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Chapter 14: The Laws of Thermodynamics

Heat and Internal Energy

• Internal Energy ($U$): Total energy of molecules in a substance including kinetic and potential energy; change in internal energy is $\Delta U$.
• Heat ($Q$): Energy transferred due to temperature difference between objects, representing energy in motion.
• Calorie (cal): Heat needed to raise 1 gram of water by 1 degree Celsius.
• Kilocalorie (kcal): Heat needed to raise 1 kilogram of water by 1 degree Celsius with $1 \text{ kcal} = 1000 \text{ cal}$ and $1 \text{ kcal} = 4186 \text{ J}$.

Heat and Work

• Both heat and work are forms of energy in transit, measured in Joules.
• Work done on a gas (like in a piston-cylinder) increases its internal energy, which has the equivalent effect of heating the gas.
• James Prescott Joule (1818-1889) determined the mechanical equivalent of heat, demonstrating heat and work as different forms of energy.

Specific Heat

• Specific Heat ($c$): Heat needed to change the temperature of 1 kg of a substance by 1 degree Celsius, measured in J/(kg⋅°C) or cal/(g⋅°C).
• The equation for heat is $Q = mc\Delta T$, where:
  • $Q$ is the heat added or removed
  • $m$ is the mass of the substance
  • $c$ is the specific heat
  • $\Delta T$ is the change in temperature.

Molar Specific Heat

• Molar Specific Heat ($C$): Heat required to change the temperature of 1 mole of a substance by 1 degree Celsius, measured in J/(mol⋅°C).
• $C = Mc$, with $M$ as the molar mass (kg/mol).
• $Q = nC\Delta T$, where $n$ is the number of moles.

Heat Transfer Mechanisms

• Conduction: Heat transfer through direct contact needing a temperature difference.
  • Thermal Conductivity ($k$): Material's ability to conduct heat, with heat current $H = \frac{Q}{t} = kA\frac{\Delta T}{L}$, where:
    • $H$ is the rate of heat flow
    • $A$ is the cross-sectional area
    • $L$ is the material length
    • $\Delta T$ is the temperature difference.
• Convection: Heat transfer through fluid movement (liquids/gases), exemplified by hot air rising, boiling water, and blood flow.
• Radiation: Heat transfer via electromagnetic waves that needs no medium.
  • All objects emit and absorb radiation.
  • Stefan-Boltzmann Law: $P = \epsilon \sigma A T^4$, where:
    • $P$ is power radiated
    • $\epsilon$ is emissivity (0 to 1)
    • $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \text{ W/(m}^2 \cdot \text{K}^4)$)
    • $A$ is surface area
    • $T$ is temperature in Kelvin
  • Net power radiated is $P_{\text{net}} = \epsilon \sigma A (T^4 - T_0^4)$, with $T_0$ as the surroundings' temperature.

14.2 The First Law of Thermodynamics

Statement of the First Law

• First Law of Thermodynamics: The change in internal energy of a system equals the heat added minus the work done by the system with $\Delta U = Q - W$ where:
  • $\Delta U$ is the change in internal energy
  • $Q$ is heat added to the system
  • $W$ is work done by the system.

Thermodynamic Processes

• Adiabatic: No heat exchange ($Q = 0$), resulting in $\Delta U = -W$.
• Isobaric: Constant pressure, work done is $W = P\Delta V$.
• Isochoric (Isovolumetric): Constant volume, no work done ($W = 0$), so $\Delta U = Q$.
• Isothermal: Constant temperature, meaning $\Delta U = 0$ and $Q = W$.

Applications of the First Law

• Cyclic Process: Returns to the initial state, $\Delta U = 0$ and $Q = W$.
• Heat Engine: Converts heat into work, with efficiency $e = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} = 1 - \frac{Q_C}{Q_H}$ where:
  • $Q_H$ is heat absorbed from the hot reservoir
  • $Q_C$ is heat released to the cold reservoir
• Refrigerator: Transfers heat from cold to hot reservoir.
  • Coefficient of Performance (COP): $COP = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C}$.

