Heat and Temperature, Thermal Expansion, Specific Heat Capacity, Calorimetry, Change of State and Latent Heat

1. Heat and Temperature

Understanding the concepts of heat and temperature is fundamental to grasping thermodynamics. While often used interchangeably in everyday language, they represent distinct physical quantities.

1.1 Temperature

Temperature is a measure of the average kinetic energy of the particles (atoms or molecules) within a substance. It quantifies how hot or cold an object is. The higher the average kinetic energy, the higher the temperature. It is an intensive property, meaning it does not depend on the amount of substance. Temperature is typically measured using a thermometer and is expressed in degrees Celsius (°C), Fahrenheit (°F), or Kelvin (K).

1.2 Heat

Heat, on the other hand, is the transfer of thermal energy between systems due to a temperature difference. It is energy in transit. Heat flows spontaneously from a region of higher temperature to a region of lower temperature. It is an extensive property, meaning it depends on the amount of substance. The SI unit of heat is the Joule (J). Other common units include the calorie (cal) and the kilocalorie (kcal).

Key Distinction: Temperature is a measure of the internal energy of a system, while heat is the transfer of that energy. A substance has a temperature, but it does not "contain" heat; rather, heat is exchanged between substances.

1.3 Scales of Temperature Measurement

Different scales are used to measure temperature, each with its own reference points and units.

• Celsius (°C): Water freezes at 0°C and boils at 100°C at standard atmospheric pressure.
• Fahrenheit (°F): Water freezes at 32°F and boils at 212°F at standard atmospheric pressure.
• Kelvin (K): This is the SI unit of thermodynamic temperature. It is an absolute scale, meaning 0 K (absolute zero) is the theoretical temperature at which particles have minimum motion. Water freezes at 273.15 K and boils at 373.15 K.

1.4 Relationship between Temperature Scales

The following formulas allow conversion between the scales:

• To convert Celsius to Fahrenheit: $F = \frac{9}{5}C + 32$
• To convert Fahrenheit to Celsius: $C = \frac{5}{9}(F - 32)$
• To convert Celsius to Kelvin: $K = C + 273.15$
• To convert Kelvin to Celsius: $C = K - 273.15$

2. Thermal Expansion

Most substances expand when heated and contract when cooled. This phenomenon is known as thermal expansion. The extent of expansion depends on the material, the change in temperature, and the original dimensions of the object.

2.1 Linear Expansion

Linear expansion occurs in one dimension, typically for solids with one length dimension significantly larger than the others (like rods or wires). The change in length ($\Delta L$) is proportional to the original length ($L_0$) and the change in temperature ($\Delta T$).

The formula is: $\Delta L = \alpha L_0 \Delta T$

Where:

• $\Delta L$ is the change in length.
• $L_0$ is the original length.
• $\Delta T$ is the change in temperature.
• $\alpha$ is the coefficient of linear expansion, a material property. Its units are per degree Celsius (or Kelvin), i.e., $°C^{-1}$ or $K^{-1}$.

The new length ($L$) after expansion is given by: $L = L_0 (1 + \alpha \Delta T)$.

2.2 Area (Superficial) Expansion

Area expansion occurs in two dimensions, applicable to thin plates or sheets. The change in area ($\Delta A$) is proportional to the original area ($A_0$) and the change in temperature ($\Delta T$).

The formula is: $\Delta A = \beta A_0 \Delta T$

Where:

• $\Delta A$ is the change in area.
• $A_0$ is the original area.
• $\Delta T$ is the change in temperature.
• $\beta$ is the coefficient of area expansion. For isotropic materials (expanding uniformly in all directions), $\beta \approx 2\alpha$.

The new area ($A$) is given by: $A = A_0 (1 + \beta \Delta T)$.

2.3 Volume (Cubical) Expansion

Volume expansion occurs in three dimensions and applies to solids, liquids, and gases. The change in volume ($\Delta V$) is proportional to the original volume ($V_0$) and the change in temperature ($\Delta T$).

The formula is: $\Delta V = \gamma V_0 \Delta T$

Where:

• $\Delta V$ is the change in volume.
• $V_0$ is the original volume.
• $\Delta T$ is the change in temperature.
• $\gamma$ is the coefficient of volume expansion. For isotropic solids, $\gamma \approx 3\alpha$. For liquids, only the coefficient of volume expansion is relevant.

The new volume ($V$) is given by: $V = V_0 (1 + \gamma \Delta T)$.

2.4 Anomalous Expansion of Water

Water exhibits unusual behavior between 0°C and 4°C. Instead of expanding uniformly upon heating, it contracts as its temperature increases from 0°C to 4°C. Its maximum density occurs at 4°C. Above 4°C, water expands normally upon heating.

