Thermal Properties of Matter

Learning Objectives

• Understand the derivation and application of the ideal-gas equation $PV = \mu RT$ and its integration of Boyle's and Charles' laws to describe gas behavior.
• Calculate the effects of thermal expansion in materials, using coefficients of linear, area, and volume expansion to predict changes under thermal stress.
• Determine specific heat capacity and its role in heat transfer, expressed in $J/kg\cdot K$, and solve related numerical problems.
• Analyze the concept of latent heat during phase changes, such as melting and boiling, and calculate the latent heat of fusion and vaporization.
• Explore the mechanisms of heat transfer: conduction, convection, and radiation, and apply these concepts to solve complex problems involving different materials and conditions.
• Apply Stefan-Boltzmann Law $H = A\sigma T^4$ to quantify radiative heat transfer from black bodies and understand its implications in thermal systems.
• Investigate Newton's Law of Cooling to model the rate of temperature change in bodies exposed to different ambient conditions.
• Convert temperatures between Celsius, Fahrenheit, and Kelvin scales using appropriate formulas and understand the significance of fixed points.
• Solve calorimetry problems using the principle of heat exchange, ensuring heat lost equals heat gained in isolated systems.
• Interpret phase diagrams and understand the conditions for phase changes, including the triple point and its significance in thermodynamics.
• Examine the concept of thermal conductivity in conduction, using the formula $H = KA(T_1−T_2)/L$ to analyze heat flow through composite rods and materials.
• Understand the principles of thermal radiation, black body behavior, and the greenhouse effect, applying these concepts to real-world examples.

Chapter Notes

Ideal-Gas Equation and Absolute Temperature

• Ideal-Gas Equation: $PV = \mu RT$
  • Variables:
    • $P$: Pressure
    • $V$: Volume
    • $T$: Temperature in Kelvin
    • $\mu$: Number of moles
    • $R$: Universal gas constant ($8.31 \text{ J mol}^{-1} \text{ K}^{-1}$)
  • Combines: Boyle's Law ($PV = \text{constant}$) and Charles' Law ($V/T = \text{constant}$).
  • Absolute Zero: Extrapolated from gas behavior, $-273.15 ^\circ C$ or $0 \text{ K}$.

Thermal Expansion

• Linear Expansion: $\frac{\Delta l}{l} = \alpha_1 \Delta T$
• Area Expansion: $\frac{\Delta A}{A} = 2\alpha_1 \Delta T$
• Volume Expansion: $\frac{\Delta V}{V} = 3\alpha_1 \Delta T$
• Coefficients:
  • Linear Expansion ($\alpha_1$): Characteristic of material.
  • Volume Expansion ($\alpha_V$): Generally temperature-dependent.

Specific Heat Capacity

• Definition: Amount of heat required to change the temperature of unit mass by one degree.
• Formula: $s = \frac{Q}{m \Delta T}$
  • $Q$: Heat absorbed or released
  • $m$: Mass
  • $\Delta T$: Temperature change
• Units: J/kg·K

Latent Heat

• Definition: Heat absorbed or released during a phase change without temperature change.
• Types:
  • Fusion: Solid to liquid
  • Vaporization: Liquid to gas
• Formula: $Q = mL$
  • $L$: Latent heat
  • Units: J/kg

Heat Transfer Mechanisms

• Conduction: Transfer through a medium without movement of the medium.
  • Formula: $H = \frac{KA(T_1 - T_2)}{L}$
  • $K$: Thermal conductivity
• Convection: Transfer through fluid motion.
  • Natural Convection: Due to density differences.
  • Forced Convection: External forces like pumps.
• Radiation: Transfer through electromagnetic waves.
  • Stefan-Boltzmann Law: $H = A\sigma T^4$
  • $\sigma$: Stefan-Boltzmann constant ($5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$)

Stefan-Boltzmann Law

• Formula: $H = A\sigma T^4$
  • $A$: Surface area
  • $T$: Absolute temperature
  • Emissivity ($e$): Fraction of energy radiated compared to a perfect black body.

Newton's Law of Cooling

• Formula: $H = \sigma A (T^4 - T_s^4)$
  • $T_s$: Surrounding temperature
  • Describes rate of temperature change of a body in relation to its surroundings.

Temperature, Heat and Temperature Scale

• Temperature: Measure of hotness or coldness.
• Heat: Energy transfer due to temperature difference.
• Scales: Celsius, Fahrenheit, Kelvin
  • Conversions:
    • $t_F = \frac{9}{5}t_C + 32$
    • $T_K = t_C + 273.15$

Calorimetry and Principle of Heat Exchange

• Principle: Heat lost by hot body = Heat gained by cold body
• Calorimeter: Device to measure heat exchange.

Change of State and Phase Diagrams

• Phase Changes: Melting, freezing, boiling, sublimation.
• Triple Point: Temperature and pressure where all three phases coexist.

Conduction and Thermal Conductivity

• Formula: $H = \frac{KA(T_1 - T_2)}{L}$
  • $K$: Thermal conductivity
  • Materials: Metals have high $K$, insulators have low $K$.

Thermal Radiation and Black Body Concepts

• Black Body: Perfect absorber and emitter of radiation.
• Wien's Displacement Law: $\lambda_m T = \text{constant}$
  • Describes peak wavelength shift with temperature.