Gas Laws and Ideal Gases

Temperature, Energy, and Moles

• Kelvin is related to the kinetic energy of a molecule; temperature measures molecular speed and energy.
• Moles quantify the amount of a substance.
• Combined Gas Law will be discussed, where subscripts 1 and 2 denote initial and final states.
  • State 1: Initial conditions are recorded.
  • State 2: Final conditions are recorded, but one value is unknown, which needs to be solved.

Basic Properties of Gases

• The discussion will cover gas properties and their relationships, relating to scenarios like low tire pressure.

Pressure Measurement

• Pressure is the quantity of stuff hitting the walls of a container, measured by barometers and manometers.
• More molecular motion means more pressure.

Molar Mass

• Molar mass of molecular nitrogen (N₂) is $28$ grams per mole.

• Molar mass of water is $18$ grams per mole.

• A drop in barometric pressure indicates a higher chance of rain due to increased water content in the air.
  

  Evangelista Torricelli (early 1600s-1700s):

• Units for pulse measurement (e.g., $120/80$) are in millimeters of mercury.

• Torricelli measured pressure by inverting a tube of mercury in a vat of mercury.

• At sea level, the mercury column was $760$ millimeters high.

• $1 \, \text{Torr} = 1 \, \text{mm Hg}$, (named to honor Torricelli).

Manometers

• Measure the difference between gas pressure and atmospheric pressure.
• $P<em>{\text{gas}} = P</em>{\text{atm}} + h$, where $h$ is the height difference in the mercury column.

Units of Pressure

• One atmosphere (atm) is the pressure at sea level, equivalent to $760$ mm Hg.
• Pascal (Pa) is the SI unit of pressure: $1 \, \text{N/m}^2 = 1 \, \text{Pa}$.
• Key units to know: millimeters of mercury, atmospheres, and torr.

Pressure Conversion Example

• Convert $785$ mm Hg to atmospheres using dimensional analysis.
• $785 \, \text{mm Hg} \times \frac{1 \, \text{atm}}{760 \, \text{mm Hg}} = 1.03 \, \text{atm}$

Dimensional Analysis

• Essential for stoichiometry; involves using equalities to convert units.
• Example: periodic table provides mole-mass relationships, which are equalities.

Gas Laws

• Summarize observations mathematically.
• Boyle's Law: Pressure is inversely proportional to volume (at constant temperature and moles).
  • Mathematical Representation: $P \propto \frac{1}{V}$
  • If Pressure is proportional to $\frac{1}{V}$, then: $P \times V = k$ ($k$ is a constant)
  • For different states: $P<em>1V</em>1 = P<em>2V</em>2$
• Example: If pressure doubles, volume halves.

Boyle's Law Application

• A balloon with $2.5$ liters of helium at sea level is moved to Mount Everest, where pressure is $270$ torr.
• Convert pressure to atmospheres.
  

$P<em>1V</em>1 = P<em>2V</em>2$

Steps to Solve Gas Problems

1. Write down knowns (e.g., $V<em>1, P</em>1, P_2$).
2. Convert units if necessary.
3. Apply gas law (e.g., Boyle's Law).
4. Solve for the unknown volume.
5. Verify units make sense and math aligns with the scenario.

Charles's Law

• Volume and temperature are directly proportional (at constant pressure and moles).
  • Mathematical Representation: $V \propto T$
  • $V/T = k$ (a constant)
  • At different states: $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$
• Increasing temperature increases volume; this is used in hot air balloons.
• Always use Kelvin for temperature in gas law calculations.

Charles's Law Application

• A balloon is cooled from $44°C$ to $21°C$, with initial volume $3.2$ liters.
  

$V<em>1 = 3.2 \, \text{L}$$T</em>1 = 44°C + 273 = 317 \, \text{K}$

$T_2 = 21°C + 273 = 294 \, \text{K}$

• Solve for the final volume using Charles's Law.

