Heat

Introduction to Heat

1. Definition:

   • Heat is a form of energy transferred between two bodies due to a temperature difference.
   • SI Unit: Joule ($J$).
   • Common Unit: Calorie ($cal$), where $1 \, cal = 4.186 \, J$.

2. Temperature:

   • A measure of the average kinetic energy of particles in a substance.
   • SI Unit: Kelvin ($K$).
   • Other Units: Celsius ($^\circ C$), Fahrenheit ($^\circ F$).
   • Conversion: $T(K) = T(^{\circ}C) + 273.15$ $T(^{\circ}F) = \frac{9}{5} T(^{\circ}C) + 32$

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Modes of Heat Transfer

1. Conduction:

   • Transfer of heat through a substance without the movement of particles.
   • Governing Law: $Q = \frac{kA \Delta T}{d} \cdot t$
     • $Q$: Heat transferred.
     • $k$: Thermal conductivity.
     • $A$: Cross-sectional area.
     • $\Delta T$: Temperature difference.
     • $d$: Thickness.
     • $t$: Time.

2. Convection:

   • Transfer of heat through the movement of fluids (liquids or gases).
   • Types:
     • Natural Convection: Due to density differences.
     • Forced Convection: Using external forces like fans or pumps.

3. Radiation:

   • Transfer of heat in the form of electromagnetic waves.
   • Does not require a medium.
   • Governing Law (Stefan-Boltzmann Law): $P = \sigma A T^4$
     • $\sigma$: Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, W/m^2 K^4$).
     • $A$: Surface area.
     • $T$: Absolute temperature in Kelvin.

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Specific Heat Capacity

1. Definition:

   • The amount of heat required to raise the temperature of $1 \, kg$ of a substance by $1 \, K$.
   • Formula: $Q = mc\Delta T$
     • $m$: Mass.
     • $c$: Specific heat capacity.
     • $\Delta T$: Change in temperature.

2. Units:

   • SI Unit: $J/(kg \, K)$.

3. Applications:

   • Water has a high specific heat capacity, making it ideal for cooling systems.

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Latent Heat

1. Definition:

   • The amount of heat required to change the state of a unit mass of a substance without changing its temperature.
   • Formula: $Q = mL$
     • $m$: Mass.
     • $L$: Latent heat.

2. Types:

   • Latent Heat of Fusion: Heat required to convert a solid into a liquid.
   • Latent Heat of Vaporization: Heat required to convert a liquid into a gas.

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Numerical Example

1. Example 1: Calculate the heat required to raise the temperature of $2 \, kg$ of water from $20^\circ C$ to $80^\circ C$. ($c_{\text{water}} = 4200 \, J/kg \, K$)

   • Formula: $Q = mc\Delta T$
   • Substituting values: $Q = 2 \cdot 4200 \cdot (80 - 20)$ $Q = 2 \cdot 4200 \cdot 60 = 504,000 \, J$

2. Example 2: Calculate the heat required to convert $1 \, kg$ of ice at $0^\circ C$ to water at $0^\circ C$. ($L_f = 334,000 \, J/kg$)

   • Formula: $Q = mL$
   • Substituting values: $Q = 1 \cdot 334,000 = 334,000 \, J$

Thermal Expansion

1. Definition:

   • When a substance is heated, its dimensions increase due to an increase in the average kinetic energy of its particles.

2. Types of Expansion:

   • Linear Expansion:
     • Change in length: $\Delta L = L_0 \alpha \Delta T$
       • $\alpha$: Coefficient of linear expansion ($K^{-1}$).
       • $L_0$: Original length.
   • Area Expansion:
     • Change in area: $\Delta A = A_0 \beta \Delta T$
       • $\beta = 2\alpha$: Coefficient of area expansion.
       • $A_0$: Original area.
   • Volume Expansion:
     • Change in volume: $\Delta V = V_0 \gamma \Delta T$
       • $\gamma = 3\alpha$: Coefficient of volume expansion.
       • $V_0$: Original volume.

