4 Physics and Chemistry chapter
Paragraph 1: Calculus & Differentiation
In mathematical analysis, the derivative of a function \( f(x) \) with respect to \( x \) is denoted as \( f'(x) \) or \( \dv{f}{x} \). The chain rule states that if \( y = g(f(x)) \), then the derivative is given by \( \dv{y}{x} = \dv{g}{f} \cdot \dv{f}{x} \). Additionally, the second derivative, which represents the curvature, is written as \( \pdv[2]{f}{x} \).
For instance, the integral of a polynomial function can be computed using the formula: \( f(x) \)
\[
\int x^n \dd{x} = \frac{x^{n+1}}{n+1} + C
\]
where \( C \) is the constant of integration.
Paragraph 2: Linear Algebra & Matrices
In linear algebra, a vector **\( \mathbf{v} \)** in \( \mathbb{R}^n \) can be represented as \( \mathbf{v} = (v_1, v_2, \dots, v_n) \). The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i \), and the determinant of a \( 2 \times 2 \) matrix is computed as:
\[
\det \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad – bc
\]
An eigenvalue \( \lambda \) of a matrix \( A \) satisfies the equation \( \det(A – \lambda I) = 0 \), where \( I \) is the identity matrix.
Physics & Quantum Mechanics
In quantum mechanics, the expectation value of an operator \( A \) in a given quantum state \( \ket{\psi} \) is represented as \( \expval{A} = \bra{\psi} A \ket{\psi} \). The Schrödinger equation, which governs the time evolution of a quantum state, is given by:
\[
i \hbar \pdv{\ket{\psi}}{t} = H \ket{\psi}
\]
where \( H \) is the Hamiltonian operator. The Heisenberg uncertainty principle states that the product of uncertainties in position \( \Delta x \) and momentum \( \Delta p \) must satisfy \( \Delta x \Delta p \geq \frac{\hbar}{2} \)
Chemistry Paragraphs
Chemical Reactions and Stoichiometry
A common example of a redox reaction is the reaction between hydrogen and oxygen to form water:
\[ \ce{2H2 + O2 -> 2H2O} \]
The law of conservation of mass states that the total number of atoms of each element remains constant before and after the reaction. The stoichiometric coefficients in the balanced equation ensure that mass and charge are conserved.
Acid-Base Reactions and pH
The pH of a solution is calculated using the negative logarithm of the hydrogen ion concentration:
\[ \text{pH} = -\log \left( \ce{[H+]} \right) \]
For strong acids like hydrochloric acid (\(\ce{HCl}\)), the dissociation is nearly complete:
HELL
\[ \ce{HCl -> H+ + Cl-} \]
OH
The pH of a 0.01 M solution of \(\ce{HCl}\) is calculated as:
\[ \text{pH} = -\log (0.01) = 2 \]
With breaklines
\[
\ce{HCl -> H+ + Cl-}
\]
\( \ce{HCl -> H+ + Cl-} \)
[latex]f(x)[/latex]
MCHEM Crazy
\[
\ce{Zn^2+ <=>[+ 2OH-][+ 2H+] $\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}$ <=>[+ 2OH-][+ 2H+] $\underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}$}
\]
