Chapter 15: Thermal Properties of Matter

15.1 Avogadro's Number and Molar Mass

This section introduces the concept of the mole, Avogadro's number, and the molar mass, which are fundamental for quantifying substances in thermodynamics and kinetic theory.

• Avogadro's Number (NA): One mole of a substance contains exactly elementary entities (atoms, molecules, etc.).

• Molar Mass (M): The mass of one mole of a substance. It is calculated as , where is the mass of a single molecule.

• Total Mass: The total mass of a sample is , where is the number of moles.

• Example: 2 moles of water (H2O) have a mass of g = 36.04 g.

15.2 Equations of State and Ideal Gases

Equations of state relate the macroscopic properties of substances, such as pressure, volume, temperature, and amount of substance. The ideal gas law is a key example.

• Equation of State: Relates pressure (p), volume (V), temperature (T), and molar number (n) for substances in thermodynamic equilibrium.

• Ideal Gas: Assumes only elastic collisions between point-like particles, with no long-range forces.

• Standard Temperature and Pressure (STP): C (273 K) and 1 atm ( Pa).

• Ideal Gas Law: , where J/(mol·K) is the ideal gas constant.

• Density Form: , where is the mass density and is the molar mass.

• Comparing States: For a fixed amount of gas, (combined gas law).

• Isobaric Process: At constant pressure, .

• pV Diagrams: Plots of pressure vs. volume for an ideal gas at different temperatures (isotherms) show that at constant T.

• Phase Diagrams: Show the phases (solid, liquid, gas) of a substance as a function of pressure and temperature. The triple point and critical point are key features.

• Example: Tire pressure changes with temperature at constant volume, illustrating the direct relationship between pressure and temperature for a fixed volume of gas.

15.3 Kinetic Theory of Gases

Kinetic theory provides a microscopic explanation for the macroscopic properties of gases, relating temperature to the motion of molecules.

• Assumptions:

  • Gas molecules are point particles in random motion.

  • Collisions with container walls are perfectly elastic.

  • No long-range forces between molecules.

• Total Translational Kinetic Energy:

• Average Kinetic Energy per Molecule: , where J/K is Boltzmann's constant.

• Ideal Gas Law (per particle): , where is the number of molecules.

• Root-Mean-Square Speed:

• Speed Distribution: Molecular speeds follow the Maxwell-Boltzmann distribution, with most molecules near the average speed but some much faster or slower.

• Example: At higher temperatures, the average speed and spread of molecular speeds increase.

15.4 (Molar) Heat Capacities

Heat capacity quantifies the amount of heat required to change a substance's temperature. Molar heat capacity is the heat capacity per mole.

• Heat Added: , where is the molar heat capacity.

• Relationship to Specific Heat: , where is the specific heat (per kg).

• Monatomic Ideal Gas: J/(mol·K)

• Diatomic Ideal Gas:

• Degrees of Freedom: Each degree of freedom contributes to the average kinetic energy per molecule.

15.5 The First Law of Thermodynamics

The first law relates changes in internal energy to heat and work, providing a foundation for energy conservation in thermodynamic systems.

• Thermodynamic System: A defined region that can exchange energy (heat, work) and matter with its surroundings.

• First Law: , where is the change in internal energy, is heat added to the system, and is work done by the system.

• Sign Conventions:

  • : Heat enters the system.

  • : Work done by the system (energy leaves the system).

• Work by Volume Change: For constant pressure, .

• pV Diagrams: The area under the curve on a pV diagram represents the work done during a process.

• Cyclic Processes: For a process that returns to its initial state, , so .

• Example: Compressing a gas at constant pressure from 9.00 L to 2.00 L with 400 J of heat leaving the gas:

  • Work done by the gas: J

  • Change in internal energy: J

15.6 Thermodynamic Processes

Thermodynamic processes describe how a system changes state, often under specific constraints such as constant pressure, volume, or temperature.

• Adiabatic: No heat exchange (). Occurs in well-insulated systems or very rapid processes.

• Isochoric: Constant volume (). No work is done ().

• Isobaric: Constant pressure. Work done is .

• Isothermal: Constant temperature (). For an ideal gas, , so . Work done is .

15.7 Properties of Ideal Gases

This section summarizes the thermodynamic properties and relationships for ideal gases, including the connection between heat capacities and the behavior of gases under different processes.

• First Law for Ideal Gases: and .

• Constant Volume: (since ).

• Constant Pressure: and .

• Adiabatic Processes: . The adiabatic exponent .

• Adiabatic Relations:

• Example: Heating 3.50 mol of a diatomic ideal gas at constant pressure from 200 K to 350 K:

  • Heat added: 15.3 kJ

  • Expansion work: 4.4 kJ

  • Change in internal energy: 10.9 kJ

Additional info: The notes include references to the Maxwell-Boltzmann distribution, phase diagrams for water, and the use of pV diagrams for visualizing thermodynamic processes. These are standard topics in a college-level physics course on thermal properties of matter.