Comprehensive Study Notes on Stoichiometry, Gases, Physical Properties, and Solutions

Nomenclature of Chemical Hydrates and Compounds

Naming hydrates requires identifying the anhydrous salt followed by a prefix indicating the number of water molecules attached to the crystal lattice. For $\text{Ba}(\text{OH})_2 \cdot 8\text{H}_2\text{O}$, the name is Barium Hydroxide Octahydrate. For $\text{CoCl}_2 \cdot 6\text{H}_2\text{O}$, the name is Cobalt (II) Chloride hexahydrate.

To write the chemical formula from a name, the charges of the ions must be balanced, and the water of hydration is appended with a dot. Iron (III) phosphate tetrahydrate is written as $\text{FePO}_4 \cdot 4\text{H}_2\text{O}$. Sodium carbonate decahydrate is written as $\text{Na}_2\text{CO}_3 \cdot 10\text{H}_2\text{O}$.

Stoichiometry and Mole-Mass Conversions

Calculating the mass of a substance from a large quantity of formula units involves using Avogadro\u2019s number, which is $6.02 \times 10^{23}\text{ formula units/mol}$. In a scenario where one counts $4.87 \times 10^{21}$ formula units of aluminum oxide ($\text{Al}_2\text{O}_3$), the molar amount is calculated as follows:

$\text{Moles} = \frac{4.87 \times 10^{21}}{6.02 \times 10^{23}} = 0.0809\text{ moles}$

The molar mass of aluminum oxide is calculated using the atomic masses of Aluminum ($26.98\text{ g/mol}$) and Oxygen ($16.03\text{ g/mol}$ or $16.00\text{ g/mol}$), resulting in a molar mass of approximately $101.96\text{ g/mol}$. The final mass is:

$\text{Mass} = 0.0809\text{ moles} \times 101.96\text{ g/mol} = 8.25\text{ g of Aluminum Oxide}$

In gas stoichiometry involving iron and water, the unbalanced equation is $3\text{Fe} + 4\text{H}_2\text{O} \rightarrow \text{Fe}_3\text{O}_4 + 4\text{H}_2$. If $2.12\text{ moles}$ of iron react with excess water, the moles of hydrogen gas produced are determined by the molar ratio ($4:3$):

$\text{Moles of H}_2 = 2.12\text{ mol Fe} \times \frac{4\text{ mol H}_2}{3\text{ mol Fe}} = 2.827\text{ mol H}_2$

At Standard Temperature and Pressure (STP), one mole of any gas occupies $22.4\text{ L}$. Therefore, the volume of hydrogen gas produced is:

$\text{Volume} = 2.827\text{ mol} \times 22.4\text{ L/mol} = 63.32\text{ L of H}_2$

Percent Yield in Chemical Reactions

Percent yield is the ratio of the experimentally determined yield to the theoretical yield, expressed as a percentage. In a reaction where Iron reacts with Oxygen to form Iron (III) Oxide ($4\text{Fe} + 3\text{O}_2 \rightarrow 2\text{Fe}_2\text{O}_3$), given $1.46 \times 10^{24}$ atoms of Iron:

$\text{Moles of Fe} = \frac{1.46 \times 10^{24}}{6.02 \times 10^{23}} = 2.4252\text{ mol Fe}$

Using the stoichiometric ratio ($2:4$ or $1:2$), the theoretical moles of $\text{Fe}_2\text{O}_3$ are $1.2126\text{ mol}$. Given the molar mass of $\text{Fe}_2\text{O}_3$ is approximately $159.7\text{ g/mol}$, the theoretical mass is:

$\text{Theoretical Mass} = 1.2126\text{ mol} \times 159.7\text{ g/mol} = 193.65\text{ g}$

If the experimental yield is $179\text{ g}$, the percent yield is:

$\text{Percent Yield} = \frac{179\text{ g}}{193.65\text{ g}} \times 100 = 92.4\%$

Empirical and Molecular Formula Determination

To determine the formula of a hydrate, such as magnesium carbonate, the mass of water lost during heating must be calculated. If a $15.67\text{ g}$ sample is heated and $7.58\text{ g}$ of anhydrous magnesium carbonate remains, the mass of water lost is $8.09\text{ g}$. The molar ratio is found by dividing the moles of water ($8.09 / 18.02 = 0.4489\text{ mol}$) by the moles of the anhydrous salt ($7.58 / 84.32 = 0.08989\text{ mol}$). The ratio is approximately $5$, giving the formula $\text{MgCO}_3 \cdot 5\text{H}_2\text{O}$ (Magnesium carbonate pentahydrate).

