Problem 52 - An Introduction to Thermal Physics

Question

Plot the van der Waals isotherm for \(T / T_{c}=0.95,\) working in terms of reduced variables. Perform the Maxwell construction (either graphically or numerically) to obtain the vapor pressure. Then plot the Gibbs free energy (in units of \(N k T_{c}\) ) as a function of pressure for this same temperature and check that this graph predicts the same value for the vapor pressure.

Step-by-Step Solution

Answer

Perform both the Maxwell construction and Gibbs free energy plot to find consistent vapor pressure at the given reduced temperature.

Step 1: Define Reduced Variables and Isotherm

The van der Waals equation in reduced form is given by \[ \left( P + \frac{3}{v^2} \right)(3v - 1) = 8T \] where \( P, v, \) and \( T \) are reduced pressure, volume, and temperature, respectively. For \( T/T_c = 0.95 \), substitute \( T = 0.95 \) into the equation to find the isotherm curve.

Step 2: Plot the Isotherm

Using the reduced equation, plot \( P \, \text{vs} \, v \) to observe the behavior of the curve. This curve should have a loop, representing liquid, vapor, and unstable states.

Step 3: Perform the Maxwell Construction

For the Maxwell construction, identify the areas under the isotherm curve and above the horizontal line (representing equilibrium pressure) connecting two points on the curve (liquid and vapor). These areas should be equal for phase equilibrium. Adjust the horizontal line to balance these areas and determine the vapor pressure \( P_v \).

Step 4: Calculate Gibbs Free Energy

The Gibbs free energy in reduced form is \[ G = G_0 + \int_{v_1}^{v_2} v \, \mathrm{d}P \] where \( G_0 \) is a reference Gibbs energy and \( v_1 \) to \( v_2 \) is the range over which the phase change occurs. Compute \( G \) for each phase at given pressures.

Step 5: Plot Gibbs Free Energy vs Pressure

Plot the Gibbs free energy \( G \) against pressure \( P \) using the computed values. The point where the free energies of the liquid and vapor phases cross gives us the correct vapor pressure.

Step 6: Verify Consistency of Vapor Pressure

Ensure the determined vapor pressure from Steps 3 and 5 are consistent. The equilibrium pressure where the phase transition occurs should match from both the Maxwell equal-area rule and the Gibbs free energy graph.

Key Concepts

Maxwell construction

The Maxwell construction is a graphical technique used to determine the equilibrium vapor pressure of a substance. It comes into play particularly in the context of van der Waals isotherms, which can show regions of instability. These instabilities are depicted in the curve as loops. The Maxwell construction balances the areas under and over a horizontal phase line connecting the liquid and vapor phases on a pressure-volume ( P-v ) graph.

Here's how it works:

• Plot the isothermal curve for a fluid at a specific temperature below the critical temperature.
• Identify the loop that represents metastable states.
• Draw a horizontal line through the loop such that the area above the line equals the area below it. This line represents the equilibrium vapor pressure.

Maxwell's construction essentially ensures that the work done by the system during a reversible phase change is zero, preserving the equilibrium conditions necessary for phase transitions.

Gibbs free energy

Gibbs free energy is a thermodynamic potential that can help predict the behavior of substances during phase changes. It encompasses both energy changes and entropy changes in a system, making it a comprehensive measure for understanding various transitions.

In relation to van der Waals isotherms and the Maxwell construction, Gibbs free energy is crucial because:

• It allows us to understand and verify equilibrium conditions for phase transitions.
• The intersection of Gibbs free energy curves for liquid and vapor states indicates the vapor pressure.

Practically, the Gibbs free energy for each phase can be found by calculating the integral of volume over pressure in the given context:\[G = G_0 + \int_{v_1}^{v_2} v \, dP\]
The intersection point where the Gibbs free energies of different phases match offers a critical check for the vapor pressure found through Maxwell’s construction.

reduced variables

Reduced variables simplify complex systems, especially when working with equations such as the van der Waals equation. By scaling pressure, volume, and temperature to their critical values, these reduced variables eliminate the need to work with numerous constants.

Here's what each reduced variable defines:

• Reduced pressure ( P ) is the actual pressure divided by the critical pressure.
• Reduced volume ( v ) is the actual volume divided by the critical volume.
• Reduced temperature ( T ) is the actual temperature divided by the critical temperature.

Using these reduced variables in the van der Waals equation, physicists can create universally applicable plots and analyses, such as plotting the behavior of a fluid along an isotherm and facilitating techniques like the Maxwell construction.

vapor pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature. It's a critical concept when discussing phase changes and transitions between liquid and gaseous states.

In the context of the van der Waals isotherm:

• The vapor pressure is found using the Maxwell construction by equating the areas of the loop in a P-v plot.
• It can also be verified through Gibbs free energy calculations by finding where free energies of different phases become equal.

Finding the vapor pressure is essential to understanding the boiling point of a substance under specific conditions and assessing how various substances behave under temperature and pressure changes, making it a fundamental aspect of thermodynamic studies.