Temperature and Kinetic Energy

Temperature is directly related to the average kinetic energy of gas molecules. The average kinetic energy per molecule is given by:

\[ \langle E_k \rangle = \frac{1}{2} m \langle v^2 \rangle \]

And since temperature is proportional to the average kinetic energy of the particles:

\[ \langle E_k \rangle = \frac{3}{2} k_B T \]

Where:

  • \(\langle E_k \rangle\) is the average kinetic energy.
  • \(k_B\) is Boltzmann’s constant.
  • \(T\) is the temperature in Kelvin.

 

Ideal Gas Law Derivation

The Ideal Gas Law can be derived from the Kinetic Theory of Gases, which relates the macroscopic properties of gases (pressure, volume, temperature) to the microscopic behavior of molecules (velocity and kinetic energy).

1. The total pressure $p$ is the force exerted by all molecules on the walls per unit area. For a large number of molecules, this results in the equation derived in the previous chapter:

\[ p = \frac{1}{3} \frac{N m \langle v^2 \rangle}{V} \]

2. Using the relationship between kinetic energy and temperature:

\[\langle E \rangle = \frac{1}{2} m \langle v^2 \rangle = \frac{3}{2} k_B T\]

we can rewrite the pressure equation as:

\[pV = N k_B T\]

3. Finally, using the relationship between the number of molecules $N$ and the number of moles $n$ (where $N = n N_A$), we arrive at the Ideal Gas Law:

\[pV = nRT\]

where $R = N_A k_B$ is the gas constant.

 

Examples

Lightning and Sound Speed

This example uses the difference in the speed of light and the speed of sound to estimate the distance from a lightning strike.

The speed of light is so fast that we consider it instantaneous, while the speed of sound at room temperature is approximately:

\[v_{\text{sound}} = 343 \, \text{m/s}\]
.

Given a delay of 5 seconds between seeing lightning and hearing thunder, the distance to the lightning strike is:

\[ \text{Distance} = 343 \, \text{m/s} \times 5 \, \text{s} = 1715 \, \text{m} = 1.7 \, \text{km} \]
.

Energy at Different Speeds

Consider two gases at the same temperature but with different molecular masses, like hydrogen and oxygen. Since the average kinetic energy is proportional to temperature, the lighter hydrogen molecules will have a higher average speed than the heavier oxygen molecules.

The average kinetic energy is:

\[ \langle E \rangle = \frac{3}{2} k_B T \]
,

which is the same for both gases. However, the relation between speed and mass is:

\[ \langle v \rangle \propto \frac{1}{\sqrt{m}} \]
.

Therefore, hydrogen molecules move faster than oxygen molecules at the same temperature because their mass is smaller.

 

Absolute Zero (T = 0K)

At absolute zero ($T = 0 K$), the kinetic energy of molecules reaches its minimum. According to classical mechanics, the velocity of all molecules would be zero, meaning that they would stop moving entirely.

However, in quantum mechanics, the Heisenberg Uncertainty Principle implies that even at absolute zero, molecules still possess some residual energy known as zero-point energy. This is the lowest possible energy that a system can have.

1. Classical Case: In classical mechanics, at $T = 0 K$, the velocity of molecules is zero, and they would have no kinetic energy.

2. Quantum Mechanical Case: In quantum mechanics, due to the uncertainty principle, particles exhibit fluctuations even at absolute zero. Thus, the concept of zero-point energy emerges.

At absolute zero, while molecular motion is minimal, the molecules still retain some energy due to quantum mechanical effects.