Significance: This anomalous expansion is crucial for aquatic life in cold climates. When the surface of a lake freezes, the water at 4°C sinks to the bottom, preventing the entire body of water from freezing solid and allowing aquatic organisms to survive.

2.5 Applications of Thermal Expansion

• Expansion Joints: Bridges and railway tracks have expansion gaps to prevent buckling due to thermal expansion.
• Bimetallic Strips: Used in thermostats and fire alarms. Two metals with different coefficients of expansion are bonded together. When heated, the strip bends towards the metal with the lower coefficient of expansion.
• Thermometers: The expansion of mercury or alcohol in a glass tube is used to measure temperature.
• Overhead Power Lines: Wires are strung loosely to allow for contraction in winter and expansion in summer without breaking.

3. Specific Heat Capacity

Specific heat capacity is a physical property of a substance that quantifies the amount of heat energy required to raise the temperature of one unit of mass of the substance by one degree Celsius (or Kelvin).

3.1 Definition and Formula

The heat energy ($Q$) absorbed or released by a substance is directly proportional to its mass ($m$), its specific heat capacity ($c$), and the change in temperature ($\Delta T$).

The formula is: $Q = mc \Delta T$

Where:

• $Q$ is the heat energy transferred (in Joules).
• $m$ is the mass of the substance (in kilograms).
• $c$ is the specific heat capacity of the substance (in $J kg^{-1} K^{-1}$ or $J kg^{-1} °C^{-1}$).
• $\Delta T$ is the change in temperature (in Kelvin or Celsius).

The specific heat capacity ($c$) can be expressed as: $c = \frac{Q}{m \Delta T}$

3.2 Units and Values

The SI unit for specific heat capacity is Joules per kilogram per Kelvin ($J kg^{-1} K^{-1}$). Another common unit is calories per gram per degree Celsius ($cal g^{-1} °C^{-1}$).

Different substances have different specific heat capacities. For example:

• Water has a very high specific heat capacity (approximately $4186 \, J kg^{-1} K^{-1}$ or $1 \, cal g^{-1} °C^{-1}$). This means it takes a lot of energy to heat water, and it also loses heat slowly.
• Metals generally have low specific heat capacities. For instance, iron has a specific heat capacity of about $470 \, J kg^{-1} K^{-1}$.

3.3 Significance of High Specific Heat Capacity

Substances with high specific heat capacities, like water, can absorb or release large amounts of heat with only small changes in temperature. This property is vital in many natural and technological applications:

• Climate Regulation: Large bodies of water moderate coastal climates because they absorb heat during the day/summer and release it at night/winter.
• Coolant: Water is an excellent coolant in engines and power plants due to its ability to absorb significant amounts of heat.
• Body Temperature: The high water content in living organisms helps maintain a stable body temperature.

3.4 Molar Heat Capacity

Molar heat capacity ($C_m$) is the heat required to raise the temperature of one mole of a substance by one degree Celsius (or Kelvin). Its unit is $J mol^{-1} K^{-1}$.

It is related to specific heat capacity by: $C_m = M c$, where $M$ is the molar mass of the substance.

4. Calorimetry

Calorimetry is the science of measuring the heat transferred during a physical or chemical process. A device used for this purpose is called a calorimeter.

4.1 Principle of Calorimetry

The principle of calorimetry is based on the conservation of energy. When a hot body is brought into thermal contact with a cold body within an isolated system (the calorimeter), heat flows from the hot body to the cold body until they reach a common final temperature. Assuming no heat is lost to the surroundings:

Heat lost by the hot body = Heat gained by the cold body

$Q_{lost} = Q_{gained}$

This principle is used to determine the specific heat capacity of solids, liquids, or to find the resulting temperature when substances of different temperatures are mixed.

4.2 Calorimeter and Water Equivalent

A typical calorimeter consists of a sealed container (often made of metal like copper or aluminum) placed inside an insulating jacket. A thermometer and a stirrer are usually inserted through the lid.

The calorimeter itself has a heat capacity. To simplify calculations, the concept of "water equivalent" ($W$) is used. The water equivalent of a calorimeter is the mass of water that would absorb the same amount of heat as the calorimeter for the same temperature change. It is given by $W = m_{cal} c_{cal}$, where $m_{cal}$ and $c_{cal}$ are the mass and specific heat capacity of the calorimeter material, respectively. The unit of $W$ is kg or g.

The equation for calorimetry including the calorimeter becomes:

Heat lost by hot body = Heat gained by cold body + Heat gained by calorimeter

$m_h c_h (T_h - T_f) = m_c c_c (T_f - T_c) + W (T_f - T_c)$

Or, using water equivalent: $m_h c_h (T_h - T_f) = (m_c + W) c_w (T_f - T_c)$, where $c_w$ is the specific heat of water.