Avogadro's Law

• Volume is proportional to the number of moles (at constant temperature and pressure).
• $V \propto n$, where $n$ is the number of moles.
• $\frac{V<em>1}{n</em>1} = \frac{V<em>2}{n</em>2}$

Combined Gas Law

• Combines Boyle's, Charles's, and Avogadro's Laws.
• $\frac{P<em>1V</em>1}{T<em>1} = \frac{P</em>2V<em>2}{T</em>2}$
• Example application: propane tanks use this principle.

Combined Gas Law Application

• $78$ liters of gas at $927$ Torr and $35°C$ is heated to $115°C$ at constant volume.
  

$V<em>1 = 78 \, \text{L}$$P</em>1 = 927 \, \text{Torr}$

$T<em>1 = 35°C + 273 = 308 \, \text{K}$$T</em>2 = 115°C + 273 = 388 \, \text{K}$

• Since volume is constant, $V<em>1$ and $V</em>2$ cancel out in the equation.
• Solve for $P_2$.

Practice Problem

• Initial conditions: $V<em>1 = 7.85 \, \text{L}$, $P</em>1 = P<em>2$ (constant pressure), $T</em>1 = 65°C$.
• Final conditions: $V_2 = 5.28 \, \text{L}$.
• Find $T_2$.
• Conversion: $0°C = 273 \, \text{K}$

Ideal Gas Law

• $PV = nRT$, where:
  • $P$ is pressure.
  • $V$ is volume.
  • $n$ is the number of moles.
  • $R$ is the ideal gas constant.
  • $T$ is temperature in Kelvin.

Ideal Gas Law Constant

• $R = 0.0821 \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}}$
• Useful for many applications, like propane tank pressure calculations.

Ideal Gas Law Application

• $2.8$ moles of dimetrotin monoxide in a $20$ liter tank at $23°C$.
• Calculate the pressure in the tank.
• Convert Celsius to Kelvin: $T = 23 + 273 = 296 \, \text{K}$.
• Rearrange $PV = nRT$ to solve for $P$.

Volume Calculation

• Given $5$ grams of neon at $256$ mm Hg and $35°C$, calculate the volume.
• Convert grams to moles and pressure to atmospheres.
• Use $PV = nRT$ to solve for $V$.

Units and Problem Solving

• It's important for all the units to be the same, and if you aren't given the pressure or ther temperature you have to be told to use standard temperature and pressue or that they are constant.
• Dimensional analysis is critical for solving problems; track units to ensure they cancel out correctly.
• It is important to see the flow of units to make sure they cross off appropriately .

Standard Temperature and Pressure (STP)

• STP conditions: $1$ atm pressure and $273.15 \, \text{K}$ ($0°C$).
• One mole of any gas occupies $22.4$ liters at STP.
• STP is a standard for comparison; real-world conditions may vary due to altitude and temperature differences.

Ideal Gas Law and STP

• At STP, $1$ mole of gas equals $22.4$ liters.
• Using $PV = nRT$ and knowing that R equals constant, all the constants can be dervied at STP.

Using STP for Calculations

• If the STP volume of a gas is $4.56$ liters, calculate the volume at $1.86$ atm and $-35°C$.
• Given the conditions are at STP there are certain numbers to use.
• Use the combined gas law to solve for the unknown volume.

Conditions with Acronyms

• When given something with an acronym like STP it has set contitions allowing you to use them during calculation, and allows you to carry on with the problem.

Molar Mass and Density

• Molar mass (M) can be calculated using the ideal gas law and density.
• Equation: $M = \frac{m}{n}$, where $m$ is mass and $n$ is the number of moles.
  

$\text{PV} = \frac{m}{M} \text{RT}$

$\text{M} = \frac{m \text{RT}}{\text{PV}}$

Density of the gas used to find molar mass: density $\rho = \frac{m}{V}$

$\text{M} = \frac{\rho \text{RT}}{\text{P}}$

Calculating Gas Density

• Calculate the density of ammonia at $100°C$ and $1.15$ atm.