Van der Waals Equation

van der Waals Equation: Motivation

• The ideal gas law ($PV = nRT$) fails at high pressures and low temperatures.
• Ideal gas law assumptions: gas molecules have no volume and do not interact.
• The van der Waals equation modifies the ideal gas law, accounting for molecular volume and interaction.

van der Waals Equation: Formula

• The van der Waals equation is: $\left(P + a\left(\frac{n}{V}\right)^2\right)(V-nb) = nRT$
• a and b are empirical constants specific to each gas.

van der Waals Equation: Volume Correction

• The term $nb$ corrects for the volume of gas molecules.
• b is the van der Waals volume, representing the excluded volume per mole of gas.
• The available volume for gas molecules is calculated as $V-nb$.

van der Waals Equation: Pressure Correction

• The term $a\left(\frac{n}{V}\right)^2$ corrects for attractive forces between gas molecules.
• Attractive forces reduce pressure by pulling molecules inward, away from container walls.
• a is the van der Waals attraction parameter measuring the strength of attractive forces.
• The actual pressure of the gas is then $P + a\left(\frac{n}{V}\right)^2$.

van der Waals Constants: Significance

• Constant a: represents the strength of intermolecular attractions.
• High a values relate to stronger attractions.
• Constant b: represents the excluded volume per mole of gas.
• Larger molecules have larger b values.

Virial Expansion

• The virial expansion accounts for non-ideal behavior of real gases.
• Formula: $Z = \frac{PV_m}{RT} = 1 + \frac{B}{V_m} + \frac{C}{V_m^2} +...$
• Z is the compression factor.
• $V_m$ is the molar volume.
• B, C, etc., are virial coefficients which are temperature-dependent, and account for molecular interactions.

Relation to van der Waals Equation

• Second virial coefficient B is given by $B = b - \frac{a}{RT}$

Example Calculation

• Calculating the pressure exerted by 2.4 mol of $N_2$ gas in a 10.0 L vessel at 298 K can be done using 2 formulas:
• Using the ideal gas law: $P = \frac{nRT}{V} = \frac{2.4 \text{ mol} \cdot 0.08206 \frac{\text{L atm}}{\text{mol K}} \cdot 298 \text{ K}}{10.0 \text{ L}} = 5.87 \text{ atm}$
• Using the van der Waals equation ($a = 1.39 \text{ L}^2 \text{ atm mol}^{-2}$, $b = 0.0391 \text{ L mol}^{-1}$): $P = \frac{2.4 \text{ mol} \cdot 0.08206 \frac{\text{L atm}}{\text{mol K}} \cdot 298 \text{ K}}{10.0 \text{ L} - 2.4 \text{ mol} \cdot 0.0391 \text{ L mol}^{-1}} - 1.39 \text{ L}^2 \text{ atm mol}^{-2} \left(\frac{2.4 \text{ mol}}{10.0 \text{ L}}\right)^2= 5.74 \text{ atm}$
• The corrected pressure using van der Waals is lower than the pressure using the ideal gas law.

Summary: Ideal Gas Law vs. van der Waals Equation

• Molecular Volume: Ideal Gas Law says No, van der Waals says Yes (b)
• Intermolecular Force: Ideal Gas Law says No, van der Waals says Yes (a)
• Equation: Ideal Gas Law is $PV = nRT$, van der Waals Equation is $\left(P + a\left(\frac{n}{V}\right)^2\right)(V-nb) = nRT$

Quantum Mechanics: Wave-Particle Duality

• Light and matter exhibit both wave-like and particle-like properties.
• de Broglie wavelength relates a particle's momentum (p) to its wavelength ($\lambda$): $\lambda = \frac{h}{p}$, where h is Planck's constant.

Quantum Mechanics: Uncertainty Principle

• Heisenberg's uncertainty principle dictates the impossibility of knowing both position and momentum with perfect accuracy.
• Formula: $\Delta x \Delta p \geq \frac{\hbar}{2}$
• $\Delta x$ and $\Delta p$ represent uncertainties in position and momentum, respectively.
• $\hbar = \frac{h}{2\pi}$ is the reduced Planck constant.

Quantum Mechanics: Wave Functions

• Quantum system state is described by a wave function, $\Psi(r, t)$.
• $\Psi(r, t)$ contains all information about the system.
• Probability density of finding a particle is given by $|\Psi(r, t)|^2$.
• Wave function must satisfy the Schrödinger Equation.

Quantum Mechanics: Schrödinger Equation

• The time-dependent Schrödinger equation describes how the wave function evolves: $i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi$
• $\hat{H}$ is the Hamiltonian operator, representing total system energy.
• The time-independent Schrödinger equation is for stationary states: $\hat{H}\psi = E\psi$, where Eis the energy of the state.