3. Applications:

   • Gaps in railway tracks to accommodate expansion.
   • Expansion joints in bridges.

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Laws of Thermodynamics

1. Zeroth Law of Thermodynamics:

   • If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
   • Forms the basis for temperature measurement.

2. First Law of Thermodynamics:

   • Law of energy conservation: $\Delta Q = \Delta U + W$
     • $\Delta Q$: Heat added to the system.
     • $\Delta U$: Change in internal energy.
     • $W$: Work done by the system.

3. Second Law of Thermodynamics:

   • Heat cannot spontaneously flow from a colder body to a hotter body without external work.
   • Clausius Statement: It is impossible to construct a device that operates in a cycle and transfers heat from a cold body to a hot body without external energy input.
   • Kelvin-Planck Statement: No engine can convert all the heat it absorbs into work.

4. Third Law of Thermodynamics:

   • As the temperature of a system approaches absolute zero, the entropy of the system approaches a constant minimum.

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Heat Engines

1. Definition:

   • A heat engine is a device that converts heat energy into mechanical work.

2. Working Principle:

   • Operates between two heat reservoirs: a hot reservoir and a cold reservoir.
   • Efficiency ($\eta$): $\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$
     • $Q_H$: Heat absorbed from the hot reservoir.
     • $Q_C$: Heat rejected to the cold reservoir.

3. Carnot Engine:

   • An ideal heat engine with maximum efficiency.
   • Efficiency: $\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$
     • $T_H$: Temperature of the hot reservoir.
     • $T_C$: Temperature of the cold reservoir.

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Specific Heat Capacity at Constant Pressure and Volume

1. Molar Specific Heat:

   • At constant volume ($C_V$): $C_V = \left( \frac{\partial U}{\partial T} \right)_V$
   • At constant pressure ($C_P$): $C_P = \left( \frac{\partial U}{\partial T} \right)_P$

2. Relation Between $C_P$ and $C_V$:

   • For an ideal gas: $C_P - C_V = R$
     • $R$: Universal gas constant.

3. Applications:

   • Gases behave differently under constant volume and pressure.

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Thermal Conductivity

1. Thermal Resistance:

   • Reciprocal of thermal conductivity.
   • Formula for heat flow: $Q = \frac{kA\Delta T}{d} \cdot t$
     • $k$: Thermal conductivity.
     • $d$: Thickness of the material.

2. Composite Slabs:

   • Heat flow through multiple layers: $\frac{\Delta T}{R_{\text{total}}} = Q$
     • $R_{\text{total}}$: Sum of thermal resistances.

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Numerical Examples

1. Example 1: A metal rod of length $2 \, m$ expands by $1.6 \, mm$ when heated from $20^\circ C$ to $100^\circ C$. Find the coefficient of linear expansion ($\alpha$).

   • Formula: $\Delta L = L_0 \alpha \Delta T$
   • Substituting values: $1.6 \times 10^{-3} = 2 \cdot \alpha \cdot (100 - 20)$ $\alpha = \frac{1.6 \times 10^{-3}}{2 \cdot 80} = 1 \times 10^{-5} \, K^{-1}$

2. Example 2: A Carnot engine operates between $T_H = 500 \, K$ and $T_C = 300 \, K$. Find its efficiency.

   • Formula: $\eta = 1 - \frac{T_C}{T_H}$
   • Substituting values: $\eta = 1 - \frac{300}{500} = 1 - 0.6 = 0.4 \, \text{or} \, 40\%$

Heat Capacity and Calorimetry

1. Heat Capacity:

   • The amount of heat required to raise the temperature of a substance by 1 degree.
   • Formula: $C = mc$
     • $C$: Heat capacity.
     • $m$: Mass of the substance.
     • $c$: Specific heat capacity.

2. Calorimetry:

   • The study of heat transfer during physical or chemical changes.

