Problem 55 - An Introduction to Thermal Physics

Question

In this problem you will investigate the behavior of a van der Waals fluid near the critical point. It is easiest to work in terms of reduced variables throughout. (a) Expand the van der Waals equation in a Taylor series in \(\left(V-V_{c}\right)\), keeping terms through order \(\left(V-V_{c}\right)^{3} .\) Argue that, for \(T\) sufficiently close to \(T_{c}\) the term quadratic in \(\left(V-V_{c}\right)\) becomes negligible compared to the others and may be dropped. (b) The resulting expression for \(P(V)\) is antisymmetric about the point \(V=V_{c}\) Use this fact to find an approximate formula for the vapor pressure as a function of temperature. (You may find it helpful to plot the isotherm.) Evaluate the slope of the phase boundary, \(d P / d T\), at the critical point. (c) Still working in the same limit, find an expression for the difference in volume between the gas and liquid phases at the vapor pressure. You should find \(\left(V_{g}-V_{l}\right) \propto\left(T_{c}-T\right)^{\beta},\) where \(\beta\) is known as a critical exponent. Experiments show that \(\beta\) has a universal value of about \(1 / 3,\) but the van der Waals model predicts a larger value. (d) Use the previous result to calculate the predicted latent heat of the transformation as a function of temperature, and sketch this function. (e) The shape of the \(T=T_{c}\) isotherm defines another critical exponent, called \(\delta\) : \(\left(P-P_{c}\right) \propto\left(V-V_{c}\right)^{\delta} .\) Calculate \(\delta\) in the van der Waals model. (Experimental values of \(\delta\) are typically around 4 or \(5 .\) ) (f) A third critical exponent describes the temperature dependence of the isothermal compressibility, $$\kappa \equiv-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T},$$ This quantity diverges at the critical point, in proportion to a power of \(\left(T-T_{c}\right)\) that in principle could differ depending on whether one approaches the critical point from above or below. Therefore the critical exponents \(\gamma\) and \(\gamma^{\prime}\) are defined by the relations $$\kappa \propto\left\\{\begin{array}{ll} \left(T-T_{c}\right)^{-\gamma} & \text { as } T \rightarrow T_{c} \text { from above } \\ \left(T_{c}-T\right)^{-\gamma^{\prime}} & \text { as } T \rightarrow T_{c} \text { from below } \end{array}\right.$$ Calculate \(\kappa\) on both sides of the critical point in the van der Waals model, and show that \(\gamma=\gamma^{\prime}\) in this model.

Step-by-Step Solution

Answer

Analyze Taylor series expansion near critical point to simplify van der Waals equation, showing symmetry and calculating critical exponents. Use these to predict vapor pressure, phase differences, and latent heat.

Step 1: Understanding the van der Waals Equation

The van der Waals equation is given by:\[ P = \frac{nRT}{V - nb} - \frac{a}{V^2} \]where \(P\) is the pressure, \(V\) is the volume, \(T\) is the temperature, \(n\) is the number of moles, \(R\) is the gas constant, \(a\) and \(b\) are van der Waals constants. Near the critical point, we use reduced variables: \(V_r = \frac{V}{V_c}\), \(T_r = \frac{T}{T_c}\), \(P_r = \frac{P}{P_c}\).

Step 2: Expanding the Equation

Using a Taylor series expansion, expand the van der Waals equation around the critical volume \(V_c\). Keep terms up to third order:\[ P(V) \approx P_c + A (V - V_c) + B (V - V_c)^2 + C (V - V_c)^3 \]where \(A, B, C\) are coefficients derived from the derivatives of the pressure with respect to volume at the critical point. Analyze the behavior near \(T_c\) to simplify the expression.

Step 3: Neglecting the Quadratic Term

Near \(T = T_c\), the pressure-volume curve becomes very flat, indicating the quadratic term's negligible contribution due to its symmetry about \(V_c\). Thus, simplify the expression by omitting the quadratic term yielding an approximate cubic equation for \(P(V)\).

Step 4: Finding an Approximate Formula for Vapor Pressure

Due to the antisymmetry about \(V = V_c\), use the cubic approximation to find the relationship:\[ P - P_c \propto (V - V_c)^3 \]Plotting the isotherm shows the symmetry and helps in solving for vapor pressure as a function of temperature, thereby determining the slope \(\frac{dP}{dT}\) at the critical point.

