Calculating Heat, Phase Transitions, and Solid State Structures

Thermodynamic Calculations for Heating Water and Phase Changes

• Total Heat Calculation Model:     * To calculate the total heat ($Q_{\text{total}}$) required to transform a substance across different temperatures and phases, the individual heats for each step ($q_1, q_2, q_3, \dots$) must be summed.     * Formula for temperature change: $q = m \times c \times \Delta T$.     * Formula for phase change: $q = n \times \Delta H$.

• Molar Mass and Conversion:     * To perform phase change calculations, mass in grams must be converted to moles ($n$) using the molar mass.     * Example: For $135\,g$ of water ($H_2O$):         * Molar mass of water: $\approx 18.015\,g/mol$.         * $n = \frac{135\,g}{18.015\,g/mol} = 7.5\text{ moles}$.     * Note: It is assumed the mass remains constant in a closed container during these processes.

• Step-by-Step Heat Calculation Example (135g Ice at -15°C to Steam at 120°C):     * Step 1 ($q_1$): Heating Ice from $-15^\circ\text{C}$ to $0^\circ\text{C}$:         * $q_1 = m \times c_{\text{ice}} \times \Delta T$         * $q_1 = 135\,g \times 2.09\,J/g^\circ\text{C} \times (0 - (-15))^\circ\text{C}$     * Step 2 ($q_2$): Melting Ice at $0^\circ\text{C}$:         * $q_2 = n \times \Delta H_{\text{fusion}}$         * $\Delta H_{\text{fusion}} = 6.01\,kJ/mol = 6.01 \times 10^3\,J/mole$         * $q_2 = 7.5\,moles \times 6010\,J/mole$     * Step 3 ($q_3$): Heating Liquid Water from $0^\circ\text{C}$ to $100^\circ\text{C}$:         * $q_3 = m \times c_{\text{liquid}} \times \Delta T$         * $q_3 = 135\,g \times 4.18\,J/g^\circ\text{C} \times (100 - 0)^\circ\text{C}$     * Step 4 ($q_4$): Boiling Water at $100^\circ\text{C}$:         * $q_4 = n \times \Delta H_{\text{vap}}$         * $\Delta H_{\text{vap}} = 40.67\,kJ/mol = 40.67 \times 10^3\,J/mole$         * $q_4 = 7.5\,moles \times 40670\,J/mole$     * Step 5 ($q_5$): Heating Steam from $100^\circ\text{C}$ to $120^\circ\text{C}$:         * $q_5 = m \times c_{\text{gas}} \times \Delta T$         * $q_5 = 135\,g \times 1.86\,J/g^\circ\text{C} \times (120 - 100)^\circ\text{C}$     * Final Result:         * Summing all values yields a $Q_{\text{total}}$ of $416,000\,J$ or $416\,kJ$.

• Enthalpy Considerations:     * $\Delta H_{\text{vap}}$ for water ($40.67\,kJ/mol$) is specifically measured at the boiling point ($100^\circ\text{C}$). Standard vaporization values at different temperatures may vary slightly but typically don't significantly alter the end result for these types of problems.

Phase Diagrams and Phase Equilibrium

• Core Concepts:     * Phase Diagram: A plot of Pressure vs. Temperature that maps out the physical states of a substance under specific conditions.     * Phase Boundaries (Solid Lines): Represent conditions where two phases exist in equilibrium (e.g., melting point/freezing point, vaporization/condensation, sublimation/deposition).     * Temperature during Phase Changes: Remains constant while energy is used to overcome or form intermolecular forces.

• Specific Points and Regions:     * Triple Point: The specific temperature and pressure where solid, liquid, and gas phases all coexist in thermodynamic equilibrium.     * Critical Point: The end point of the liquid-gas boundary line. Beyond this point, the phases merge into a supercritical fluid.     * Solids Region: Typically found at lower temperatures and varied pressures.     * Liquids Region: Typically found at higher temperatures and higher pressures.     * Gases Region: Typically found at higher temperatures and lower pressures.

