Gas Laws II and Stoichiometry in Gas-Phase Reactions: Study Notes

Gas Laws II

Combined Gas Law

The combined gas law relates the pressure, volume, and temperature of a fixed amount of gas. It is useful for predicting the behavior of a gas when conditions change, assuming the number of moles remains constant.

• Equation: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

• Variables: $P$ = pressure, $V$ = volume, $T$ = temperature (in Kelvin)

• Application: Used to calculate the new volume of a gas when it is moved to a location with different pressure and temperature.

• Example: A 3.0-L helium balloon at 645 mm Hg and 30.7°C is taken to a summit at 509 mm Hg and 10°C. Find the new volume, assuming $n$ is constant.

Ideal Gas Law: Density and Molar Mass

The ideal gas law can be rearranged to solve for the density and molar mass of a gas. This is especially useful for identifying unknown gases and comparing their densities to air.

• Equation: $PV = nRT$

• Density Formula: $d = \frac{PM}{RT}$, where $d$ = density, $P$ = pressure, $M$ = molar mass, $R$ = gas constant, $T$ = temperature

• Molar Mass Calculation: $M = \frac{mRT}{PV}$, where $m$ = mass of gas

• Example: 5.59 g of an ideal gas in a 5.00-L flask at 1.15 atm and 350 K. Calculate the molar mass using the above formula.

Density of Gases at STP

Some gases are more dense than air at standard temperature and pressure (STP). The density depends on the molar mass and the conditions.

• Air at STP: Average molar mass ≈ 29 g/mol

• Gases more dense than air: Gases with molar mass greater than air (e.g., Radon (Rn))

• Example: Rn is more dense than air at STP; Ne, NH3, and N2 are less dense.

Radon Accumulation in Buildings

Radon is a radioactive noble gas that is denser than air. In multilevel houses, it tends to accumulate in lower areas due to its density.

• Greater concentration: In the basement

• Reason: Heavier gases settle in lower parts of buildings.

Dalton's Law of Partial Pressures

Dalton's Law

Dalton's Law states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of each individual gas.

• Equation: $P_{\text{total}} = P_1 + P_2 + P_3 + \ldots$

• Partial Pressure: $P_i = X_i P_{\text{total}}$, where $X_i$ is the mole fraction of gas $i$

• Mole Fraction: $X_i = \frac{n_i}{n_{\text{total}}}$

• Example: In a simulation with equal numbers of red and blue gas particles at the same temperature and volume, both exert the same pressure.

Partial Pressure and Mole Fraction (Simulation)

Simulations can illustrate how partial pressure depends on the number of particles (moles) of each gas in a mixture.

• Equal particles: Equal partial pressures

• Different particles: Partial pressure is proportional to the number of particles

• Example: 30 red and 10 blue particles, total pressure 48 mm Hg. Pressure from blue particles: $P_{\text{blue}} = \frac{10}{40} \times 48 = 12$ mm Hg

Stoichiometry in Gas-Phase Reactions

Volume Changes in Gas Reactions

When gases react, the total number of moles (and thus the volume at constant temperature and pressure) can change according to the stoichiometry of the reaction.

• Example: A balloon with 8 L of NO2(g); half reacts to form N2O4(g) at constant T and P. The final volume depends on the change in moles of gas.

Calculating Partial Pressures of Products

In reactions producing multiple gaseous products, the partial pressure of each can be calculated using stoichiometry and the ideal gas law.

• Example: TNT (C7H5N3O6) reacts with O2 to form CO2, H2O, and N2. If 5 kg TNT reacts at 300 K in a 10.0-L box, use stoichiometry to find moles of each product, then $P = \frac{nRT}{V}$ for each.

Limiting Reactant and Yield in Gas Reactions

When two gases react, the limiting reactant determines the amount of product formed. The total number of moles after reaction can be calculated, accounting for percent yield.

• Equation: $\text{O}_2(g) + 2\text{H}_2(g) \rightarrow 2\text{H}_2\text{O}(g)$

• Example: 0.40 mol H2 and 0.20 mol O2 added to a cylinder. If H2O yield is 80%, calculate total moles of gas at the end.

Calculating Final Pressure After Reaction

If the temperature changes after a reaction in a constant-volume container, the final pressure can be calculated using the ideal gas law.

• Equation: $P = \frac{nRT}{V}$

• Example: After reaction, with final temperature 225°C, use the total moles of gas and the new temperature to find the final pressure.

Additional info:

• STP refers to Standard Temperature and Pressure: 0°C (273.15 K) and 1 atm.

• Gas constant $R = 0.0821$ L·atm/(mol·K) for calculations in these units.

• Percent yield is used to account for incomplete reactions: $\text{Actual yield} = \text{Theoretical yield} \times \text{Percent yield}$