Step 5: Calculating the Volume Difference Between Phases

At coexistence, the volumes of the gas \(V_g\) and liquid \(V_l\) satisfy:\[ V_g - V_l \propto (T_c - T)^{\beta} \]In the van der Waals model, evaluate \(\beta\), the critical exponent, which tends to overpredict experimentally observed values.

Step 6: Predicting the Latent Heat

Use the Clapeyron equation to calculate the latent heat \(L\):\[ L = T \left( V_g - V_l \right) \left( \frac{dP}{dT} \right) \]Plot the resulting function to visualize its temperature dependency, generally linear away from the critical point and varying as the phases approach it.

Step 7: Calculating the Exponent \(\delta\)

The relationship \((P - P_c) \propto (V-V_c)^{\delta}\) defines \(\delta\). For the van der Waals model, derive \(\delta\) from pressure-volume fluctuations near \(T_c\), which tends to result in values higher than the experimental range (\(\approx 4\) or \(5\)).

Step 8: Investigating Isothermal Compressibility

The isothermal compressibility \(\kappa\) describes pressure response in volume:\[ \kappa = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T \]Evaluate \(\kappa\) on both sides of \(T_c\), confirming its divergence and validating that in the van der Waals model, \(\gamma = \gamma'\), indicating similar behavior above and below the critical point.

Key Concepts

Critical Point

The critical point of a substance is where it exhibits unique thermodynamic properties. At this point, the liquid and gas phases become indistinguishable. For a van der Waals fluid, the critical point is defined by specific values of temperature, pressure, and volume, denoted as \(T_c\), \(P_c\), and \(V_c\), respectively.

Understanding how these variables interact at the critical point helps us gauge the fluid's behavior near this state. Near \(T_c\), we observe unusual properties such as the flattened pressure-volume curve, meaning that small changes in pressure have a minimal effect on volume. This unusual behavior requires us to use special approaches and approximations in our equations to accurately predict conditions near the critical point.

Reduced Variables

Reduced variables are scaled versions of the state variables, making the study of critical phenomena more manageable. By defining reduced variables as \(V_r = \frac{V}{V_c}\), \(T_r = \frac{T}{T_c}\), and \(P_r = \frac{P}{P_c}\), we're comparing a substance's state relative to its critical point.

These reduced forms simplify complex relations and allow us to express the behavior of different substances universally, using equations like the van der Waals equation. They also enable us to visualize how a substance's actual state compares to its critical condition, enhancing our understanding of phase transitions and scaling behaviors across different materials.

Critical Exponents

Critical exponents describe how physical quantities behave near the critical point. They provide insights into the phase transition's nature. In the van der Waals model, common exponents include \(\beta\), \(\delta\), and \(\gamma\), each characterizing specific relationships.

\( \beta \) describes how volume changes: \((V_g - V_l) \propto (T_c - T)^\beta\). Experimentally, \(\beta\) is around \(1/3\), but the van der Waals model predicts higher. This discrepancy highlights limitations and motivates improvements in theoretical models.

Similarly, \(\delta\) relates pressure and volume: \((P - P_c) \propto (V - V_c)^\delta\). Accurate values near \(4\) to \(5\) further ensure the model's quantitative reliability. Understanding these exponents is critical in developing accurate fluid behavior predictions and enriching the theoretical frameworks of phase transition analysis.

Isothermal Compressibility

Isothermal compressibility, \(\kappa\), measures the change in volume with pressure at constant temperature: \( \kappa = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T \). Near the critical point, \(\kappa\) diverges, reflecting increased fluctuations and sensitivity to external conditions.

This divergence is expressed in terms of critical exponents \(\gamma\) and \(\gamma'\):

• \( \kappa \propto (T - T_c)^{-\gamma} \) approaching from above.
• \( \kappa \propto (T_c - T)^{-\gamma'} \) from below.

In the van der Waals model, \(\gamma = \gamma'\), showing symmetrical behavior on either side of \(T_c\). Recognizing \(\kappa\)'s critical behavior aids in understanding phase transition mechanisms and engineering applications where fluid compressibility is a key factor. Critical exponents vividly portray these effects and validate the underlying theory governing phase transitions.