Problem 11 - A Modern Course in Statistical Physics

Question

For a van der Waals gas, plot the isotherms in the \(\bar{P}-\bar{V}\) plane \((\bar{P}\) and \(\bar{V}\) are the reduced pressure and volume) for reduced temperatures \(\bar{T}=0.5, \bar{T}=1.0\), and \(\bar{T}=1.5\). For \(\bar{T}=0.5\), is \(\bar{P}=0.1\) the equilibrium pressure of the liquid-gas coexistence region?

Step-by-Step Solution

Answer

Plot isotherms for \bar{T}=0.5, 1.0, 1.5; equilibrium pressure is determined from the critical point data.

Step 1 - Understand the van der Waals equation in reduced terms

The van der Waals equation in reduced form describes the behavior of a real gas. The equation is given by: The reduced form equation is then expressed as where: where

Step 2 - Plot the isotherms for given reduced temperatures

For reduced temperatures plot the reduced pressure in plane. Use the van der Waals equation in reduced form. get the curves.

Key Concepts

isotherms

Isotherms are curves on a graph that represent the relationship between pressure and volume at constant temperature. For a van der Waals gas, this relationship is more complex compared to an ideal gas because it considers molecular size and interactions.
In the \(\bar{P} - \bar{V}\) plane, we plot the reduced pressure \(\bar{P}\) against the reduced volume \(\bar{V}\) for different reduced temperatures \(\bar{T}\).
Each curve on the plot corresponds to a specific temperature.

• For higher temperatures, the isotherms resemble the hyperbolic shape typical of an ideal gas.
• For lower temperatures, the isotherms show regions where liquid and gas coexist, known as the phase transition region.

This region is crucial for understanding the equilibrium between liquid and gas states in real gases.

reduced pressure

Reduced pressure \(\bar{P}\) is a dimensionless form of pressure, scaled by a characteristic pressure of the substance. It is given by:
\[\bar{P} = \frac{P}{P_c} \]
where:

• \(P\) is the actual pressure of the gas.
• \(P_c\) is the critical pressure of the gas.

Using reduced pressure allows us to compare different gases on a common scale, making it easier to study their behaviors.
Plotting \(\bar{P}\) versus reduced volume simplifies the analysis of phase transitions and other properties, revealing insights that would be obscured by using raw pressure values alone.

reduced temperature

Reduced temperature \(\bar{T}\) is another dimensionless quantity, which scales the actual temperature with the critical temperature of the gas. It is given by:
\[\bar{T} = \frac{T}{T_c} \]
where:

• \(T\) is the actual temperature of the gas.
• \(T_c\) is the critical temperature of the gas.

When \(\bar{T} = 1.0\), the gas is at its critical temperature.
For \(\bar{T} < 1.0\), the gas can exhibit phase transitions from gas to liquid.
For \(\bar{T} > 1.0\), the gas behaves more like an ideal gas, with less deviation seen in the van der Waals isotherms.
Understanding reduced temperature is crucial in predicting the behavior of gases under different thermal conditions.

van der Waals equation

The van der Waals equation accounts for real gas behavior by modifying the ideal gas law. It incorporates two factors:

• Molecular volume: It adjusts the volume available to gas molecules.
• Intermolecular forces: It accounts for the attraction between gas molecules.

In reduced form, the van der Waals equation is:
\[ ( \bar{P} + \frac{3} { \bar{V}^2 } ) ( 3\bar{V} - 1 ) = 8\bar{T} \]
Here, each variable is scaled by its critical value:

• \(\bar{P} = \frac{P}{P_c}\)
• \(\bar{T} = \frac{T}{T_c}\)
• \(\bar{V} = \frac{V}{V_c}\)

This form of the equation helps in visualizing and evaluating the behavior of real gases under various conditions by comparing them to their critical properties.