For an unknown compound with $17.56\%\text{ Sodium}$, $39.69\%\text{ Chromium}$, and $42.75\%\text{ Oxygen}$, the empirical formula is found by calculating the moles of each element in a $100\text{ g}$ sample:

$\text{Na}: \frac{17.56\text{ g}}{22.99\text{ g/mol}} = 0.7638\text{ mol}$$\text{Cr}: \frac{39.69\text{ g}}{52.00\text{ g/mol}} = 0.7633\text{ mol}$$\text{O}: \frac{42.75\text{ g}}{16.00\text{ g/mol}} = 2.6718\text{ mol}$

Dividing by the smallest value ($0.7633$) gives a ratio of $1:1:3.5$. Multiplying by $2$ yields the empirical formula $\text{Na}_2\text{Cr}_2\text{O}_7$. If the total molar mass is found to be $523.94\text{ g/mol}$ and the empirical mass is $139\text{ g/mol}$, the multiplier is approximately $4$, resulting in the molecular formula $\text{Na}_4\text{Cr}_4\text{O}_{16}$.

Limiting Reagents and Excess Reactants

In the reaction $4\text{CH}_4 + \text{S}_8 \rightarrow 4\text{CS}_2 + 8\text{H}_2\text{S}$, the limiting reagent is determined by comparing the moles of product each reactant can produce. Given $81\text{ L}$ of methane and $398\text{ g}$ of sulfur, methane produces less product and is the limiting reagent, while sulfur remains in excess.

In another reaction, $\text{S}_8 + 4\text{Cl}_2 \rightarrow 4\text{S}_2\text{Cl}_2$, with $300.0\text{ g}$ of sulfur and $510.0\text{ g}$ of product, the determination shows that Chlorine gas is the limiting reagent. For $\text{S}_8$, the calculation is as follows:

$\text{Moles S}_8 = \frac{300\text{ g}}{256.56\text{ g/mol}} = 1.1693\text{ mol}$

Applying stoichiometric ratios to find the excess reagent left over, the calculation reveals $0.43\text{ mol}$ of $\text{S}_8$ remains.

Physical Properties: Vapor Pressure and Boiling Points

Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system. It varies with temperature because temperature represents average kinetic energy; the faster particles move, the more molecules have enough energy to break free of Intermolecular Forces (IMFs) to become a gas.

A substance boils when its vapor pressure equals atmospheric pressure. In boiling point studies of liquids like $\text{CHCl}_3$, $\text{CCl}_4$, and $\text{H}_2\text{O}$, the liquid with the strongest IMFs is Water ($\text{H}_2\text{O}$), as it requires higher temperatures to achieve the same vapor pressure. If pressure is reduced to $90\text{ kPa}$ at $60 \, ^\circ\text{C}$, $\text{CHCl}_3$ would boil. If pressure is further reduced to $50\text{ kPa}$, both $\text{CHCl}_3$ and $\text{CCl}_4$ would boil.

At high altitudes where atmospheric pressure is lower, water boils at a lower temperature. Conversely, a pressure cooker increases internal pressure, which increases the boiling point of the liquid, allowing food to cook faster at a higher temperature.

Molecular Properties and Intermolecular Forces

Electronegativity is defined as how strongly an atom pulls electrons to its nucleus and how much the shells pull these electrons away. For example, oxygen gas ($\text{O}_2$) is nonpolar, whereas water ($\text{H}_2\text{O}$) is a liquid at room temperature due to extreme electronegativity differences causing strong polar bonds and Intermolecular Forces.

Evaporation occurs because particles within a substance have varying kinetic energies. Some surface particles possess higher energy than others, allowing them to break the IMFs holding them together and escape into the gas phase. This process occurs even below the boiling point. At absolute zero ($0\text{ K}$), all particles have the same kinetic energy because there is no particle movement.

Gas Laws and Calculations

The behavior of gases is governed by relationships between pressure ($P$), volume ($V$), and temperature ($T$). Boyle\u2019s Law states that pressure and volume are inversely related ($P \propto 1/V$). Charles\u2019 Law states that volume and temperature are directly related ($V \propto T$). Pressure and temperature also have a direct relationship.