• Water’s Anomalous Phase Diagram:     * Most substances have a positive slope for the solid-liquid boundary (higher pressure = higher melting point).     * Water has a negative slope: the melting point decreases as pressure increases.     * Real-world application: This allows glaciers to move, as the high pressure at the base of the glacier creates a thin layer of liquid water that acts as a lubricant.

• Freeze-Drying (Lyophilization):     * Process involves freezing the substance (e.g., food) to make water crystalline, then decreasing pressure via vacuum.     * Low pressure forces the water to sublime directly from solid to gas without becoming liquid.     * Retains about $97\%$ of nutritional value and allows for a 25-year shelf life if stored away from oxygen and moisture (e.g., in Mylar bags or glass jars).

Supercritical Fluids

• Definition: A state of matter that exists beyond the critical temperature ($T_c$) and critical pressure ($P_c$) where the distinction between liquid and gas disappears.

• Properties:     * They expand to fill their container like a gas.     * They possess a higher density than a gas (approximately intermediate between liquid and gas).     * They have no surface tension and very low viscosity, allowing them to flow easily through solids.

• Scientific and Industrial Use:     * Decaffeination: Supercritical $CO_2$ is used as a solvent to extract caffeine from coffee beans. This replaced the use of methylene chloride, which is toxic.

• Critical Temperature Example ($CO_2$):     * On a cool day ($18^\circ\text{C}$, which is below the critical point), liquid $CO_2$ can be heard in a fire extinguisher.     * On a hot day ($35^\circ\text{C}$, which is above the critical point), no liquid exists regardless of pressure; the substance is either a gas or a supercritical fluid.

The Solid State of Matter

• Classification by Order:     * Crystalline Solids: Atoms, molecules, or ions arranged in a definite, repeating pattern (lattice). They exhibit precise melting points because all intermolecular forces are of equal strength.     * Amorphous Solids (Glass): Non-crystalline solids where particles are disordered (chaos). They have a distribution of attractive forces, leading to a range of melting temperatures rather than a single point. Examples: glass, candle wax, jello.

• Types of Crystalline Solids:     * Ionic Solids: Held by strong electrostatic attractions between positive and negative ions. High melting points, hard, brittle, non-conductive as solids but conductive when melted or dissolved (as ions become mobile).     * Metallic Solids: Formed by metal atoms with nuclei in a "sea of delocalized electrons." Exhibit metallic luster, are malleable (hammerable), and ductile (can be drawn into wires). Bonding strength varies (e.g., $Hg$ is liquid at RT; $Ga$ has a low melting point).     * Covalent Network Solids: Atoms held together by a continuous network of covalent bonds. Extremely hard with very high melting points. Examples: Diamond (melts above $3500^\circ\text{C}$, 10 on the Mohs scale), Silicon, Silicon Dioxide ($SiO_2$ or quartz), and Silicon Carbide ($SiC$ or carborundum used in sandpaper).     * Molecular Solids: Composed of neutral molecules held together by intermolecular forces (Van der Waals). Properties depend on size and polarity.         * Example: $CO_2$ (small, nonpolar) has a melting point of $-78^\circ\text{C}$. $I_2$ (larger, nonpolar) has a higher melting point of $114^\circ\text{C}$ due to stronger London Dispersion Forces ($LDF$).

• Carbon Allotropes:     * Graphite: Layers of carbon arranged in hexagons. Held together by weak $LDF$, allowing layers to rub off (pencil lead). A single layer is called graphene.     * Carbon Nanotubes: Graphene sheets rolled into a tube.     * Buckyballs (Buckminsterfullerene): $C_{60}$, 60 carbon atoms in a spherical soccer-ball shape.

Crystal Lattice Structures and Unit Cells

• Unit Cell: The simplest repeating unit of a crystalline solid.     * Lattice Points: Represent the locations of atoms or ions.     * Coordination Number: The number of neighbor particles a single particle contacts.