   • Principle of Calorimetry:

     • Heat lost by a hot body = Heat gained by a cold body. $Q_{\text{lost}} = Q_{\text{gained}}$

   • Formula for heat exchange: $m_1c_1(T_1 - T_f) = m_2c_2(T_f - T_2)$

     • $T_1$, $T_2$: Initial temperatures of bodies.
     • $T_f$: Final equilibrium temperature.

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Change of State and Phase Diagram

1. Phase Diagram:

   • A graph showing different phases of a substance as a function of temperature and pressure.
   • Key Points:
     • Triple Point: All three phases coexist.
     • Critical Point: Above this, the substance exists as a supercritical fluid.

2. Cooling Curve:

   • Represents how the temperature of a substance changes as it loses heat.
   • Plateaus indicate phase changes (e.g., melting, boiling).

3. Supercooling and Superheating:

   • Supercooling: A liquid is cooled below its freezing point without solidifying.
   • Superheating: A liquid is heated above its boiling point without vaporizing.

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Thermal Efficiency and Insulation

1. Thermal Insulators:

   • Materials that resist the flow of heat.
   • Examples: Wood, glass wool, and Styrofoam.

2. Thermal Efficiency:

   • Efficiency of a heat transfer process: $\eta = \frac{\text{Useful heat transfer}}{\text{Total heat input}} \cdot 100$

3. Applications:

   • Thermos flasks use insulation to reduce heat exchange.
   • Double-glazed windows reduce heat loss in buildings.

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Blackbody Radiation

1. Blackbody:

   • An idealized object that absorbs all incident radiation and emits radiation based on its temperature.

2. Planck’s Law:

   • Describes the spectral distribution of radiation emitted by a blackbody.

3. Stefan-Boltzmann Law:

   • Total energy radiated per unit surface area is proportional to the fourth power of temperature: $P = \sigma A T^4$

4. Wien’s Displacement Law:

   • Wavelength at which the intensity of radiation is maximum: $\lambda_{\text{max}} T = b$
     • $b$: Wien’s constant ($2.897 \times 10^{-3} \, m \cdot K$).

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Heat Engines and Refrigerators

1. Heat Engines:

   • Convert heat energy into mechanical work.
   • Efficiency: $\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$

2. Refrigerators:

   • Transfer heat from a colder region to a hotter region.
   • Coefficient of Performance (COP): $COP = \frac{Q_C}{W}$
     • $Q_C$: Heat extracted from the cold reservoir.
     • $W$: Work done.

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Numerical Examples

1. Example 1: A $500 \, g$ aluminum block ($c_{\text{Al}} = 900 \, J/kg \cdot K$) is heated from $20^\circ C$ to $100^\circ C$. Calculate the heat absorbed.

   • Formula: $Q = mc\Delta T$
   • Substituting values: $Q = 0.5 \cdot 900 \cdot (100 - 20)$ $Q = 0.5 \cdot 900 \cdot 80 = 36,000 \, J$

2. Example 2: A refrigerator extracts $500 \, J$ of heat from its interior using $100 \, J$ of work. Find its COP.

   • Formula: $COP = \frac{Q_C}{W}$
   • Substituting values: $COP = \frac{500}{100} = 5$

3. Example 3: Calculate the peak wavelength of radiation emitted by a blackbody at $T = 6000 \, K$.

   • Formula: $\lambda_{\text{max}} T = b$
     • Substituting values: $\lambda_{\text{max}} = \frac{b}{T} = \frac{2.897 \times 10^{-3}}{6000}$ $\lambda_{\text{max}} = 4.83 \times 10^{-7} \, m = 483 \, nm$

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Recap: Key Points to Remember

• Heat transfer occurs via conduction, convection, or radiation.
• Thermal expansion is significant in practical applications like bridges and railway tracks.
• Laws of thermodynamics govern heat, energy, and work in physical processes.
• Heat engines and refrigerators operate based on thermodynamic principles.