Quantum Mechanics: Quantum Numbers

• Quantum numbers describe atomic orbital properties.
• They distinguish one electron from another.
• Types of quantum numbers: Principal (n), Angular Momentum (l), Magnetic (m_l), and Spin (m_s).
• Principal Quantum Number (n): describes the energy level of an electron.
• Angular Momentum or Azimuthal Quantum Number (l): describes the shape of the electron's orbital and the number of angular nodes.
• Magnetic Quantum Number (m_l): describes the orientation of the orbital in space.
• Spin Quantum Number (m_s): describes the intrinsic angular momentum of the electron.

Quantum Mechanics: Operators

• In quantum mechanics, physical quantities are represented by operators.
• Momentum operator in one dimension: $\hat{p} = -i\hbar \frac{\partial}{\partial x}$
• Expected value of an operator $\hat{A}$ for a system in state $\Psi$: $\langle A \rangle = \int \Psi^* \hat{A} \Psi d\tau$

Poisson Process: Definition

• N(t) counts events in [0, t].
• {N(t), t ≥ 0} is a Poisson process with rate λ > 0 if: N(0) = 0, has independent increments, and the number of events in any interval of length t follows a Poisson distribution with mean λt.
• $P{N(t+s) - N(s) = n} = e^{-\lambda t} \frac{(\lambda t)^n}{n!}, \quad n = 0, 1, \dots$

Poisson Process: Properties

• Memoryless Property: the process "forgets" past events.
• Arrival Times: $T_i$ is the time of the i-th event; $T_1, T_2, \dots$ are arrival times.
• Interarrival Times: $X_i = T_i - T_{i-1}$ is the time between (i-1)-th and i-th event, $T_0 = 0$.
• $X_1, X_2, \dots$ are interarrival times.
• Interarrival times $X_i$ are iid exponential random variables with mean $1/\lambda$.

Poisson Process: Variations

• The Non-Homogeneous Poisson Process has a rate λ that varies with time: λ(t).
• Expected number of events in $[0, t]$ is $m(t) = \int_0^t \lambda(s) ds$.
• Number of events in (s, t] is Poisson($m(t) - m(s)$).
• The Compound Poisson Process has a random size per event.
• $Y_i$ is the size of the i-th event.
• $S(t) = \sum_{i=1}^{N(t)} Y_i$, $N(t)$ is a Poisson process.

Poisson Process: Applications

• Arrivals in queueing systems
• Reliability analysis
• Finance

Partial Differential Equations: Separation of Variables for Heat Equation

• Heat equation: $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} \quad 0 0$
• Case 1: $\lambda = 0$
  • Solutions are $X(x) = c_1x + c_2$.
  • Applying $X(0) = 0$ yields $c_2 = 0$.
  • Applying $X(L) = 0$ yields $c_1 = 0$.
  • Thus, $X(x) = 0$ (Trivial Solution).
• Case 2: $\lambda < 0$, set $\lambda = -\alpha^2, \alpha > 0$.
  • Solutions are $X(x) = c_1e^{\alpha x} + c_2e^{-\alpha x}$.
  • Applying $X(0) = 0$ yields $c_2 = -c_1$.
  • Applying $X(L) = 0$ yields $c_1 = 0$, so $X(x) = 0$ (Trivial Solution).
• Case 3: $\lambda > 0$, set $\lambda = \alpha^2, \alpha > 0$.
  • Solutions are $X(x) = c_1 cos(\alpha x) + c_2 sin(\alpha x)$.
  • Applying $X(0) = 0$ yields $c_1 = 0$.
  • Applying $X(L) = 0$ requires $sin(\alpha L) = 0$ for non-trivial solutions.
  • $\alpha_n = \frac{n\pi}{L}, \quad n = 1, 2, 3,...$
  • $\lambda_n = \alpha_n^2 = (\frac{n\pi}{L})^2, \quad n = 1, 2, 3,...$
  • Eigenfunctions: $X_n(x) = sin(\frac{n\pi}{L}x), \quad n = 1, 2, 3,...$
• Solve $T'(t) + k\lambda T(t) = 0$:
  • $T_n(t) = e^{-k(\frac{n\pi}{L})^2t}, \quad n = 1, 2, 3,...$
• Thus, $u_n(x,t) = X_n(x)T_n(t) = sin(\frac{n\pi}{L}x)e^{-k(\frac{n\pi}{L})^2t}, \quad n = 1, 2, 3,...$
• Superposition Principle:
  • $u(x,t) = \sum_{n=1}^{\infty} b_n sin(\frac{n\pi}{L}x)e^{-k(\frac{n\pi}{L})^2t}$
• Initial Condition & Fourier Coefficients:
  • $u(x,0) = f(x) = \sum_{n=1}^{\infty} b_n sin(\frac{n\pi}{L}x)$
  • $b_n = \frac{2}{L} \int_{0}^{L} f(x) sin(\frac{n\pi}{L}x) dx, \quad n = 1, 2, 3,...$

Linear Algebra: Vectors - Definition

• Vectors are math entities defined by direction, sense, and magnitude.