• Major Metallic Unit Cells:     * Simple Cubic Structure (SCS):         * Atoms at 8 corners only. Each corner is $\frac{1}{8}$ of an atom.         * Total atoms: $1\text{ atom per cell}$.         * Coordination number: $6$.         * Packing efficiency: $\approx 52\%$         * Example: Polonium ($Po$).         * Edge length ($l$): $2 \times r$.     * Body-Centered Cubic (BCC):         * Atoms at 8 corners plus one full atom in the center.         * Total atoms: $2\text{ atoms per cell}$.         * Coordination number: $8$.         * Packing efficiency: $\approx 68\%$         * Examples: $K, Ba, Cr, Mo, W, Fe$.     * Face-Centered Cubic (FCC) / Cubic Closest Packing (CCP):         * Atoms at 8 corners and the centers of all 6 faces ($6 \times \frac{1}{2} = 3$ atoms).         * Total atoms: $4\text{ atoms per cell}$.         * Coordination number: $12$.         * Packing efficiency: $\approx 74\%$         * Examples: $Al, Cu, Pb$.         * Formula for diagonal: $4 \times r = \sqrt{2} \times a$.

• Calculation Example (Density of Polonium):     1. Radius ($r$): $1/2 \times \text{edge length}$.     2. Mass: Find mass of one atom using Molar Mass and Avogadro's number ($6.022 \times 10^{23}$).     3. Volume: $l^3$.     4. Density: $\text{Mass} / \text{Volume}$. For Polonium ($l = 336\,pm$), result is $9.16\,g/cm^3$.

Ionic Crystal Structures

• Arrangement: Cations and anions are usually different sizes. Typically, large anions form a closest-packed array, and smaller cations fit into "holes."

• Types of Holes:     * Tetrahedral Holes: Smaller, created between 4 anions. Up to 2 holes per anion.     * Octahedral Holes: Larger, created between 6 anions (3 in one layer, 3 in another). 1 hole per anion.     * Cubic Holes: Occur in simple cubic arrays.

• Stoichiometry Examples:     * Zinc Oxide: Zinc occupies half of tetrahedral holes in a CCP of sulfide. Since there are 2 tetrahedral holes per anion, $1/2 \times 2 = 1\text{ Zn per S}$. Formula: $ZnS$.     * Lithium Selenide: Lithium fills all tetrahedral holes. Formula: $Li_2Se$.     * Aluminum Oxide (Sapphire): Aluminum ions in $2/3$ of octahedral holes. Formula: $Al_2O_3$.

X-Ray Crystallography

• Bragg Equation: Used to determine the spacing ($d$) between layers of atoms based on X-ray diffraction patterns.     * $n \times \lambda = 2 \times d \times \sin(\theta)$     * $\lambda$ = wavelength of X-ray.     * $n$ = integer (order of diffraction).     * $\theta$ = angle of diffraction.     * Constructive interference (peaks lining up) creates high-intensity beams captured by a diffractometer.

• Calculation Example:     * Given: $n = 1$, $\lambda = 0.1315\,nm$, $\theta = 25.25^\circ$.     * $d = \frac{1 \times 0.1315}{2 \times \sin(25.25^\circ)} = 0.154\,nm$.

• Historical Note: Rosalind Franklin (along with Raymond Gosling) used X-ray diffraction to discover the double-helix structure of DNA (Form A and Form B), a discovery for which Watson and Crick later received the Nobel Prize.

Questions & Discussion

• Question: What phases are present at the triple point?

• Response: All of them. Solid, liquid, and gas coexist.

• Question: Is the triple point like Jell-O?

• Response: No, it is a specific thermodynamic equilibrium point.

• Question: What is in the bubbles of boiling water?

• Response: It's not air; it is gaseous water (water vapor).

• Discussion on Decaffeination: Discussed methylene chloride vs supercritical $CO_2$. California has outlawed methylene chloride.

• Anecdote on Metals/Jewelry: The instructor mentioned silver being malleable and soft, and warning a friend against an opal engagement ring because opal is too soft and brittle (roughly 5.5-6.5 on Mohs, vs 10 for Diamond).