Thermodynamics: Systems and Properties

Thermodynamics

• Thermodynamics involves the study of energy, its transformations, and its relationship to matter.
• Its principles are key for understanding all physical systems.

Fundamental Concepts

• A system refers to a defined region of space or quantity of matter under consideration.
• The surroundings include everything outside the system that can affect its behavior.
• The boundary is the surface that separates the system from its surroundings.

Types of Systems

• Isolated system: No mass or energy exchange occurs with the surroundings.
• Closed system: Energy is exchanged, but not mass, with the surroundings.
• Open system: Both mass and energy are exchanged with the surroundings.

Thermodynamic Properties

• These are characteristics of a system that can be quantified to describe its state.

Intensive Properties

• Intensive properties do not depend on system size.
• Temperature (T) is a measure of the average kinetic energy of particles in a system.
• Pressure (P) is the force exerted per unit area by a system on its surroundings.
• Density ($\rho$) is the mass per unit volume of a substance.

Extensive Properties

• Extensive properties depend on system size.
• Volume (V) is the amount of space occupied by a system.
• Mass (m) is the amount of matter in a system.
• Energy (E) is the capacity to do work.

Thermodynamic Processes

• Change in the state of a system due to changes in its properties.

Types of Processes

• Isothermal Process: Occurs at a constant temperature (T = constant).
• Isobaric Process: Occurs at constant pressure (P = constant).
• Isochoric (or Isovolumetric) Process: Occurs at constant volume (V = constant).
• Adiabatic Process: No heat is exchanged with the surroundings (Q = 0).

Zeroth Law of Thermodynamics

• If two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other.
• Establishes temperature as a fundamental property.

First Law of Thermodynamics

• Conservation of energy states that energy cannot be created or destroyed, only transformed.
• Expressed as $\Delta U = Q - W$, where $\Delta U$ is the change in internal energy, $Q$ is heat added, & $W$ is work done.

Second Law of Thermodynamics

• The total entropy of an isolated system can only increase over time or remain constant in ideal cases.
• Introduces entropy, a measure of disorder.
• Entropy (S) measures disorder or randomness.
• Reversible Process: Can be reversed without leaving any trace; it is an idealization.
• Irreversible Process: Cannot be reversed without leaving a trace; all real-world processes are irreversible.

Third Law of Thermodynamics

• Entropy of a perfect crystal at absolute zero (0 K) is zero.
• Provides a reference point for determining entropy at different temperatures.

Power Generation

• Fundamental to the design and operation of power plants.

Refrigeration and Air Conditioning

• Relies on thermodynamic cycles to transfer heat from a cold reservoir to a hot reservoir.

Chemical Reactions

• Used to predict the spontaneity and equilibrium of chemical reactions.

Materials Science

• Thermodynamic properties are crucial for designing and optimizing materials.

Biological Systems

• Plays a vital role in understanding energy flow and metabolic processes in biological systems.

Fonction logarithme népérien

• This is the natural logarithm function.

Définition

• logartihme népérien, or ln, is defined on $]0; +\infty[$.
• ln is the reciprocal of exponential functions expressed as: $\ln(x) = y \Leftrightarrow e^y = x$

Conséquences

• $\ln(e) = 1$
• $\ln(1) = 0$
• For all $x > 0$: $e^{\ln x} = x$
• For all real $x$: $\ln(e^x) = x$

Exemples

• To solve $\ln x = 3$, $\ln x = 3 \Leftrightarrow x = e^3$
• To solve $e^x = 5$, $e^x = 5 \Leftrightarrow x = \ln 5$

Propriétés algébriques

• For all real numbers $a$ and $b$ for all real numbers $n$:
• $\ln(ab) = \ln a + \ln b$
• $\ln(\frac{1}{b}) = -\ln b$
• $\ln(\frac{a}{b}) = \ln a - \ln b$
• $\ln(a^n) = n \ln a$
• $\ln(\sqrt{a}) = \frac{1}{2} \ln a$

Étude de la fonction

• $\ln$ is defined and differentiable on $]0; +\infty[$.
• $(\ln x)' = \frac{1}{x}$
• $\ln$ is stricly increasing on $]0; +\infty[$.

Tableau de variations

Limites à connaître

• $\lim_{x \to 0^+} \ln x = -\infty$
• $\lim_{x \to +\infty} \ln x = +\infty$

Matrizenmultiplikation

• Matrix Multiplication

Definition

• Let $A = (a_{ij}){m \times n}$ and $B = (b{ij})_{n \times r}$ be Matrices.
• The product $C = A \cdot B$ is defined as the $m \times r$ matrix $C = (c_{ij})$ with
• $c_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj} = a_{i1}b_{1j} + a_{i2}b_{2j} +... + a_{in}b_{nj}$
• for all $i = 1,..., m$ and $j = 1,..., r$.

Beispiel

• Example
• $A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$, $B = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}$
• $C = A \cdot B = \begin{pmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \ 43 & 50 \end{pmatrix}$

Eigenschaften

• Properties
• Assoziativität: $(A \cdot B) \cdot C = A \cdot (B \cdot C)$
• Distributivität: $A \cdot (B + C) = A \cdot B + A \cdot C$ and $(A + B) \cdot C = A \cdot C + B \cdot C$
• Nicht kommutativ: Im Allgemeinen ist $A \cdot B \neq B \cdot A$

Bemerkungen

• Remarks
• The number of columns of A must match the number of rows of B for the product to be defined.
• Matrix multiplication is a fundamental operation in linear algebra and has applications in many areas such as graphics, physics, and computer science.

