Derive reduced equation of state from van der waal’s equation.
Answers
The van der Waals equation (or van der Waals equation of state; named after Johannes Diderik van der Waals) is an equation of state that generalizes the ideal gas law based on plausible reasons that real gases do not act ideally. The ideal gas law treats gas molecules as point particles that interact with their containers but not each other, meaning they neither take up space nor change kinetic energy during collisions.[1] The ideal gas law states that volume (V) occupied by n moles of any gas has a pressure (P) at temperature (T) in kelvins given by the following relationship, where R is the gas constant:
PV = nRT,
To account for the volume that a real gas molecule takes up, the van der Waals equation replaces V in the ideal gas law with {\displaystyle (V_{m}-b)} {\displaystyle (V_{m}-b)}, where b is the volume that is occupied by one mole of the molecules. This leads to:[1]
{\displaystyle P(V_{m}-b)=RT} {\displaystyle P(V_{m}-b)=RT}
The second modification made to the ideal gas law accounts for the fact that gas molecules do in fact attract each other and that real gases are therefore more compressible than ideal gases. Van der Waals provided for intermolecular attraction by adding to the observed pressure P in the equation of state a term {\displaystyle a/V_{m}^{2}} {\displaystyle a/V_{m}^{2}}, where a is a constant whose value depends on the gas. The van der Waals equation is therefore written as:[1]
{\displaystyle (P+a/V_{m}^{2})(V_{m}-b)=RT} {\displaystyle (P+a/V_{m}^{2})(V_{m}-b)=RT}
and can also be written as the equation below
{\displaystyle (P+an^{2}/V^{2})(V-nb)=nRT} {\displaystyle (P+an^{2}/V^{2})(V-nb)=nRT}
where Vm is the molar volume of the gas, R is the universal gas constant, T is temperature, P is pressure, and V is volume. When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the van der Waals equation reduces to the ideal gas law, PVm=RT.[1]
It is available via its traditional derivation (a mechanical equation of state), or via a derivation based in statistical thermodynamics, the latter of which provides the partition function of the system and allows thermodynamic functions to be specified. It successfully approximates the behavior of real fluids above their critical temperatures and is qualitatively reasonable for their liquid and low-pressure gaseous states at low temperatures. However, near the transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the van der Waals equation fails to accurately model observed experimental behaviour, in particular that p is a constant function of V at given temperatures. As such, the van der Waals model is not useful only for calculations intended to predict real behavior in regions near the critical point. Empirical corrections to address these predictive deficiencies have been inserted into the van der Waals model, e.g., by James Clerk Maxwell in his equal area rule, and related but distinct theoretical models, e.g., based on the principle of corresponding states, have been developed to achieve better fits to real fluid behaviour in equations of more comparable complexity.
plz mark as brainliest
Reduced Equation of State of Wander Waals equation:
The van der Waals equation can also be articulated in terms of reduced properties:
Using the critical parameters VK, pK, and TK, it is suitable to go to the dimensionless variables
φ=VVK,π=ppK,τ=TTK
And redraft the Van der Waals equation in reduced form:
V=φVK,p=πpK,T=τTK,⇒(πpK+a(φVK)2)(φVK−b)=RτTK,⇒(πa27b2+a(3b)2φ2)(3bφ−b)=Rτ⋅8a27bR,⇒(π+3φ2)(3φ−1)=8τ.
This equation is more adaptable than the original Van der Waals equation.
Hope it helped...