4.3 Example: Mixing Hot and Cold Water

Suppose we mix a mass $m_1$ of water at temperature $T_1$ with a mass $m_2$ of water at temperature $T_2$ ($T_1 > T_2$). Let the final equilibrium temperature be $T_f$. Assuming no heat loss:

Heat lost by hot water = Heat gained by cold water

$m_1 c_w (T_1 - T_f) = m_2 c_w (T_f - T_2)$

Since $c_w$ is the same on both sides, it cancels out:

$m_1 (T_1 - T_f) = m_2 (T_f - T_2)$

This equation can be solved for $T_f$.

5. Change of State and Latent Heat

Matter exists in different states: solid, liquid, and gas. A change of state (or phase transition) occurs when a substance transforms from one state to another, typically due to a change in temperature or pressure. During a change of state, the temperature of the substance remains constant, even though heat is being added or removed.

5.1 States of Matter and Transitions

• Melting (Fusion): Solid to Liquid. The temperature at which this occurs is the melting point.
• Freezing (Solidification): Liquid to Solid. Occurs at the same temperature as melting.
• Boiling (Vaporization): Liquid to Gas. The temperature at which this occurs is the boiling point.
• Condensation: Gas to Liquid. Occurs at the same temperature as boiling.
• Sublimation: Solid directly to Gas (e.g., dry ice).
• Deposition: Gas directly to Solid.

5.2 Latent Heat

Latent heat is the energy absorbed or released by a substance during a change of state at constant temperature and pressure. The word "latent" means hidden, as this heat energy does not cause a temperature change but is used to break or form intermolecular bonds.

5.3 Latent Heat of Fusion ($L_f$)

The latent heat of fusion is the amount of heat energy required to change one unit mass of a substance from solid to liquid (or liquid to solid) at its melting point.

The formula for heat absorbed or released during fusion/freezing is: $Q = m L_f$

Where:

• $Q$ is the heat energy (in Joules).
• $m$ is the mass of the substance (in kilograms).
• $L_f$ is the specific latent heat of fusion (in $J kg^{-1}$).

For example, the latent heat of fusion of ice is approximately $3.34 \times 10^5 \, J kg^{-1}$. This means $3.34 \times 10^5$ Joules of energy are needed to melt 1 kg of ice at 0°C into water at 0°C.

5.4 Latent Heat of Vaporization ($L_v$)

The latent heat of vaporization is the amount of heat energy required to change one unit mass of a substance from liquid to gas (or gas to liquid) at its boiling point.

The formula for heat absorbed or released during vaporization/condensation is: $Q = m L_v$

Where:

• $Q$ is the heat energy (in Joules).
• $m$ is the mass of the substance (in kilograms).
• $L_v$ is the specific latent heat of vaporization (in $J kg^{-1}$).

For water at its boiling point (100°C at standard pressure), the latent heat of vaporization is approximately $2.26 \times 10^6 \, J kg^{-1}$. This is much larger than the latent heat of fusion, indicating that more energy is required to convert a liquid into a gas than to melt a solid.

5.5 Heating Curve

A heating curve graphically represents the change in temperature of a substance as heat is added at a constant rate. For a substance like ice being heated to steam, the curve typically shows:

• A rising segment representing the increase in temperature of the solid (ice).
• A horizontal plateau at 0°C representing the melting of ice into water (latent heat of fusion).
• A rising segment representing the increase in temperature of the liquid (water).
• A horizontal plateau at 100°C representing the boiling of water into steam (latent heat of vaporization).
• A rising segment representing the increase in temperature of the gas (steam).

5.6 Example Calculation: Melting Ice

Calculate the total heat required to convert 2 kg of ice at -10°C to water at 20°C. Assume the specific heat of ice is $2100 \, J kg^{-1} K^{-1}$, the latent heat of fusion of ice is $3.34 \times 10^5 \, J kg^{-1}$, and the specific heat of water is $4186 \, J kg^{-1} K^{-1}$.

This process involves three steps:

1. Heating the ice from -10°C to 0°C: $Q_1 = m c_{ice} \Delta T = (2 \, kg) \times (2100 \, J kg^{-1} K^{-1}) \times (0°C - (-10°C)) = 2 \times 2100 \times 10 = 42000 \, J$
2. Melting the ice at 0°C to water at 0°C: $Q_2 = m L_f = (2 \, kg) \times (3.34 \times 10^5 \, J kg^{-1}) = 6.68 \times 10^5 \, J$
3. Heating the water from 0°C to 20°C: $Q_3 = m c_{water} \Delta T = (2 \, kg) \times (4186 \, J kg^{-1} K^{-1}) \times (20°C - 0°C) = 2 \times 4186 \times 20 = 167440 \, J$

Total heat required: $Q_{total} = Q_1 + Q_2 + Q_3 = 42000 \, J + 668000 \, J + 167440 \, J = 877440 \, J$.