Ideal Gas Law and Gas Law Relationships – Study Notes

Chapter 5: Gases

Introduction to the Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and amount of an ideal gas. It is derived from the combination of three empirical gas laws: Boyle's Law, Charles's Law, and Avogadro's Law. Understanding these relationships is essential for predicting the behavior of gases under various conditions.

• Boyle's Law: At constant temperature and amount of gas, the volume of a gas is inversely proportional to its pressure. $V \propto \frac{1}{P}$

• Charles's Law: At constant pressure and amount of gas, the volume of a gas is directly proportional to its temperature (in Kelvin). $V \propto T$

• Avogadro's Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles. $V \propto n$

• Combined Gas Law: Combining the above laws gives: $V \propto \frac{nT}{P}$

The Ideal Gas Equation

The combined relationships can be expressed as the ideal gas equation, which is applicable to ideal gases:

• Ideal Gas Law: $PV = nRT$

  • P = pressure

  • V = volume

  • n = number of moles

  • R = universal gas constant

  • T = temperature (in Kelvin)

• Gas Constant (R): The value of R depends on the units used for pressure and volume. Common values include:

  R Value	Units
  0.08206	L·atm·mol-1·K-1
  0.08314	L·bar·mol-1·K-1
  8.314	J·mol-1·K-1

• Unit Consistency: Always ensure that units for pressure, volume, temperature, and R are compatible.

Standard Temperature and Pressure (STP)

STP is a reference condition commonly used in gas calculations. It is defined as:

• Temperature: 0°C = 273.15 K

• Pressure: 1.00 bar (sometimes 1 atm)

• Molar Volume at STP: The volume occupied by 1 mole of an ideal gas at STP is 22.7 L (using 1 bar).

Example Calculation: What is the volume of 1.00 mol of gas at STP?

• Use $V = \frac{nRT}{P}$

• Substitute: $n = 1.00$ mol, $R = 0.08314$ L·bar·mol-1·K-1, $T = 273.15$ K, $P = 1.00$ bar

• Result: $V = 22.7$ L

Applications and Problem Solving with the Ideal Gas Law

The ideal gas law can be used to solve a variety of problems involving changes in pressure, volume, temperature, or amount of gas. When some variables are held constant, the law can be rearranged to relate the changing variables.

• General Relationship for Changing Conditions: $\frac{P_1 V_1}{n_1 T_1} = \frac{P_2 V_2}{n_2 T_2}$ If n is constant: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

• Example: If a cylinder contains 50.0 L of oxygen at 18.5 bar and 21°C, what will the volume be if the pressure is reduced to 1.00 bar and temperature remains constant?

  • Use $P_1 V_1 = P_2 V_2$

  • $V_2 = \frac{P_1 V_1}{P_2} = \frac{(50.0\,L)(18.5\,bar)}{1.00\,bar} = 925\,L$

• Example: A 0.50 mol sample of oxygen gas at 0°C and 1.0 bar is compressed to half its initial volume and the pressure increases to 2.2 bar. What is the final temperature?

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

  • Solve for $T_2$

Density and Molar Mass of Gases

The ideal gas law can be rearranged to relate the density and molar mass of a gas. This is useful for identifying gases and calculating their properties.

• Density Equation: $d = \frac{PM}{RT}$

  • d = density (g/L)

  • P = pressure

  • M = molar mass

  • R = gas constant

  • T = temperature (K)

• Interpretation: The density of a gas increases with its molar mass and pressure, and decreases with temperature.

Stoichiometry and Gas Reactions

Gas law calculations can be applied to chemical reactions involving gases. The ideal gas law allows determination of the amount of gas required or produced in a reaction under specified conditions.

• Example Reaction: $4\ NH_3 (g) + 5\ O_2 (g) \rightarrow 4\ NO (g) + 6\ H_2O (g)$

• To find the volume of $NH_3$ required to react with $1.00$ mol $O_2$ at $T = 850°C$ and $P = 5.00$ bar:

  • Stoichiometry: $1.00$ mol $O_2 \times \frac{4\ mol\ NH_3}{5\ mol\ O_2} = 0.8$ mol $NH_3$

  • Use $V = \frac{nRT}{P}$ to find volume

Summary Table: Common Gas Law Relationships

Law	Relationship	Variables Held Constant
Boyle's Law	$P_1 V_1 = P_2 V_2$	n, T
Charles's Law	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	n, P
Avogadro's Law	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	P, T
Combined Gas Law	$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$	n
Ideal Gas Law	$PV = nRT$	None

Key Points to Remember

• Always convert temperatures to Kelvin for gas law calculations.

• Check units for pressure, volume, and R to ensure consistency.

• The ideal gas law is an approximation; real gases may deviate under high pressure or low temperature.

• Use stoichiometry to relate moles of gases in chemical reactions.

Additional info: The notes omit derivations and manometer details, focusing on practical applications and relationships for exam preparation.