Algorithmic Complexity

• Represents a measure of resources (time, space) required by an algorithm, as a function of the input size.
• Algorithms are compared by efficiency.
• Time is Expressed using "Big O" notation: $O(f(n))$

Why?

• Used to Predict Performance.
• Used to Compare algorithms.
• Used to Design efficient solutions.
• Used to Optimize Code.

How?

• Identify operations: Determine the basic operations the algorithm performs.
• Count: Count how many times those operations are performed.
• Express as a function: Write a function that expresses the number of operations as a function of the input size ($n$).
• Simplify: Determine the dominant term and express the complexity using Big O notation.

Common Complexities

$O(1)$ - Constant

• The algorithm takes the same amount of time regardless of the input size.
• Example: Accessing an element in an array by index

$O(\log n)$ - Logarithmic

• The time taken increases logarithmically with the input size.
• Example: Binary search

$O(n)$ - Linear

• The time taken increases linearly with the input size.
• Example: Looping through an array.

$O(n \log n)$ - Linearithmic

• The time taken increases by $n \log n$ with the input size.
• Example: Merge sort, heap sort.

$O(n^2)$ - Quadratic

• The time taken increases quadratically with the input size.
• Example: Bubble sort, insertion sort.

$O(2^n)$ - Exponential

• The time taken doubles with each addition to the input size.
• Example: Trying all possible subsets of a set.

$O(n!)$ - Factorial

• The time taken is proportional to the factorial of the input size.
• Example: Trying all possible permutations of a string.

Example

• Illustrative task
• Find the maximum value in an array.

Analysis

• The algorithm iterates through the array once.
• For each element, it performs a constant number of operations (comparison, assignment).
• Therefore, the time complexity is $O(n)$, where $n$ is the size of the array.

Space Complexity

• Measure of the amount of memory required by an algorithm.
• Also expressed using Big O notation.
• Considers variables, data structures, function call stack.

Tips

• Focus on the worst-case scenario
• Ignore constant factors that don't scale ($O(2n) = O(n)$)
• Be aware of space complexity
• Optimize for performance when it matters

Chemical Kinetics

Reaction Rate

• The reaction rate is the change in concentration of a reactant or product with respect to time.
• $rate = -\frac{\Delta[A]}{\Delta t} = \frac{\Delta[B]}{\Delta t}$

Factors Affecting Reaction Rate

• Reactant concentration
• Temperature
• Surface area
• Catalysis
• Pressure (for gases)

Rate Law

• The rate law is an equation that relates the rate of a reaction to the concentrations of reactants.
• $rate = k[A]^m[B]^n$

Reaction Order

• The reaction order is the sum of the exponents in the rate law.
• $overall\ order = m + n$
• Types of Reaction Order
  • Zero order: rate = k
  • First order: rate = k[A]
  • Second order: rate = k[A]^2 or rate = k[A][B]

Integrated Rate Laws

• Illustrative zero-order reactions
  • $[A]_t = -kt + [A]_0$, where $[A]_t$ is the concentration of A at time t, and $[A]_0$ is the initial concentration of A
• Illustrative first-order reactions
  • $ln[A]_t = -kt + ln[A]_0$
• Illustrative second-order reactions
  • $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$

Half-Life

• Half-life ($t_{1/2}$) is the time required for the concentration of a reactant to decrease to one-half of its initial value.
• Half-Life Equations
  • Zero order: $t_{1/2} = \frac{[A]_0}{2k}$
  • First order: $t_{1/2} = \frac{0.693}{k}$
  • Second order: $t_{1/2} = \frac{1}{k[A]_0}$

Collision Theory

• Illustrative principles -Molecules must collide to react. -Molecules must collide with sufficient energy (activation energy).
• Molecules must collide with proper orientation.
• Activation Energy

Arrhenius Equation

• Activation Energy ($E_a$) is the minimum energy required for a reaction to occur.
  • $k = Ae^{\frac{-E_a}{RT}}$, where $k$ is the rate constant, $A$ is the frequency factor, $E_a$ is the activation energy, $R$ is the gas constant (8.314 J/mol·K), and $T$ is the temperature in Kelvin.
• Determining Activation Energy can be done via this formula.
  • $ln(\frac{k_2}{k_1}) = \frac{E_a}{R}(\frac{1}{T_1} - \frac{1}{T_2})$

Reaction Mechanisms

• A reaction mechanism is the series of elementary steps that make up an overall reaction.
• Elementary steps are single-step reactions that cannot be broken down into simpler steps.
• The rate-determining step is the slowest step in a reaction mechanism

Catalysis

• Catalysis is the process of increasing the rate of a reaction by adding a catalyst.
• Types of Catalysis
  • Homogeneous catalysis: catalyst is in the same phase as the reactants
  • Heterogeneous catalysis: catalyst is in a different phase from the reactants
  • Enzyme catalysis: biological catalysts (enzymes) speed up biochemical reactions

Temperature Dependence

• Increasing temperature generally increases the reaction rate.
• Higher temperatures provide more molecules with necessary activation energy.