Gas Laws: Boyle's, Charles', and Combined Gas Law – Study Notes

Gas Laws

Introduction

Gas laws describe the relationships between pressure, volume, and temperature of gases. Understanding these laws is fundamental in General Chemistry, as they allow prediction and calculation of gas behavior under varying conditions.

Pressure Unit Conversions

• Atmosphere (atm): Standard unit of pressure.

• Torr: 1 atm = 760 torr.

• Kilopascal (kPa): 1 atm = 101.325 kPa.

• Conversion Example: To convert 648 torr to atm:

$\text{atm} = \frac{648\ \text{torr}}{760\ \text{torr/atm}} = 0.853\ \text{atm}$

Boyle's Law

Boyle's Law states that the volume of a fixed amount of gas is inversely proportional to its pressure at constant temperature.

• Formula: $P_1V_1 = P_2V_2$

• Key Points:

  • If pressure increases, volume decreases (and vice versa), provided temperature is constant.

  • Used to calculate new volume or pressure after a change.

• Example: A 3.50 L sample of hydrogen gas has a pressure of 0.950 atm. What is its volume if the pressure is increased to 1.50 atm at constant temperature?

$P_1 = 0.950\ \text{atm}$ $V_1 = 3.50\ \text{L}$ $P_2 = 1.50\ \text{atm}$ $V_2 = ?$ $V_2 = \frac{P_1V_1}{P_2} = \frac{(0.950)(3.50)}{1.50} = 2.22\ \text{L}$

Charles' Law

Charles' Law states that the volume of a fixed amount of gas is directly proportional to its absolute temperature (in Kelvin) at constant pressure.

• Formula: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Key Points:

  • Temperature must be in Kelvin.

  • If temperature increases, volume increases (and vice versa), provided pressure is constant.

• Example: A 0.684 L sample of helium gas, initially at 0°C (273 K), is heated at constant pressure. If the final volume is 0.986 L, what is its final temperature in °C?

$V_1 = 0.684\ \text{L}$ $T_1 = 273\ \text{K}$ $V_2 = 0.986\ \text{L}$ $T_2 = ?$ $T_2 = \frac{V_2 \times T_1}{V_1} = \frac{0.986 \times 273}{0.684} = 393\ \text{K}$ $T_2$ in °C: $393 - 273 = 120\ ^\circ\text{C}$

Combined Gas Law

The Combined Gas Law combines Boyle's and Charles' Laws to relate pressure, volume, and temperature of a fixed amount of gas.

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

• Key Points:

  • Used when all three variables (pressure, volume, temperature) change.

  • Temperature must be in Kelvin.

• Example: Calculate the volume of 0.896 mol hydrogen gas that originally occupies 908 mL at 717 torr and 20°C. After its temperature and pressure are changed to 20°C and 1.07 atm, respectively, what is the new volume?

$P_1 = 717\ \text{torr} = 0.943\ \text{atm}$ $V_1 = 908\ \text{mL}$ $T_1 = 293\ \text{K}$ $P_2 = 1.07\ \text{atm}$ $T_2 = 293\ \text{K}$ $V_2 = ?$ $V_2 = \frac{P_1V_1T_2}{P_2T_1} = \frac{(0.943)(908)(293)}{(1.07)(293)} = 801\ \text{mL}$

Summary Table: Gas Law Relationships

Law	Relationship	Formula	Variables Held Constant
Boyle's Law	Pressure & Volume (inverse)	$P_1V_1 = P_2V_2$	Temperature, Amount of Gas
Charles' Law	Volume & Temperature (direct)	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	Pressure, Amount of Gas
Combined Gas Law	Pressure, Volume, Temperature	$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$	Amount of Gas

Additional info:

• Always convert temperatures to Kelvin for gas law calculations: $K = ^\circ\text{C} + 273$

• Standard Temperature and Pressure (STP): 0°C (273 K) and 1 atm.

• These laws assume ideal gas behavior, which is a good approximation under many conditions but may deviate at high pressures or low temperatures.