Step 1: Understanding the Topic
The question asks about the van der Waals equation, which is a modification of the ideal gas law ($PV=nRT$) to better describe the behavior of real gases. Real gases deviate from ideal behavior because their molecules have a finite volume and experience intermolecular attractive forces, two factors ignored by the ideal gas law.
Step 2: The Van der Waals Equation and its Correction Terms
The equation is given by:
\[
\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT
\]
It introduces two corrections:
Volume Correction ($nb$): Real gas molecules occupy a finite volume. The term '$b$' is the excluded volume per mole, so '$nb$' is the total volume occupied by the gas molecules themselves. This amount is subtracted from the container volume ($V$) to give the actual free volume in which molecules can move.
Pressure Correction ($\frac{an^2}{V^2}$): Real gas molecules attract each other. This reduces the force with which they hit the container walls, lowering the measured pressure compared to an ideal gas. The term '$a$' is a measure of the strength of these intermolecular attractions. This correction term is added back to the measured pressure ($P$) to represent the ideal pressure.
Step 3: The Van der Waals Isotherm (P–V Curve)
An isotherm is a plot of Pressure (P) vs. Volume (V) at a constant Temperature (T). The shape of the van der Waals isotherm changes significantly with temperature:
High Temperature ($T>T_c$): At temperatures well above the critical temperature ($T_c$), the kinetic energy of the molecules is high, and the gas behaves much like an ideal gas. The P-V curve is a smooth hyperbola.
Critical Temperature ($T = T_c$): At a specific temperature called the critical temperature, the curve shows a point of inflection. At this critical point, the gas can be liquefied by compression without a distinct phase transition.
Low Temperature ($T<T_c$): Below the critical temperature, the isotherm shows an S-shaped oscillating region. This unphysical region is mathematically corrected (using the Maxwell construction) to a horizontal line. This flat portion represents the liquid-vapor equilibrium, where condensation occurs at a constant pressure (the vapor pressure).
Step 4: Final Answer Summary
The van der Waals equation modifies the ideal gas law to account for molecular volume and intermolecular forces:
\[
\boxed{
\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT
}
\]
Its corresponding P-V isotherm accurately predicts the deviation of real gases from ideal behavior and illustrates the conditions for the liquefaction of gases below the critical temperature.