Charles's law Knowledge-Based Study

Study notes

1. Charles's law describes the relationship between the volume and temperature of a gas when the pressure and the amount of gas are held constant.

   Discovered by French scientist Jacques Charles in the late 18th century, the law states that the volume of a given mass of an ideal gas is directly proportional to its absolute temperature at constant pressure.

   • Constant variables: Pressure (P) and amount of substance (n).
   • Changing variables: Volume (V) and absolute temperature (T).
   • Observed behavior: As the temperature increases, the gas expands; as the temperature decreases, the gas contracts.

   A critical requirement when applying Charles's law is expressing temperature on an absolute scale, specifically in kelvins (K).

   If you plot the volume of a gas against its temperature in Celsius at constant pressure, the data points form a straight line. Extrapolating this line backward to zero volume yields a temperature of approximately -273.15 °C, which defines absolute zero (0 K).

   To convert from Celsius to Kelvin, use the relationship:

   T(K) = T(°C) + 273.15

   Using temperatures in Celsius in gas law equations leads to incorrect results because volume does not scale directly with degrees Celsius.

   Mathematically, Charles's law is expressed as:

   $V \propto T$   or   $\frac{V}{T} = k$ (where k is a constant)

   For a fixed amount of gas under constant pressure changing from an initial state (1) to a final state (2), the equation becomes:

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

   Example Calculation

   A flexible balloon has a volume of 2.00 L at 27 °C. What is its volume if the temperature is raised to 127 °C at constant pressure?

   1. Convert temperatures to Kelvin:
      T1 = 27 + 273 = 300 K
      T2 = 127 + 273 = 400 K
   2. Apply the formula:
      $V_2 = V_1 \times \frac{T_2}{T_1} = 2.00 \text{ L} \times \frac{400 \text{ K}}{300 \text{ K}} = 2.67 \text{ L}$

   The kinetic molecular theory explains why gas volume increases with temperature:

   • Heating a gas increases the average kinetic energy of its particles, causing them to move faster.
   • More energetic particles collide with container walls more frequently and with greater force.
   • To maintain a constant pressure, the container must expand, increasing the overall volume until internal and external pressures balance.

   Real-World Applications:

   • Hot air balloons: Heating the air inside the envelope increases its volume and decreases its density relative to the surrounding air, creating buoyant lift.
   • Bicycle tires in winter: Leaving a tire outside in cold weather causes the air inside to contract, resulting in a noticeably softer tire.