Relate the temperature of a gas to the average kinetic energy of the particles; recall and use the equation T (in K) = θ (in °C) + 273

2.1.2 Particle Model

Objective

Relate the temperature of a gas to the average kinetic energy of its particles and recall the conversion between Celsius and Kelvin:

\$ T\ (\text{K}) = \theta\ (\text{°C}) + 273 \$

Key Concepts of the Particle Model

• All matter is made up of tiny particles (atoms, molecules, ions).
• Particles are in constant random motion.
• Temperature is a measure of the average kinetic energy of the particles.
• Increasing temperature increases the average speed of the particles.

Temperature and Average Kinetic Energy

The average translational kinetic energy of a particle in an ideal gas is given by:

\$ E{\text{avg}} = \frac{3}{2}\,k{\mathrm{B}}\,T \$

where:

• \$E_{\text{avg}}\$ = average kinetic energy per particle (J)
• \$k_{\mathrm{B}}\$ = Boltzmann constant \$=1.38\times10^{-23}\ \text{J K}^{-1}\$
• \$T\$ = absolute temperature in kelvin (K)

Thus, as \$T\$ increases, \$E_{\text{avg}}\$ increases proportionally.

Converting Between Celsius and Kelvin

The Kelvin scale starts at absolute zero (0 K). To convert a temperature from degrees Celsius to kelvin, add 273:

\$ T\ (\text{K}) = \theta\ (\text{°C}) + 273 \$

Conversely, to convert from kelvin to Celsius, subtract 273.

Worked Example

1. Find the absolute temperature of a gas at \$25\ ^\circ\text{C}\$.
2. Calculate its average kinetic energy per particle.

Step 1 – Convert to Kelvin

\$ T = 25 + 273 = 298\ \text{K} \$

Step 2 – Use the kinetic energy formula

\$ E_{\text{avg}} = \frac{3}{2}\,(1.38\times10^{-23})\,(298) \$

\$ E_{\text{avg}} \approx 6.2\times10^{-21}\ \text{J} \$

Summary

• The temperature of a gas is directly proportional to the average kinetic energy of its particles.
• Absolute temperature must be used (kelvin) when applying the kinetic‑energy equation.
• Conversion between Celsius and Kelvin is simple: add or subtract 273.