Gases and Vapors: Properties, Laws, and Calculations

Gases and Vapors

Introduction: Gas vs. Vapor

Understanding the distinction between gases and vapors is fundamental in chemistry and engineering. Both are in the gaseous state, but their definitions and behaviors differ based on standard conditions and phase changes.

• Gas: A state of matter where the substance exists in the gaseous phase under standard conditions (e.g., oxygen, nitrogen at room temperature).

• Vapor: The gaseous form of a substance that is typically a liquid or solid under standard conditions (e.g., water vapor, iodine vapor).

• Vapors are formed through evaporation (liquid to gas) or sublimation (solid to gas).

• Gas properties can be described using the Ideal Gas Law.

• Vapor behavior is governed by phase change and equilibrium conditions, and in closed systems, vapor contributes to the total pressure.

• Vapors often deviate from ideal gas behavior more than permanent gases.

Gas Laws

The Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and amount of gas in a system. It is a foundational equation for understanding gas behavior.

• Equation: $PV = nRT$

• Where:

  • $P$ = absolute pressure of the gas

  • $V$ = total volume of the gas

  • $n$ = number of moles of the gas

  • $R$ = ideal (universal) gas constant

  • $T$ = absolute temperature of the gas (in Kelvin)

• Values of R:

  • $0.08206 \; \frac{L \cdot atm}{mol \cdot K}$

  • $8.314 \; \frac{J}{mol \cdot K}$

  • $8.314 \; \frac{Pa \cdot m^3}{mol \cdot K}$

Other Gas Laws

• Boyle's Law: At constant temperature, the pressure and volume of a gas are inversely proportional. $P_1V_1 = P_2V_2$

• Charles's Law: At constant pressure, the volume of a gas is directly proportional to its absolute temperature. $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Gay-Lussac’s (Amonton’s) Law: At constant volume, the pressure of a gas is directly proportional to its absolute temperature. $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Combined Gas Law

The Combined Gas Law is used when a gas undergoes changes involving pressure, volume, and temperature, but the amount of gas remains constant.

• Equation: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

• Useful for comparing two states of a gas and eliminates the need for the gas constant $R$.

Properties of Gases

Standard Temperature and Pressure (STP)

• Standard Temperature: $T = 273 \; K = 0^\circ C = 32^\circ F = 492^\circ R$

• Standard Pressure: $P = 1 \; atm = 760 \; mmHg = 101325 \; Pa = 101.325 \; kPa = 1.01325 \; bar = 760 \; Torr = 14.7 \; psi$

Density of a Gas

• Equation: $\rho = \frac{PM}{RT}$

• Where $\rho$ is the density, $P$ is pressure, $M$ is molar mass, $R$ is the gas constant, and $T$ is temperature in Kelvin.

Specific Gravity of a Gas

• Equation: $SG = \frac{\text{density of gas}}{\text{density of air (reference gas)}}$

Pressure Relationships

• Absolute Pressure: $P_{abs} = P_{atm} + P_{gauge}$

• Vacuum Pressure: $P_{vac} = P_{atm} - P_{abs}$

Molar Volume at STP

• At STP, 1 mole of an ideal gas occupies:

  • $22.4 \; L$

  • $22,400 \; cm^3$

  • $0.0224 \; m^3$

  • $359 \; ft^3$ (per lb-mol)

• General Equation for Gas at Non-Standard Conditions: $\frac{P_G V_G}{T_G} = \frac{n V_m P_S}{T_S}$

• Where $V_s = nV_m$ is the volume occupied by gas at STP.

Sample Problems

Sample Problem 1

Calculate the volume (in $cm^3$) occupied by 2 kg-mol $N_2$ at $40^\circ F$ and $100 \; kPa$.

Sample Problem 2

Butane gas is stored in a cylinder at a pressure of 120 psig and at a temperature of $27^\circ C$. Some of the butane gas was used and after some time, the pressure had gone down to 55 psig. If the temperature is $27^\circ C$, what fraction of butane had been used?

Sample Problem 3

A tank contains $N_2$ at 8 atm and $33^\circ C$. When 2.2 kg of the gas is removed, the pressure goes down to 7 atm and the temperature to $28^\circ C$. What is the volume of the tank?

Sample Problem 4

The gas acetylene is produced according to the following reaction by treating calcium carbide with water:

$CaC_2 + H_2O \rightarrow C_2H_2 + Ca(OH)_2$

Calculate the number of hours of service that can be derived from 3.5 kg of carbide in an acetylene lamp burning $0.08 \; m^3$ of gas per hour at a temperature of $26^\circ C$ and a pressure of 745 torr.

Additional info:

• These sample problems require application of the gas laws, unit conversions, and stoichiometry for chemical reactions involving gases.