If \(P, v\), and \(T\) are the pressure, molar volume, and temperature of a gas and \(P_{C}, v_{C}\), and \(T_{C}\) are the critical pressure, critical molar volume, and critical temperature, then the reduced pressure \(P_{R}\), the reduced molar volume \(v_{R}\), and the reduced temperature \(T_{R}\) are defined as $$ P_{R}=\frac{P}{P_{C}}, \quad \nu_{R}=\frac{v}{v_{C}}, \quad T_{R}=\frac{T}{T_{C}} $$ (a) Show that, in terms of reduced quantities, the van der Waals equation becomes $$ \left(P_{R}+\frac{3}{v_{R}^{2}}\right)\left(v_{R}-\frac{1}{3}\right)=\frac{8}{3} T_{R} $$ When the van der Waals equation is in this form, the material constants \(a\) and \(b\) do not appear explicitly. Thus, all gases that obey the van der Waals equation may be considered in the same state when the values of \(P_{R}, v_{R}\), and \(T_{R}\) are the same (i.e., each gas is measured in units of its particular values of \(P_{c}, v c\), and \(T_{C}\) ). This is the principle of corresponding states, which is a principle of universal similarity established first by van der Waals.. (b) Plot three curves for \(P_{R}\) as a function of \(v_{R}\), one for \(T=\frac{1}{2} T_{C}\), one for \(T=T_{C}\) and one for \(T=2 T_{c}\). What happens physically when the equation indicates three allowed values of \(v_{R}\) for a single \(P_{R}\) and \(T\) ?

Step by step solution

01

Understand initial equations and definitions

Given the definitions, we aim to reframe the van der Waals equation in terms of the reduced quantities. Recall the van der Waals equation:\[\left(P + \frac{a}{v^2}\right)(v - b) = RT\]

02

Substitute reduced variables into the equation

Using the reduced quantities definitions:\[P = P_{R} P_{C}, \quad v = v_{R} v_{C}, \quad T = T_{R} T_{C}\]Substitute these into the van der Waals equation.

03

Replace critical conditions

Recall the critical conditions defined as follows:\[P_{C} = \frac{a}{27b^2}, \quad v_{C} = 3b, \quad T_{C} = \frac{8a}{27Rb}\]These critical points will simplify the relationship between the constants and the variables.

04

Simplify the resulting equation

Substitute the critical conditions back into the rearranged equation, simplify terms, and rearrange. This will result in:\[ \left(P_{R} + \frac{3}{v_{R}^2}\right)\left(v_{R} - \frac{1}{3}\right) = \frac{8}{3}T_{R} \]

05

Plot curves using the reformulated equation

To plot the curves of \(P_{R}\) versus \(v_{R}\) for different temperatures, substitute \(T_{R}= \frac{1}{2}, 1, \) and \(2\) in the reformulated equation and solve for \(P_{R}\) in terms of \(v_{R}\).

06

Interpret the physical meaning

When the graph yields three values for \(v_{R}\) given a single \(P_{R}\) and \(T_{R}\), it indicates a phase transition where the gas can exist in multiple phases (liquid, gas) at equilibrium.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reduced Quantities

Reduced quantities are a way to normalize the state variables of a gas—pressure \(P\), volume \(v\), and temperature \(T\)—by their critical values (critical pressure \(P_C\), critical volume \(v_C\), and critical temperature \(T_C\). By using these reduced forms, we define the reduced pressure \(P_R\), reduced volume \(v_R\), and reduced temperature \(T_R\) as follows:
\[ P_{R} = \frac{P}{P_{C}}, \quad v_{R} = \frac{v}{v_{C}}, \quad T_{R} = \frac{T}{T_{C}} \] These reduced quantities make it easier to compare different gases, as they scale the numbers relative to a standard state.
The benefit is that we can then examine gases under a universal framework by considering the values of \(P_R, v_R,\) and \(T_R\) instead of their actual physical measurements.

Critical Conditions

Critical conditions are the unique points where distinct phases (like liquid and gas) of a substance become indistinguishable. These include the critical pressure \(P_C\), critical volume \(v_C\), and critical temperature \(T_C\):
\[ P_{C} = \frac{a}{27b^2}, \quad v_{C} = 3b, \quad T_{C} = \frac{8a}{27Rb} \] At these critical points, critical constants \(a\) and \(b\) from the van der Waals equation play a key role. They relate to intermolecular attractions and the volume occupied by gas molecules.
Understanding these conditions is crucial for plotting and analyzing gas behavior and phase transitions under various temperatures and pressures.

Principle of Corresponding States

The principle of corresponding states, first proposed by van der Waals, asserts that all gases behave similarly when compared within their reduced quantities. When expressed in terms of reduced quantities, the van der Waals equation becomes similar across different gases:
\[ \left(P_{R} + \frac{3}{v_{R}^2}\right)\left(v_{R} - \frac{1}{3}\right) = \frac{8}{3}T_{R} \] This standard form eliminates the constants \(a\) and \(b\), highlighting that gases at the same reduced state exhibit equivalent behavior. In research and practical applications, this principle helps in simplifying complex calculations and draws general conclusions on gas behaviors under different conditions.

Phase Transition

A phase transition occurs when a substance changes from one state of matter to another, like from liquid to gas. The plot of reduced pressure \(P_R\) as a function of reduced volume \(v_R\) at different reduced temperatures \(T_R\) can illustrate these changes.
When the plot shows three allowed values of \(v_R\) for a single \(P_R\) and \(T_R\), it indicates phase coexistence, typically within the liquid and gas states. This phase equilibrium signifies that under those conditions, both phases have the same free energy, and the substance can exist in either phase.
Such a scenario is critical for explaining phenomena like boiling, condensation, and understanding critical points in temperature-pressure phase diagrams.

Molar Volume

Molar volume \(v\) is the volume occupied by one mole of a gas. For critical conditions, it switches to critical molar volume \(v_C\). By defining reduced molar volume \(v_R\) as the ratio \[ v_{R} = \frac{v}{v_{C}} \] we can then better handle calculations and graphical interpretations for a gas under varying conditions.
Using reduced molar volumes helps standardize properties and simplifies the interpretation of experimental results and theoretical predictions. It's indispensable in fields like thermodynamics and material science where precise behavior of gases under varying volumes is crucial.