States of Matter: Solid, Liquid, and Gas

Chapter 3: States of Matter

• Matter has mass and volume.
• Mass refers to the amount of matter in an object.
• Volume refers to the amount of space an object occupies.
• The four states of matter are solid, liquid, gas, and plasma.

Solids

• Have definite shape and volume.
• Particles are tightly packed and don't move freely.

Types of Solids

• Crystalline solids have atoms, ions, or molecules arranged in a repeating pattern; examples include salt, sugar and diamond.
• Amorphous solids are arranged randomly; examples include glass, rubber and plastic.

Liquids

• Have definite volume, but no definite shape.
• Particles are close together but can move.
• Viscosity is the liquid's resistance to flow.
• Surface tension is the force that causes a liquid's surface to contract.

Gases

• Have no definite shape or volume.
• Particles are far apart and move freely.
• Gas pressure refers to the force exerted by a gas per unit area.
• As temperature increases, gas pressure increases.
• As volume decreases, gas pressure increases.
• As the number of particles increases, gas pressure increases.

Changes of State

• Melting is the change from solid to liquid.
• Freezing is the change from liquid to solid.
• Vaporization is the change from liquid to gas.
  • Boiling is vaporization throughout a liquid.
  • Evaporation is vaporization at the surface of a liquid.
• Condensation is the change from gas to liquid.
• Sublimation is the change from solid to gas.
• Deposition is the change from gas to solid.

Plasma

• Plasma is similar to gas but contains freely moving ions and electrons.
• Plasma is electrically conductive.
• Magnetic fields affect plasma.
• Plasma emits light.
• Stars, lightning, and neon signs are examples of plasma.

Electric Potential Energy

• Electrostatic force is conservative.
• Change in potential energy = negative work done by a force: ΔU = -W
• Work done: W = ∫ q E ⋅ d s
• Potential energy difference: ΔU = -q ∫ E ⋅ d s
• U = 0 is the zero point.

Electric Potential

• Represents electric potential energy per unit charge: V = U/q
• Change in electric potential: ΔV = ΔU/q = -∫ E ⋅ d s
• V = 0 is the zero point because only changes in electric potential matter.
• Electric potential is a scalar.
• Unit: 1 V = 1 J/C
• Electron-volt: 1 eV = (1.60 x 10-19 C)(1 V) = 1.60 x 10-19 J

Potential Difference

• The potential difference between points A and B is expressed as: ΔV = VB - VA = -∫AB E ⋅ d s

Electric Potential due to Point Charges

• Electric potential due to a point charge q at distance r: V = kq/r
• Electric potential due to multiple point charges: V = Σi kqi/ri

Electric Potential due to Continuous Charge Distribution

• Electric potential is V = ∫ kdq/r
• Where dq = λ dl = σ dA = ρ dV

Equipotential Surfaces

• Electric potential is constant on these surfaces.
• Electric fields are perpendicular to equipotential surfaces.
• Moving a charge between two points on the surface requires no work.

Obtaining E from V

• In 1D: Ex = -dV/dx
• In 3D: E = -∇V = -(∂V/∂x i + ∂V/∂y j + ∂V/∂z k)

Electric Potential Energy of a System of Charges

• The work needed to assemble the charges from infinity is the system's electric potential energy.
• Two charges: U = q2V1 = kq1q2/r12
• Multiple charges: See text for formula

Vector Fields

• Assigns a vector to each point in $\mathbb{R}^2$ with $\overrightarrow{F}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$
• An example is $\overrightarrow{F}(x,y) = \langle -y, x \rangle$.

Line Integrals

• Integrals over a curve C that has a function $f(x,y)$.
• If C is parametrized by $\overrightarrow{r}(t)$ for $a \leq t \leq b$, then the line integral of $f$ along $C$ is $\qquad \int_C f(x,y) ds = \int_a^b f(\overrightarrow{r}(t)) |\overrightarrow{r},'(t)| dt$, where $ds = |\overrightarrow{r},'(t)| dt$ is the arc length element.
• See text for example of computing $\int_C x y^4 ds$ where $C$ is the right half of the circle $x^2 + y^2 = 16$.

Line Integrals of Vector Fields

• Describes line integrals of vector fields
• If $\overrightarrow{F}$ is a vector field on $\mathbb{R}^2$, and $C$ is a curve parametrized by $\overrightarrow{r}(t)$ for $a \leq t \leq b$, then the line integral of $\overrightarrow{F}$ along $C$ is $\qquad \int_C \overrightarrow{F} \cdot d\overrightarrow{r} = \int_a^b \overrightarrow{F}(\overrightarrow{r}(t)) \cdot \overrightarrow{r},'(t) dt$
• See text for example of computing $\int_C \overrightarrow{F} \cdot d\overrightarrow{r}$ where $\overrightarrow{F}(x,y) = \langle x, y \rangle$ and $C$ is parametrized by $\overrightarrow{r}(t) = \langle t, t^2 \rangle$ for $0 \leq t \leq 1$.

