Effect on Gas Pressure in Terms of Particles

(a) Change of Temperature at Constant Volume

According to the kinetic theory, gas particles are in constant, random motion and exert pressure by colliding with the walls of their container.

When the temperature increases/ while the volume remains constant:

• The average kinetic energy of gas particles increases because temperature is directly proportional to kinetic energy: \( E_k \propto T \)
• Particles move faster and strike the walls more frequently and with greater force.
• This leads to a greater total force exerted per unit area (i.e., higher pressure).

When the temperature decreases at constant volume:

• The average kinetic energy of the gas particles decreases.
• Particles move more slowly, collide with the walls less often and with less force.
• This causes the pressure to decrease.

This relationship is described by Gay-Lussac’s Law (constant volume):

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

Conclusion: At constant volume, pressure increases with temperature and decreases when temperature falls.

(b) Change of Volume at Constant Temperature

At constant temperature, the kinetic energy of gas particles remains unchanged.

When the volume increases:

• Gas particles have more space to move around.
• They travel longer distances before hitting the walls.
• The frequency of collisions with the walls decreases.
• This results in a decrease in pressure.

When the volume decreases:

• Gas particles are confined to a smaller space.
• They hit the container walls more frequently.
• This results in an increase in pressure.

This relationship is described by Boyle’s Law:

\( P \propto \frac{1}{V} \quad \text{(at constant temperature)} \)

Conclusion: At constant temperature, pressure decreases when volume increases, and increases when volume decreases.

Boyle’s Law:

 Boyle’s Law states that for a fixed mass of gas kept at constant temperature, the pressure of the gas is inversely proportional to its volume.

• Mathematically:
  

  \( p \propto \frac{1}{V} \quad \text{(at constant temperature)} \)

• This means:
  

  \( pV = \text{constant} \)

• If pressure increases, volume decreases; if volume increases, pressure decreases — as long as temperature remains constant.
• This law is derived from the kinetic theory of gases, assuming ideal behavior.

Using the equation:

If a gas changes from pressure \( p_1 \), volume \( V_1 \) to new conditions \( p_2 \), \( V_2 \) at constant temperature, then:

\( p_1 V_1 = p_2 V_2 \)

Graphical Representations:

• Graph 1: Pressure \( p \) vs. Volume \( V \)
  
  • This is a curve (hyperbola).
  • It shows inverse proportionality: as \( V \) increases, \( p \) decreases.
• Graph 2: Pressure \( p \) vs. \( \frac{1}{V} \)
  
  • This is a straight line through the origin.
  • It shows a direct proportionality between pressure and inverse volume.