Ideal Gas Law Derivations and Applications

Gases

Ideal Gas Law Derivations

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and amount of a gas. By rearranging the Ideal Gas Law, we can derive new equations to solve for unknown variables when a gas sample undergoes changes in state.

• Key Variables: Pressure (P), Volume (V), Temperature (T), and Amount (n, in moles)

• These derivations are required when we have variables at two sets of different states (initial and final conditions).

Ideal Gas Law Equation

The Ideal Gas Law is expressed as:

$PV = nRT$

• P = Pressure (in atm, or other units as specified)

• V = Volume (in liters, L)

• n = Amount of gas (in moles)

• R = Ideal gas constant ($0.0821\ \text{L} \cdot \text{atm} / \text{mol} \cdot \text{K}$)

• T = Temperature (in Kelvin, K)

Stepwise Approach to Ideal Gas Law Problems

1. Write out the Ideal Gas Law formula: $PV = nRT$

2. Circle the variables in the Ideal Gas Law formula that have two sets of different values (initial and final states).

3. Cross out the variables in the Ideal Gas Law formula that are not discussed or are remaining the same.

4. Since R is constant and the value is the same, it can always be ignored in ratio problems.

5. Algebraically move the circled variables to the left side of the Ideal Gas Law formula.

6. Make sure all variables are in the correct units (e.g., temperature in Kelvin, volume in liters, pressure in atm or as specified).

Additional info: Temperature must always be converted to Kelvin by adding 273.15 to the Celsius value.

Example Problem

Example: A sample of sulfur hexachloride gas occupies 5.30 L at 202 °C. Assuming that the pressure remains constant, what temperature (in °C) is needed to decrease the volume to 6.25 L?

• Step 1: Write the combined gas law for constant pressure: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• 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 1: A sample of nitrogen dioxide gas at 120 °C and 315 torr occupies a volume of 500 mL. What will the gas pressure be if the volume is reduced to 325 mL at 120 °C? Solution outline: Use Boyle's Law ($P_1V_1 = P_2V_2$) since temperature and amount of gas are constant.

• Practice 2: A cylinder with a movable piston contains 0.611 moles of gas and has a volume of 295 mL. What will its volume be if 0.103 moles of gas escape? Solution outline: Use Avogadro's Law ($\frac{V_1}{n_1} = \frac{V_2}{n_2}$) since pressure and temperature are constant.

• Practice 3: 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 the new pressure (in torr) in the can be? Solution outline: Use Gay-Lussac's Law ($\frac{P_1}{T_1} = \frac{P_2}{T_2}$) since volume and amount of gas are constant.

Summary Table: Gas Law Relationships

Law	Variables Held Constant	Equation	Relationship
Boyle's Law	n, T	$P_1V_1 = P_2V_2$	Pressure and volume are inversely proportional
Charles's Law	n, P	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	Volume and temperature are directly proportional
Gay-Lussac's Law	n, V	$\frac{P_1}{T_1} = \frac{P_2}{T_2}$	Pressure and temperature are directly proportional
Avogadro's Law	P, T	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	Volume and amount of gas are directly proportional

Additional info: These relationships are special cases of the Ideal Gas Law when certain variables are held constant.