Definition of the Fourier Transform

• Defined as $\qquad F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$
  • $f(t)$ is a function of time $t$
  • $F(\omega)$ is a function of frequency $\omega$
  • $j = \sqrt{-1}$ is the imaginary unit
• The inverse Fourier Transform is defined as: $\qquad f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} d\omega$

Important Properties

• Linearity: $\qquad \mathcal{F}{af(t) + bg(t)} = a\mathcal{F}{f(t)} + b\mathcal{F}{g(t)} = aF(\omega) + bG(\omega)$
• Time Scaling: $\qquad \mathcal{F}{f(at)} = \frac{1}{|a|}F(\frac{\omega}{a})$
• Time Shifting: $\qquad \mathcal{F}{f(t - t_0)} = e^{-j\omega t_0}F(\omega)$
• Frequency Shifting: $\qquad \mathcal{F}{e^{j\omega_0 t}f(t)} = F(\omega - \omega_0)$
• Differentiation in Time: $\qquad \mathcal{F}{\frac{df(t)}{dt}} = j\omega F(\omega)$
• Differentiation in Frequency: $\qquad \mathcal{F}{tf(t)} = j\frac{dF(\omega)}{d\omega}$
• Convolution: $\qquad \mathcal{F}{(f * g)(t)} = F(\omega)G(\omega)$
• Multiplication: $\qquad \mathcal{F}{f(t)g(t)} = \frac{1}{2\pi}(F * G)(\omega)$
• Parseval's Theorem: $\qquad \int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(\omega)|^2 d\omega$
• Duality: If $\mathcal{F}{f(t)} = F(\omega)$, then $\mathcal{F}{F(t)} = 2\pi f(-\omega)$

Common Fourier Transform Pairs

• There is a table of common pairs in the source text

Properties of the Fourier Transform

• Some example common properties include:
  • Linearity
  • Time Scaling
  • Time Shifting
  • Frequency Shifting
  • Differentiation in Time
  • Differentiation in Frequency
  • Convolution
  • Multiplication

The Schrödinger Equation

• The equation: $-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\Psi(x) + V(x)\Psi(x) = E\Psi(x)$
• $\hbar$ is the reduced planck constant
• $m$ is the mass
• $V(x)$ is the potential
• $E$ is the energy

Simple Case: Free Particle $(V(x)=0)$

• $-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\Psi(x) = E\Psi(x)$
• $\frac{d^2}{dx^2}\Psi(x) = -\frac{2mE}{\hbar^2}\Psi(x)$
• $\Psi(x) = A e^{ikx} + B e^{-ikx}$, where $k = \sqrt{\frac{2mE}{\hbar^2}}$

Particle in a Box (Infinite Square Well)

• $V(x) = \begin{cases} 0, & 0 < x < a \ \infty, & \text{otherwise} \end{cases}$

Solutions inside the box:

• $\Psi(x) = A \sin(kx) + B \cos(kx)$, where $k = \sqrt{\frac{2mE}{\hbar^2}}$

Boundary conditions:

• $\Psi(0) = \Psi(a) = 0$

Quantized energy levels:

• $E_n = \frac{n^2\pi^2\hbar^2}{2ma^2}$, where $n = 1, 2, 3,...$

Normalized wave functions:

• $\Psi_n(x) = \sqrt{\frac{2}{a}} \sin(\frac{n\pi x}{a})$

¿Qué es Markdown?

• Light markup language with plain text syntax created by John Gruber in 2004.
• Easy to read and write.
• Can be converted to HTML, PDF, etc.
• Portable and can be opened in any text editor.

Sintaxis básica

• The text uses several markdown examples in the source text, including:
  • Enclosed Text
  • Markdown
  • Lists
  • Links
  • etc.

Física

Vectors are a line segment consisting of direction and orientation

• Vectors are characterized by:
• magnitude
• direction
• and orientation

Types of vectors

• Free Vectors: Does not have a specific application point.
• Sliding Vectors: Application point slides across the line of action.
• Fixed Vectors: Has a single point of application.
• Unit Vectors: Values ​​whose modulus equals 1.

Vector Components

• Vectors can be shown as the sum of two other vectors with perpendicular directions. These directions are usually horizontal and vertical.

$\vec{v} = \vec{v}_x + \vec{v}_y$

$v_x = v \cdot \cos{\theta}$

$v_y = v \cdot \sin{\theta}$

$\vec{v} = (v_x, v_y) = (v \cdot \cos{\theta}, v \cdot \sin{\theta})$

Addition and subtraction of vectors

To sum or subtract, add (or subtract) their components.

$\vec{v} + \vec{w} = (v_x, v_y) + (w_x, w_y) = (v_x + w_x, v_y + w_y)$

Scalar product of two vectors

• The scalar product of two vectors results in scalar form:

$\vec{v} \cdot \vec{w} = v \cdot w \cdot \cos{\theta}$

• May be calculated as:

$\vec{v} \cdot \vec{w} = v_x \cdot w_x + v_y \cdot w_y$

Vectorial product of two vectors

• Vector multiplication of other factors in another perpendicular vector, which module applies to:

$|\vec{v} \times \vec{w}| = v \cdot w \cdot \sin{\theta}$

• Orientation may be obtained via the right hand rule