Linear Algebra: Vectors - Representations

• Vectors in n-dimensional space consist of n ordered real numbers: $\vec{v} = (v_1, v_2,..., v_n) \in \mathbb{R}^n$

Linear Algebra: Vectors - Operations, Addition

• Addition of vectors $\vec{u}$ and $\vec{v}$: $\vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2,..., u_n + v_n)$

Linear Algebra: Vectors - Operations, Scalar Multiplication

• Multiplication of vector $\vec{v}$ by a scalar c: $c\vec{v} = (cv_1, cv_2,..., cv_n)$

Linear Algebra: Vectors - Dot Product

• Dot product of vectors $\vec{u}$ and $\vec{v}$: $\vec{u} \cdot \vec{v} = \sum_{i=1}^{n} u_i v_i = u_1v_1 + u_2v_2 +... + u_nv_n$

Linear Algebra: Vectors - Norm

• Norm (magnitude) of vector $\vec{v}$: $||\vec{v}|| = \sqrt{\vec{v} \cdot \vec{v}} = \sqrt{\sum_{i=1}^{n} v_i^2}$

Linear Algebra: Vectors - Distance

• Distance between vectors $\vec{u}$ and $\vec{v}$: $d(\vec{u}, \vec{v}) = ||\vec{u} - \vec{v}||$

Linear Algebra: Matrices - Definition

• Matrices are rectangular arrays of real numbers organized into m rows and n columns: $\qquad A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$

Linear Algebra: Matrices - Operations, Addition

• Addition of matrices A and B (same dimensions): $(A + B){ij} = A{ij} + B_{ij}$

Linear Algebra: Matrices - Operations, Scalar Multiplication

• Multiplication of matrix A by scalar c: $(cA){ij} = cA{ij}$

Linear Algebra: Matrices - Operations, Matrix Multiplication

• Multiplication of matrix A ($m \times p$) by matrix B ($p \times n$): $(AB){ij} = \sum{k=1}^{p} A_{ik}B_{kj}$

Linear Algebra: Matrices - Transposition

• Transpose of matrix A: $(A^T){ij} = A{ji}$

Linear Algebra: Matrices - Identity Matrix

• The identity matrix I is a square matrix with 1s on the diagonal and 0s elsewhere.

Linear Algebra: Matrices - Inverse

• The Inverse of matrix A (if it exists) is a matrix $A^{-1}$ such that: $\qquad AA^{-1} = A^{-1}A = I$

Linear Algebra: Matrices - Determinant

• The determinant of a square matrix A, noted det(A) or |A|, is a scalar.

Linear Algebra: Matrices - Rank

• Rank of matrix A is the maximum number of linearly independent columns (or rows).

Linear Algebra: Systems of Linear Equations - Representation

• A system of linear equations can be represented in matrix form: $Ax = b$
  • A is the coefficient matrix
  • x is the unknown vector
  • b is the constant vector

Linear Algebra: Systems of Linear Equations - Resolution

• Resolution of linear equations refers to finding the vector x that satisfies the equation Ax = b.

Linear Algebra: Systems of Linear Equations - Methods

• Gauss Elimination
• LU Factorization
• Cramer's Rule

Linear Algebra: Vector Spaces - Definition

• A vector space is a set of objects (vectors) with two operations that satisfy certain properties (addition and scalar multiplication).

Linear Algebra: Vector Spaces - Subspace

• A subspace is a subset of a vector space that is itself a vector space.

Linear Algebra: Vector Spaces - Basis

• A basis of a vector space is a set of linearly independent vectors that span the vector space.

Linear Algebra: Vector Spaces - Dimension

• The dimension of a vector space is the number of vectors in a basis of that vector space.

Linear Algebra: Linear Transformations

• A linear transformation between vector spaces preserves addition and scalar multiplication.

Linear Algebra: Linear Transformations - Matrix Representation

• A linear transformation can be represented by a matrix.

Linear Algebra: Eigenvalues and Eigenvectors

• An eigenvector v of matrix A satisfies $Av = \lambda v$, where $\lambda$ is an eigenvalue of A.

Thermodynamics: Zeroth Law

• If two thermodynamic systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other.
• It allows the definition of temperature.

Thermodynamics: First Law

• Energy is conserved; it can only change forms.
• In any process, the total energy of the universe remains the same.
• For a thermodynamic cycle the net heat supplied to the system equals the net work done by the system.
• Formula: $\Delta U = Q - W$
  • U = internal energy of the system
  • Q = heat exchanged between system & surroundings
  • W = work done by the system

Thermodynamics: Second Law

• The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.
• Formula: $\Delta S \geq 0$
  • S = the entropy of the system.

Thermodynamics: Third Law

• As temperature approaches absolute zero, the entropy of a system approaches a minimum or zero value.
• $T \to 0, S \to 0$
• It allows for the definition of absolute entropy.