Gas Laws and the Ideal Gas Law: Applications and Problem Solving

Gas Laws and the Ideal Gas Law

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 a gas. It is widely used to solve problems involving gases under various conditions.

• Formula: $PV = nRT$

• P = Pressure (usually in atm or torr)

• V = Volume (usually in liters, L)

• n = Amount of gas (in moles)

• R = Universal gas constant ($0.08206$ L·atm/(mol·K))

• T = Temperature (in Kelvin, K)

Note: Always convert temperature to Kelvin by adding 273.15 to the Celsius value.

Manipulating the Ideal Gas Law

When a gas undergoes a change in conditions (such as pressure, volume, or temperature), the Ideal Gas Law can be rearranged to compare two different states:

• Combined Gas Law: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

• This equation is used when the amount of gas (n) is constant.

Stepwise Problem-Solving Approach

1. Write out the Ideal Gas Law formula.

2. Circle the variables that have two sets of different values (e.g., initial and final conditions).

3. Cross out the variables that remain constant or are not discussed.

4. Algebraically move the circled variables to one side to isolate the unknown.

5. Substitute the known values and solve for the unknown.

Example Problem

Example: A sample of sulfur hexachloride gas occupies 8.30 L at 202 °C. Assuming that the pressure remains constant, what temperature (in °C) is needed to decrease the volume to 5.25 L?

• Step 1: Use the combined gas law since pressure and amount of gas are constant:

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Step 2: Convert temperatures to Kelvin: $T_1 = 202 + 273.15 = 475.15$ K

• Step 3: Rearrange to solve for $T_2$:

$T_2 = \frac{V_2 \times T_1}{V_1}$

• Step 4: Substitute values and solve.

Practice Problems

Practice Problem 1

Problem: A sample of nitrogen dioxide gas at 130 °C and 315 torr occupies a volume of 500 mL. What will the gas pressure be if the volume is reduced to 320 mL at 130 °C?

• Key Concept: Use Boyle's Law ($P_1 V_1 = P_2 V_2$) since temperature and amount of gas are constant.

• Solution Steps:

  1. Identify initial and final conditions: $P_1 = 315$ torr, $V_1 = 500$ mL, $V_2 = 320$ mL

  2. Rearrange: $P_2 = \frac{P_1 V_1}{V_2}$

  3. Substitute and solve.

Practice Problem 2

Problem: A cylinder with a movable piston contains 0.615 moles of gas and has a volume of 295 mL. What will its volume be if 0.103 moles of gas escaped?

• Key Concept: Use Avogadro's Law ($\frac{V_1}{n_1} = \frac{V_2}{n_2}$) if pressure and temperature are constant.

• Solution Steps:

  1. Initial moles: $n_1 = 0.615$ mol, final moles: $n_2 = 0.615 - 0.103 = 0.512$ mol

  2. Rearrange: $V_2 = \frac{n_2 V_1}{n_1}$

  3. Substitute and solve.

Practice Problem 3

Problem: On most spray cans it is advised to never expose them to fire. A spray can is used until all that remains is the propellant gas, which has a pressure of 1350 torr at 25 °C. If the can is then thrown into a fire at 455 °C, what will be the pressure (in torr) in the can?

• Key Concept: Use Gay-Lussac's Law ($\frac{P_1}{T_1} = \frac{P_2}{T_2}$) since volume and amount of gas are constant.

• Solution Steps:

  1. Convert temperatures to Kelvin: $T_1 = 25 + 273.15 = 298.15$ K, $T_2 = 455 + 273.15 = 728.15$ K

  2. Rearrange: $P_2 = \frac{P_1 T_2}{T_1}$

  3. Substitute and solve.

Summary Table: Gas Laws

Law	Formula	Variables Held Constant	Application
Boyle's Law	$P_1 V_1 = P_2 V_2$	n, T	Pressure-Volume relationship
Charles's Law	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	n, P	Volume-Temperature relationship
Gay-Lussac's Law	$\frac{P_1}{T_1} = \frac{P_2}{T_2}$	n, V	Pressure-Temperature relationship
Avogadro's Law	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	P, T	Volume-Mole relationship
Ideal Gas Law	$PV = nRT$	None	General gas behavior

Additional info: These problems and explanations are foundational for GOB Chemistry students, especially for understanding how gases behave under changing conditions and how to apply the gas laws to real-world and laboratory scenarios.