Problem Statement
State and derive the law of corresponding states from the van der Waals equation.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: Introduce reduced variables: $\pi = p/p_c$, $\phi = V/V_c$, $\tau = T/T_c$.
Step 2 — Apply the relevant physical law or equation: Substituting $p = \pi p_c$, $V = \phi V_c = 3\phi b$, $T = \tau T_c = 8\tau a/(27Rb)$ into the van der Waals equation:
Step 3 — Solve algebraically for the unknown: $$\left(\pi + \frac{3}{\phi^2}\right)\left(\phi – \frac{1}{3}\right) = \frac{8\tau}{3}$$
Step 4 — Substitute numerical values with units: This is the reduced van der Waals equation — it contains no substance-specific constants! All van der Waals gases obey the same equation in terms of reduced variables.
Step 5 — Compute and check the result: Law of corresponding states: All substances (approximately) have the same reduced equation of state, i.e., at the same $\pi$ and $\tau$ they have the same $\phi$.
Step 6: This law works remarkably well for simple molecules (noble gases, diatomics) and is used in engineering correlations.
Worked Calculation
$$\left(\pi + \frac{3}{\phi^2}\right)\left(\phi – \frac{1}{3}\right) = \frac{8\tau}{3}$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{\left(\pi + \frac{3}{\phi^2}\right)\left(\phi – \frac{1}{3}\right) = \frac{8\tau}{3}}$$
Introduce reduced variables: $\pi = p/p_c$, $\phi = V/V_c$, $\tau = T/T_c$.
Substituting $p = \pi p_c$, $V = \phi V_c = 3\phi b$, $T = \tau T_c = 8\tau a/(27Rb)$ into the van der Waals equation:
$$\left(\pi + \frac{3}{\phi^2}\right)\left(\phi – \frac{1}{3}\right) = \frac{8\tau}{3}$$
This is the reduced van der Waals equation — it contains no substance-specific constants! All van der Waals gases obey the same equation in terms of reduced variables.
Law of corresponding states: All substances (approximately) have the same reduced equation of state, i.e., at the same $\pi$ and $\tau$ they have the same $\phi$.
This law works remarkably well for simple molecules (noble gases, diatomics) and is used in engineering correlations.
Answer
$$\boxed{\left(\pi + \frac{3}{\phi^2}\right)\left(\phi – \frac{1}{3}\right) = \frac{8\tau}{3}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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