Combined gas law Knowledge-Based Study

Study notes

1. Unifying the Simple Gas Laws

   In the study of ideal gases, empirical laws describe how a fixed amount of gas behaves when one state variable is kept constant:

   • Boyle's Law: Pressure is inversely proportional to volume at constant temperature ($P \propto 1/V$).
   • Charles's Law: Volume is directly proportional to absolute temperature at constant pressure ($V \propto T$).
   • Gay-Lussac's Law: Pressure is directly proportional to absolute temperature at constant volume ($P \propto T$).

   In many real-world and laboratory conditions, pressure, volume, and temperature change simultaneously. By combining these three relationships for a constant amount of gas (fixed number of moles, $n$), we obtain the Combined Gas Law.

   The Combined Equation

   The Combined Gas Law states that the ratio of the product of pressure and volume to the absolute temperature of a gas remains constant as long as the amount of gas does not change:

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

   Where:

   • $P_1$ and $P_2$: Initial and final pressures (must share the same units, such as atm, kPa, or mmHg).
   • $V_1$ and $V_2$: Initial and final volumes (must share the same units, such as L, mL, or $\text{m}^3$).
   • $T_1$ and $T_2$: Initial and final temperatures, which must always be expressed in Kelvin.

   Temperature Conversion

   Calculations involving gas laws require absolute temperature because the scale starts at absolute zero. Always convert temperatures from Celsius to Kelvin using:

   $$T(\text{K}) = T(^\circ\text{C}) + 273.15$$

   General Procedure

   To solve quantitative problems involving the Combined Gas Law, follow these steps:

   1. Identify the knowns and the unknown: List $P_1, V_1, T_1$ and $P_2, V_2, T_2$.
   2. Verify units: Convert all temperatures to Kelvin. Ensure pressure and volume units match between the initial and final states.
   3. Rearrange algebraically: Solve the equation for the target variable before substituting numerical values.
   4. Substitute and solve: Calculate the final answer and ensure appropriate significant figures and units.

   Illustrative Example

   A weather balloon has a volume of 50.0 L at a pressure of 1.00 atm and a temperature of 27.0 °C. What is its volume when it ascends to an altitude where the pressure is 0.400 atm and the temperature is -23.0 °C?

   Step 1: Convert temperatures to Kelvin
   $T_1 = 27.0 + 273.15 = 300.15\text{ K}$
   $T_2 = -23.0 + 273.15 = 250.15\text{ K}$

   Step 2: Rearrange the formula for $V_2$
   $$V_2 = \frac{P_1 V_1 T_2}{P_2 T_1}$$

   Step 3: Calculate
   $$V_2 = \frac{(1.00\text{ atm})(50.0\text{ L})(250.15\text{ K})}{(0.400\text{ atm})(300.15\text{ K})} \approx 104\text{ L}$$