14.3 The Second Law of Thermodynamics

Statements of the Second Law

• Second Law of Thermodynamics: Heat flows spontaneously from hot to cold.
• Clausius Statement: No process solely transfers heat from cooler to hotter.
• Kelvin Statement: No process purely converts heat into work.

Entropy

• Entropy ($S$): Measures the disorder of a system.
• Change in entropy: $\Delta S = \frac{Q}{T}$, with $Q$ as heat added and $T$ as absolute temperature.
• Second Law in Terms of Entropy: Total entropy of an isolated system always increases spontaneously.
  • $\Delta S_{\text{total}} \ge 0$

Heat Engines and the Second Law

• Carnot Engine: Theoretical engine for maximum efficiency.
  • Carnot Efficiency: $e_C = 1 - \frac{T_C}{T_H}$, where:
    • $T_C$ is the absolute temperature of the cold reservoir
    • $T_H$ is the absolute temperature of the hot reservoir
• No engine can be more efficient than a Carnot engine between two temperatures, and efficiency is always less than 1 (100%).

Entropy and the Arrow of Time

• Arrow of Time: Direction in which time flows.
• Entropy increases define the direction of time, making processes irreversible.
• Examples of such processes: breaking a glass, mixing milk and coffee, and cooling hot coffee.

Third Law of Thermodynamics

• It is impossible to cool a system to absolute zero in a finite number of steps.
• Entropy approaches a minimum value as temperature goes to absolute zero.

Summary of Key Concepts

Laws of Thermodynamics

• First Law: Energy is conserved; ($\Delta U = Q - W$).
• Second Law: Heat flows hot to cold, entropy increases; ($\Delta S \ge 0$).
• Third Law: Impossible to reach absolute zero in finite steps; ($T \rightarrow 0 \text{ K}, S \rightarrow \text{minimum}$).

Important Equations

• Heat: $Q = mc\Delta T$
• Heat Current: $H = kA\frac{\Delta T}{L}$
• Stefan-Boltzmann: $P = \epsilon \sigma A T^4$
• Work (Isobaric): $W = P\Delta V$
• Efficiency: $e = \frac{W}{Q_H}$
• Carnot Efficiency: $e_C = 1 - \frac{T_C}{T_H}$
• Entropy Change: $\Delta S = \frac{Q}{T}$

Linear Algebra Course and Corrected Exercises

Nabil CHOUKRI

Table of Contents

Chapter 1. Vector Spaces

• Definitions and examples
• Vector subspaces
• Sum of vector subspaces
• Vector space generated
• Intersection of vector subspaces
• Linear applications
• Image and kernel of a linear application
• Rank theorem
• Linear forms and hyperplanes
• Corrected exercises

Chapter 2. Finite-Dimensional Vector Spaces

• Generators
• Linear independence
• Base of a vector space
• Dimension of a vector space
• Rank of a family of vectors
• Linear applications in finite dimension
• Matrices
• Isomorphisms
• Change of base
• Corrected exercises

Chapter 3. Matrix Calculation

• Operations on matrices
• Matrices and linear applications
• Rank of a matrix
• Invertible matrices
• Elementary matrices and equivalence
• Transposition
• Corrected exercises

Chapter 4. Determinants

• Alternating n-linear forms
• Determinant of a family of vectors
• Determinant of an endomorphism
• Determinant of a square matrix
• Properties of determinants
• Calculation of determinants
• Applications of determinants
• Corrected exercises

Chapter 5. Reduction of Endomorphisms

• Eigen elements
• Characteristic polynomial
• Diagonalization
• Trigonalization
• Cayley-Hamilton theorem
• Minimal polynomial
• Characteristic subspaces
• Dunford decomposition
• Corrected exercises

Chapter 6. Real Prehilbert Spaces

• Scalar product
• Cauchy-Schwarz inequality
• Norm associated with a scalar product
• Orthogonality
• Orthonormal bases
• Gram-Schmidt process
• Orthogonal projections
• Orthogonal subspaces
• Adjoint of an endomorphism
• Symmetric operators
• Corrected exercises

Bibliography

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