Thermodynamics

1.1 Kinetic Theory of Pressure & Temperature

• Focus Question: What causes pressure?

• Kinetic Theory: A gas is described as small particles in constant, random motion.

• Closed System: A system that does not exchange matter with the environment.

• Pressure: Force exerted on a surface per unit area.- Pressure = \frac{F}{A}

  • Units: Pascals (Pa), 1 Pa = 1 \frac{N}{m^2}

• Gas pressure in a closed container results from gas molecules colliding with the container.

• Force is the rate of change in momentum (p = mv).

• For one molecule hitting a piston:- F = \frac{\Delta p}{t} = \frac{mv - m(-v)}{t} = \frac{2mv}{t}

  • The change in velocity magnitude is 2v, accounting for direction.

• For N molecules hitting the piston per unit time t:- F = \frac{2Nmv}{t}

  • Pressure \ P = \frac{F}{A} = \frac{2Nmv}{At}

• Pressure measures how often gas molecules collide with the container.

Temperature and Kinetic Energy

• Temperature: Measure of the average kinetic energy of individual molecules.- Higher temperature means faster molecular motion.

  • Solids: Particles vibrate.

  • Liquids: Particles move across each other.

  • Gases: Particles move freely.

• Temperature and average kinetic energy:- K{avg} = \frac{3}{2} kB T

  • k_B: Boltzmann’s constant = 1.38 \times 10^{-23} \frac{J}{K}

  • T: Temperature in Kelvin.

• Molecules move at different speeds; K_{avg} is based on the average speed.

• At 0 K, molecules stop moving.

• Root Mean-Square Speed (v_{rms}):

  • K = \frac{1}{2} m v{rms}^2 = \frac{3}{2} kB T

  • v{rms} = \sqrt{\frac{3kB T}{m}}

  • m: mass of one molecule.

• Maxwell-Boltzmann distribution describes the speeds of molecules in a gas.

• v_{rms} is slightly to the right of the peak of the distribution graph.

Thermal Expansion

• Most substances expand when heated.

• Increased particle vibration leads to increased particle distance.

• Linear Expansion:

  • \Delta L = L_0 \alpha \Delta T

  • L = L_0(1 + \alpha \Delta T)

  • \alpha: coefficient of linear expansion (units: 1/°C).

• Railroad tracks have “expansion joints”.

• Constructing a Temperature Scale:

  • Use a substance with known thermal expansion (e.g., mercury).

  • Measure length at a lower temperature (e.g., freezing point of water).

  • Measure length at a higher temperature (e.g., melting point of water).

  • Write a linear equation: Temperature = slope \times length

• Area Expansion:

  • Thermal expansion of 2D or 3D objects occurs outward.

  • Holes in objects also expand.

1.2 Thermal Energy Transfer

• Focus Question: Is something with a higher temperature necessarily "hot"?

• At high temperatures, particles vibrate faster.

• Energy transfers from faster to slower particles during collisions.

• Heat: Transfer of thermal energy between objects in thermal contact due to a temperature difference.- Heat flows from high to low temperature.

  • Variable: Q, Units: Joules (J).

  • 4.186 J = 1 \text{ cal}

Zeroth Law of Thermodynamics

• Thermal Contact: Thermal energy can be exchanged between objects.

• Thermal Equilibrium: No net energy exchange occurs between objects in thermal contact.

• Zeroth Law: If A and B are both in thermal equilibrium with C, then A and B are in equilibrium with each other.

1.3 Gas Laws

• Focus Question: How do pressure, volume, and temperature affect each other?

Ideal Gas Model Assumptions

• Composition: Large amount of particles travel in random direction at various speeds.

• Distance of Particles: Particles are far apart compared to their size.

• Attractive Force: Particles interact only in collisions (no attractive forces).

• Collisions: Collisions of particles with container wall are perfectly elastic.

Gas Laws

• Relate temperature, pressure, and volume of a confined ideal gas.

• Explained by kinetic theory.

• Pressure:

  • Matter consists of small particles in motion.

  • Particles collide with the walls of the container (perfectly elastically) and bounce off, changing their direction and thus momentum.

  • A force acting over a time is required to change momentum, and this force causes pressure:

  • F = \frac{\Delta p}{\Delta t}

  • Pressure is force over area.

  • In thermodynamics, pressure is the average rate of change of momentum of particles hitting the container per unit area.

  • Higher pressure means more and faster collisions.

