# What is Van der waals equation: Definition and 15 Discussions

In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state; named after Dutch physicist 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 (i.e. all collisions are perfectly elastic). 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:

P
V
=
n
R
T

{\displaystyle 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

(

V

m

b
)

{\displaystyle (V_{m}-b)}
, where Vm is the molar volume of the gas and b is the volume that is occupied by one mole of the molecules. This leads to:

P
(

V

m

b
)
=
R
T

{\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 interact with each other (they usually experience attraction at low pressures and repulsion at high pressures) and that real gases therefore show different compressibility than ideal gases. Van der Waals provided for intermolecular interaction by adding to the observed pressure P in the equation of state a term

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:

(

P
+
a

1

V

m

2

)

(

V

m

b
)
=
R
T

{\displaystyle \left(P+a{\frac {1}{V_{m}^{2}}}\right)(V_{m}-b)=RT}
and, for n moles of gas, can also be written as the equation below:

(

P
+
a

n

2

V

2

)

(
V

n
b
)
=
n
R
T

{\displaystyle \left(P+a{\frac {n^{2}}{V^{2}}}\right)(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.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 phase 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. Corrections to address these predictive deficiencies have since been made, such as the equal area rule or the principle of corresponding states.

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1. ### Verifying properties of Van der Waals Gas

So a and b were pretty straightforward. Got stuck on part c. The question says they approximated Van der Waals in first order in a and b. So I started with that by rewriting Van der Waals eqn as ## p = \frac { N \tau } { V - Nb } - \frac {N^2a} {V^2} ## and I then Taylor approximated ## \frac...
2. ### Question in Thermal Physics (Van der Waals' Equation)

<Using the hint, I tried to find the van der Waal constants in molar form. Since STP is mentioned, I used the unitary method relationship- 22.4 L=22400cm^3=1 molar V <To find a possible conversion standard between cm^3 and mol; which turned out to be 1cm^3= 4.46*10^-5 mol. <Then I used the...
3. ### Van der Waals constant 'b' for a real gas

Doesn't volume of ideal gas include volume of the molecules? What I was taught in school is that, when two molecules collide, no other molecule can come around it (inside the green part in the figure) and that volume is excluded volume? Is it correct? If yes, why?
4. ### Finding b in Van der Waals Equation?

I've got a question that requires me to use the Van der Waals equation in the form: p(V-b)=nRT The process is isobaric, the volume changes from 1m3 to 2m3, and there is 1 mole of the unidentified gas. Ultimately, I need to find initial and final values of T. So I rearranged the formula...
5. ### Adiabatic process of real gas

Homework Statement Show that for a gas obeying the van der Waals equation ##\left(P+\frac{a}{v^2}\right)(v-b)=RT##, with ##c_v## a function of ##T## only, an equation for an adiabatic process is $$T(v-b)^{R/c_v}=constant$$ Homework Equations ##TdS=c_vdT+T\left(\frac{\partial P}{\partial...
6. ### Rearranging variables Van Der Waals EoS into new variables

The question I'm stuck on is: P = NKBT/(V-Nb) - aN2/(V2) -----> (1) Re-arrange variables in the Van Der Waals equation of state, Eq. (1), so that V always appears in the equation as V/(3Nb) and P appears as 27b2P/a. Then T should appear in the combination 27b kBT/(8a). Call these...
7. ### I Inversion and Isenthelpic Curves for van der Waals EOS

When I plot the isenthalpic curves in T-P plane, I see isenthalpic curves are not well defined in the lower temperature region in T-P plane. Inside the inversion curve Joule-Thomson coefficient is positive so gas cools and outside of the inversion curve Joule-Thomson coefficent is negative so...
8. ### Van der Waals Gas

Homework Statement Hi, I have the following task: Translated into English, that means: " For Cpm and Cvm of gases the following relationship is true: (1) a) Show with the relationship (1), that for an ideal gas Cpm - Cvm = R is valid b) Deduce from equation (1) and the tripple product rule...
9. ### Isobaric process for a Van der Waals gas

Homework Statement [/B] You are asked to calculate changes in internal energy, entropy, heat transferred and work done for each of the following process. Also you are asked to calculate "the latent heat for the isotherm in the figure". We know the a and b parameters which characterize the VdW...
10. ### How is the b term used in the van der Waals equation?

What does the following statement below mean? "The excluded volume is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times that volume. To see this, we must realize that a particle is surrounded by a sphere of radius 2r (two times the original...
11. ### How was the van der Waals equation derived?

the equation (P+a(n/V)^2)(V-nb)=nRT was derived in this manner: The pressure of a real gas is affected by intermolecular forces and so the a(n/V)^2 term must be added to the measured pressure to obtain the ideal pressure where Pmeasured+a(n/V)^2=Pideal On the other hand when they explained...
12. ### Van der Waals diameter

Homework Statement Calculate the molecular diameter of hydrogen, assuming the spherical shape on the basis of van der Waals coefficients. Van der Waals Coefficients for hydrogen a = 0.244 atm L2.mol-2 b*103 = 26.6 L.mol-1 Homework...
13. ### Van der Waals and boiling point

Homework Statement The bulb of a constant volume gas thermometer is immersed in an ice/water/water vapour mixture at equilibrium and the recorded pressure is 0.400 atm. It is then immersed in a boiling liquid and the pressure is 0.844 atm. Sufficient gas is then removed from the bulb such that...
14. ### Compressibility factor and van der Waals equation for temp

The pressure exerted on the walls of the container by a real gas is less compared to an ideal gas. This is due to the attractive forces of the gas pulling the molecules back towards the rest of the gas molecules. However, there is also a relationship whereby at lower temperatures, the z is even...
15. ### PV Work for Electrolysis Using Van Der Waals Equation

Homework Statement Solve for the work done during water electrolysis using the Van Der Waals equation.[/B] Solving for work using Ideal Gas Law: This is the system work for the electrolysis of water using the Ideal Gas Law: W = PΔV = (101.3 x 103 Pa)(1.5 moles)(22.4 x 10-3...