a change of state or phase transition. These transitions occur at a constant temperature when heat is either supplied to or removed from the substance.

Melting and Fusion

• Melting: The change of state from solid to liquid.
• Fusion: The change of state from liquid to solid.
• Melting Point: The constant temperature at which a solid and its liquid phase coexist in thermal equilibrium. It depends on pressure. The normal melting point is the melting point at standard atmospheric pressure.

Vaporization and Condensation

• Vaporization (Boiling): The change of state from liquid to gas.
• Condensation: The change of state from gas to liquid.
• Boiling Point: The constant temperature at which a liquid and its vapor phase coexist in thermal equilibrium. The boiling point increases with increasing pressure (e.g., in a pressure cooker) and decreases with decreasing pressure (e.g., at high altitudes).

Sublimation

Some substances can change directly from the solid state to the vapor state without passing through the liquid phase. This process is called sublimation. Examples include dry ice (solid CO₂) and iodine.

Latent Heat

During a phase change, the heat supplied does not change the temperature but is used to change the state. This "hidden" heat is called latent heat (L).

The heat Q required to change the state of a mass $m$ of a substance is given by:

$Q = mL$

where L is the specific latent heat of the substance for that transition.

• Latent Heat of Fusion ($L_f$): Heat per unit mass required to change a substance from solid to liquid. For water, $L_f = 3.33 \times 10^5$ J/kg.
• Latent Heat of Vaporization ($L_v$): Heat per unit mass required to change a substance from liquid to gas. For water, $L_v = 22.6 \times 10^5$ J/kg.

The very high latent heat of vaporization of water is why steam at 100 °C can cause much more severe burns than boiling water at 100 °C.

Heat Transfer

Heat is transferred from a region of higher temperature to a region of lower temperature through three distinct modes: conduction, convection, and radiation.

1. Conduction

Conduction is the transfer of heat through a material without any bulk movement of the material itself. It occurs primarily through molecular collisions. Metals are good conductors of heat because their free electrons can transport energy quickly. Gases are poor conductors.

The rate of heat flow (or heat current), $H$, through a uniform bar of length $L$ and cross-sectional area $A$, with its ends at temperatures $T_C$ and $T_D$ ($T_C > T_D$) is given by:

$H = \frac{dQ}{dt} = KA \frac{T_C - T_D}{L}$

where $K$ is the thermal conductivity of the material. Its SI unit is W m⁻¹ K⁻¹.

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2. Convection

Convection is the transfer of heat through the actual bulk movement of a fluid (liquid or gas).

• Natural Convection: Occurs due to density differences caused by heating. When a fluid is heated from below, it expands, becomes less dense, and rises. The cooler, denser fluid from above sinks to take its place, creating a convection current. Examples include sea and land breezes and trade winds.
• Forced Convection: Occurs when a fluid is forced to move by an external agent like a pump or fan. Examples include forced-air heating systems, the human circulatory system, and a car's cooling system.

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3. Radiation

Radiation is the transfer of heat in the form of electromagnetic waves. This mode of heat transfer does not require any material medium and can travel through a vacuum. This is how the Earth receives heat from the Sun.

All objects with a temperature above absolute zero emit thermal radiation. The rate at which an object radiates energy depends on its temperature, surface area, and the nature of its surface.

Stefan-Boltzmann Law

The total energy radiated per unit time (Power, $H$) by a perfect blackbody (a perfect emitter and absorber) is proportional to the fourth power of its absolute temperature ($T$) and its surface area ($A$).

$H = \sigma A T^4$

where $\sigma = 5.67 \times 10^{-8} \text{ W m}^{-2} \text{K}^{-4}$ is the Stefan-Boltzmann constant.

For a real body, the law is modified by its emissivity ($e$), a number between 0 and 1:

$H = e \sigma A T^4$

If a body at temperature $T$ is in surroundings at temperature $T_s$, the net rate of heat loss by radiation is:

$H_{net} = e \sigma A (T^4 - T_s^4)$

Wien's Displacement Law

The wavelength ($\lambda_m$) at which the radiated energy is maximum is inversely proportional to the absolute temperature ($T$) of the body.

$\lambda_m T = \text{constant}$ (Wien's constant $\approx 2.9 \times 10^{-3} \text{ m K}$)

This law explains why the color of a heated object changes from red to yellow to white as its temperature increases.

Newton’s Law of Cooling

Newton's law of cooling provides a simple model for the rate at which a hot body cools down when the temperature difference between the body and its surroundings is small.

Newton's Law of Cooling: The rate of loss of heat ($-dQ/dt$) of a body is directly proportional to the temperature difference ($\Delta T$) between the body ($T_2$) and its surroundings ($T_1$).

$-\frac{dQ}{dt} = k(T_2 - T_1)$

where $k$ is a constant that depends on the surface area and nature of the body.

Since the heat lost is also related to the temperature drop of the body ($dQ = ms dT_2$), the law can be written in terms of the rate of cooling:

$-ms \frac{dT_2}{dt} = k(T_2 - T_1) \implies \frac{dT_2}{dt} = -K(T_2 - T_1)$

where $K=k/ms$ is another constant. This differential equation shows that the rate of cooling is greatest when the body is hottest and decreases as the body's temperature approaches the temperature of the surroundings.

The solution to this equation is an exponential decay function:

$T_2(t) = T_1 + (T_{2, initial} - T_1)e^{-Kt}$

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ADDITIONAL EXERCISES

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