• Boyle’s Law: Volume of a gas is inversely proportional to pressure (constant temperature).- P1V1 = P2V2

• Charles’ Law: Volume of a gas is directly proportional to absolute temperature (constant pressure).- \frac{V1}{T1} = \frac{V2}{T2}

• 3rd Law: Pressure is directly proportional to absolute temperature (constant volume).- \frac{P1}{T1} = \frac{P2}{T2}

1.4 The First Law of Thermodynamics

• Focus Question: How does conservation of energy apply to thermodynamics processes?

Total Internal Energy

• Temperature measures average kinetic energy.

• Total internal energy is the combined energy of all molecules in the system.

• Average kinetic energy of a molecule: \frac{3}{2} kB T or \frac{3}{2} \frac{nR}{NA}T

• Internal Energy = U = N(\frac{3}{2} k_B T)

• U = (nNA) (\frac{3}{2} \frac{R}{NA} T) \rightarrow U = \frac{3}{2} nRT

• For ideal gas, T = \frac{PV}{nR}, so U = \frac{3}{2} PV

Work Done in Thermodynamics Processes

• Work Done in Thermodynamics Processes:- The initial pressure of the gas is P and the cross-sectional area of the piston is A, then the force with which one must push is PA.

  • If the piston is compressed some distance\Delta x, the work done is: W = F \Delta x \rightarrow W = PA(-\Delta x)

  • A \Delta x is volume, so W = -P \Delta V

  • Work is the area under a Pressure vs. volume graph

  • Work is only done when volume changes

The First Law of Thermodynamics

• Conservation of energy in thermal physics.

• Internal energy can be changed by doing work or adding heat.- \Delta U = Q + W

  • \Delta U - change in internal energy

  • Q - heat added to system

  • W – work done on the system

• Sign conventions:- \Delta U > 0: Temperature increases.

  • \Delta U < 0: Temperature decreases.

  • Q > 0: Heat enters the system.

  • Q < 0: Heat leaves the system.

  • W > 0: Work is done on the system; system is compressed.

  • W < 0: Work is done by the system, system expands.

Types of Processes

• Isobaric Process: Constant pressure, \Delta P = 0.

  • W = P \Delta V

  • Requires heat addition for expansion and heat removal for contraction to maintain constant pressure with temperature changes.

• Isothermal Process: Constant temperature, \Delta U = 0.

  • 0 = Q + W \rightarrow Q = -W

  • Adding heat causes expansion, removing heat causes contraction.

• Isochoric Process: Constant volume, W = 0.

  • \Delta U = -Q

  • Adding heat increases molecular speed (temperature), removing heat slows molecules down.

• Adiabatic Process: No heat enters or leaves the system, Q = 0.

  • \Delta U = W

  • Typically fast processes.

  • Adiabatic compression increases temperature, adiabatic expansion decreases temperature.

1.5 The Second Law of Thermodynamics

• Focus Question: What is entropy?

• Entropy: Measure of disorder in a system.

• Change in entropy: \Delta S = \frac{Q}{T} at constant temperature.

• 2nd Law of Thermodynamics: Entropy of the universe increases in all natural processes.

• Work can decrease entropy.

• Disorderly arrangements are more probable.

• Consequences:- Impossible for thermal energy to flow from cold to hot without work.

  • Impossible to completely convert heat into mechanical work in a cyclic process.

1.6 Specific Heat & Conductivity

• Focus Question: What factors affect the rate of thermal energy transfer?

Transfer of Energy

• Conduction: Heat transfer between objects in contact with a temperature difference.- Heat flows from hot to cold.

• Conduction occurs if there is a temperature difference between two parts of a conducting medium. The rate of heat transfer (power) through a conductor can be given by:- Q/t: rate of heat transfer

  • \frac{Q}{t} = \frac{kA(TH - TC)}{L} - k – thermal conductivity, a constant that depends on the type of material

    • The rate of thermal energy transfer is quicker for shorter lengths and larger temperature differences.

    • Materials that are good thermal conductors have a higher value of 𝑘. Good thermal conductors allow heat to easily flow through them.

• Convention: Heat transfer by movement by molecules in a fluid.- Convention occurs because a substance’s density decreases when temperature increases

  • As air gets hotter, it gets less dense and rises, colder air takes its place.

• Radiation: Heat transfer by electromagnetic waves.- Radiation is the only form of heat transfer that can occur in a vacuum since electromagnetic waves can travel in a vacuum.

• Phase Changes- As heat is added to a solid: - As heat is added, particles in the solid vibrate faster. Heat added increases the internal energy which causes temperature to increase.

  • At the melting point, temperature will remain constant despite added heat. Instead, added heat works to overcome the attractive force between