Ideal Gas Law

Gas Pressure

• Gases exert pressure on their surroundings.
• Pressure ($P$) equals force ($F$) per unit area ($A$): $P = \frac{F}{A}$.
• The SI unit of pressure is the pascal (Pa): 1 Pa = 1 N/m².
• Other common units of pressure include atmosphere (atm), millimeters of mercury (mmHg), and torr.
• Conversions: 1 atm = 101,325 Pa = 760 mmHg = 760 torr.

The Gas Laws

• Gas laws relate pressure ($P$), volume ($V$), temperature ($T$), and the number of moles ($n$) of a gas.

Boyle's Law

• At constant $T$ and $n$, a gas's volume is inversely proportional to its pressure: $P_1V_1 = P_2V_2$.

Charles's Law

• At constant $P$ and $n$, the volume of a gas is directly proportional to its absolute temperature: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$.

Avogadro's Law

• At constant $P$ and $T$, the volume of a gas is directly proportional to the number of moles: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$.

The Ideal Gas Law

• Combines Boyle's, Charles's, and Avogadro's laws: $PV = nRT$, where $R$ is the ideal gas constant.
• $R = 0.0821 \frac{\text{L atm}}{\text{mol K}} = 8.314 \frac{\text{J}}{\text{mol K}}$.

Gas Mixtures and Partial Pressures

• Dalton's Law of Partial Pressures: $P_{\text{total}} = P_1 + P_2 + P_3 +...$
• The partial pressure of a gas in a mixture is the product of its mole fraction and the total pressure: $P_i = X_i P_{\text{total}}$.
• $X_i = \frac{n_i}{n_{\text{total}}}$, where $X_i$ is the mole fraction of gas $i$.

Kinetic Molecular Theory of Gases

• Gases consist of particles in constant, random motion.
• Gas particle volume is negligible compared to the total gas volume.
• Intermolecular forces are negligible.
• The average kinetic energy ($KE$) of particles is proportional to the absolute temperature.
• Collisions between particles are elastic.
• The average kinetic energy is given by $KE = \frac{1}{2}mv^2 = \frac{3}{2}RT$, where $m$ is the mass, and $v$ is the average speed.

Diffusion and Effusion

• Diffusion: The spread of a gas through space or another gas.
• Effusion: The escape of a gas through a small hole.
• Graham's Law: $\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}$, where $M_1$ and $M_2$ are the molar masses of gas 1 and gas 2.

Real Gases

• Real gases deviate from ideal behavior at high pressures and low temperatures.
• The van der Waals equation accounts for these deviations: $(P + a(\frac{n}{V})^2)(V - nb) = nRT$, where $a$ and $b$ are gas-dependent constants.

Bernoulli's Principle

• An increase in fluid speed occurs with a decrease in pressure or potential energy.
• Formula: $P+\frac{1}{2} \rho v^{2}+\rho g h=constant$.
• $P$: absolute pressure of the fluid
• $\rho$: fluid density
• $v$: fluid velocity
• $g$: acceleration due to gravity
• $h$: height above a reference point

Applications of Bernoulli's Principle:

• Airplanes: Air flows faster over the wing, creating lower pressure above and lift.
• Race Cars: Spoilers create downforce to improve grip.
• Spray Bottles: Fast air lowers pressure, drawing liquid up the tube.
• Chimneys: Wind creates low pressure, drawing smoke out.

Venturi Effect

• Fluid flowing through a constriction speeds up, decreasing pressure.

Approximation Algorithms

• NP-hard problems lack polynomial-time algorithms (unless P=NP).

Two Approaches to Solving Large Problem Instances:

• Find optimal solutions using exponential time algorithms.
• Find near-optimal solutions in polynomial time.

Definitions for Approximation Algorithms:

• A $\rho$-approximation algorithm returns a solution within a factor of $\rho$ of the optimal value.
• $\rho$ is the approximation ratio.
  • $\rho \geq 1$ for minimization.
  • $\rho \leq 1$ for maximization.
• Absolute approximation: differences between returned solutions and optimal solutions are small.

Vertex Cover Problem

• Input: Graph $G = (V, E)$.
• Output: Subset $S \subseteq V$ where for every $(u, v) \in E$, either $u \in S$ or $v \in S$.
• Goal: Minimize $|S|$.
• NP-hard.

A 2-Approximation Vertex Cover Algorithm:

• Algorithm:
  • $S = \emptyset$
  • While $E \neq \emptyset$:
    • Pick an edge $(u, v) \in E$.
    • $S = S \cup {u, v}$.
    • Remove edges incident to $u$ or $v$.
  • Return $S$.
• Theorem: The algorithm is a 2-approximation algorithm.
• Proof:
  • Let $A$ be the set of edges picked by the algorithm.
  • Every edge in $A$ must be covered by $S^*$, the optimal vertex cover.
  • $|S^*| \geq |A|$.
  • The algorithm returns a vertex cover $S$ of size $2|A|$.
  • Therefore, $|S| = 2|A| \leq 2|S^*|$.

Traveling-Salesman Problem (TSP)

• Input: $n$ cities and cost function $c(i, j)$ to travel from city $i$ to city $j$.
• Output: A tour visiting each city once and returning to the start.
• Goal: Minimize the total cost.
• NP-hard

Two TSP Versions:

• Triangle inequality: $c(i, k) \leq c(i, j) + c(j, k)$.
• General TSP.

Algorithm for TSP with Triangle Inequality:

• Compute a minimum spanning tree $T$ of $G$.
• Let $L$ be a list of vertices visited in a preorder walk of $T$.
• Return the tour that visits the vertices in the order $L$.

Definitions:

• Preorder walk: Visit the root before visiting either of its children
• Shortcutting: remove repeated vertices

Theorem:

• The algorithm is a 2-approximation algorithm.

Proof:

• Let $W$ be the full walk of the minimum spanning tree $T$.
• Cost of full tree walk $c(W) = 2c(T)$.
• Delete repeated vertices (shortcutting).
• $H$ : tour after removing vertices from $W$ (shortcutting).
• $c(H) \leq c(W)$ triangle inequality.
• Let $H^*$ be an optimal tour.
• $c(T) \leq c(H^*)$.
• Therefore $c(H) \leq 2c(H^*)$.

Improved TSP Algorithm:

• Compute a minimum spanning tree $T$ of $G$.
• Compute a perfect matching $M$ for vertices with odd degree in $T$.
• Let $H$ be the multigraph formed by the union of $T$ and $M$.
• Form an Eulerian cycle in $H$.
• Make the cycle found previously a traveling-salesman tour by removing repeated vertices.

Theorem:

• The algorithm is a 1.5-approximation algorithm.