Temperature must always be in Kelvin for gas law calculations ($K = ^\circ\text{C} + 273.15$). For a gas heated from $27\,^\circ\text{C}$ ($300\text{ K}$) occupying $127\text{ mL}$ to a volume of $0.317\text{ L}$, the new temperature is calculated as:

$T_2 = \frac{V_2 \times T_1}{V_1} = \frac{0.317\text{ L} \times 300\text{ K}}{0.127\text{ L}} = 748.8\text{ K} \approx 476\,^\circ\text{C}$

A barometer uses a column of mercury to measure pressure. At the top of a mountain where atmospheric pressure is lower, the mercury column will be lower than the standard $760\text{ mmHg}$. To calculate new pressure in a syringe when volume is reduced to one-third, the pressure increases by a factor of three ($35.0\text{ kPa} \times 3 = 105\text{ kPa}$).

Solution Chemistry: Dissolution and Solubility

The solution process involves solvent molecules surrounding solute particles (solvating them), creating solvent cages that prevent the solute from reforming its original crystal lattice structure. The general rule is \"like dissolves like,\" meaning polar solvents dissolve polar solutes and nonpolar solvents dissolve nonpolar solutes.

Factors affecting the rate of dissolution include:

1. Surface Area: Larger surface area (smaller particles) increases the frequency of interactions with the solvent.
2. Heat/Temperature: Temperature is a measure of kinetic energy; faster-moving particles interact more frequently with fresh solvent.

Solubility curves describe the maximum amount of solute that can dissolve in a specific amount of solvent at a given temperature. An endothermic heat of solution (like $\text{KNO}_3$) means solubility increases with temperature. An exothermic heat of solution (like $\text{Ce}_2(\text{SO}_4)_3$) means solubility decreases as temperature increases. A solution containing more than the maximum amount of solute is \"supersaturated,\" while one containing the exact maximum is \"saturated.\"

Colligative Properties: Boiling Point and Freezing Point

Colligative properties depend on the concentration of solute particles, not their identity. Adding salt to roads in winter lowers the freezing point of water, making it harder for ice to form its solid lattice structure. $\text{CaCl}_2$ is superior to $\text{NaCl}$ as a road salt because it dissociates into three ions ($\text{Ca}^{2+}$ and $2\text{Cl}^-$) rather than two, providing a greater freezing point depression.

In cooking, adding salt to water elevates the boiling point. The salt ions interact with water molecules to become solvated, requiring more energy to break the ion-dipole bonds so the water can escape into the gas phase. This allows the food to cook at a higher temperature.

Molarity and Dilution Calculations

Molarity ($M$) is defined as moles of solute per liter of solution ($M = n/V$). To find the concentration of $24\text{ g}$ of $\text{NaNO}_3$ in $500\text{ mL}$ ($0.5\text{ L}$):

$\text{Moles} = \frac{24\text{ g}}{85.0\text{ g/mol}} = 0.28\text{ mol}$$\text{Molarity} = \frac{0.28\text{ mol}}{0.5\text{ L}} = 0.56\text{ M}$

Dilution calculations rely on the principle that the number of moles remains constant: $C_1V_1 = C_2V_2$. To prepare $5\text{ L}$ of $2\text{ M H}_2\text{SO}_4$ from an $18\text{ M}$ stock solution:

$V_{\text{stock}} = \frac{5\text{ L} \times 2 \text{ M}}{18\text{ M}} = 0.556\text{ L} \text{ (approximately } 0.56\text{ L or } 4.41\text{ L depending on specifics)}$

Questions & Discussion

Q: Why does a straw work?A: Drinking through a straw works due to pressure differences. Sucking creates lower pressure inside the straw. The higher atmospheric pressure pushing down on the liquid in the cup forces the liquid up the straw toward the lower pressure in your mouth. You cannot drink from a $40\text{ ft}$ straw because the weight of the water column becomes equal to the atmospheric pressure pushing it up, reaching a physical limit.

Q: What is the pressure in the National Microbiology Lab in Winnipeg?A: The lab is kept at a lower pressure than the external environment to ensure that if a leak occurs, air flows into the building rather than pathogens escaping out.

Q: How does a Coke bottling facility achieve carbonation?A: Carbonation is achieved by increasing the pressure of $\text{CO}_2$ gas over the liquid mixture of sugar and water. This increases the solubility of the gas. They often do this at cold temperatures to maximize solubility.