# An approximation of the ideal gas law for real gases

• pentazoid
RT} = b(b + \frac{a}{RT} ) In summary, the virial expansion is a systematic way to account for deviations from ideal gas behavior, with functions B(T), C(T), etc. as the virial coefficients. Any equation relating P, V, and T is called an equation of state, such as the ideal gas law and the van der Waals equation. The second and third virial coefficients for a gas obeying the van der Waals equation are B(T) = b - a/RT and C(T) = b(b + a/RT).
pentazoid

## Homework Statement

Even at low density, real gases don't quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion,

PV=nRT(1+B(T)/(V/n) + C(T)/(V/n)^2+...)

where functions B(T), C(T) and so on are called the virial coefficients.

Any proposed relation between P,V, and T , like the ideal gas law or the virial equationm is called an equation of state. Another famous equation of state , which is qualitative accurate even for dense fluids, is the van der waals equations

(P+an^2/V^2)(V-nb)=nRT

where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients(B and C) for a gas obeying the van der waals equation , and terms of a and b. (hint: The binomial expansion says that (1+x)^p=1+px+1/2p(p-1)x^2, provided that abs(px)<<1. Apply this approximation to the quantity [1-nb/V]^-1.)

## The Attempt at a Solution

(P+an^2/V^2)(V-nb)=nRT==> PV-Pnb-an^3b/V^2+an^2/V=nRT==> PV=nRT+Pnb+an^3/V^2-an^2/V ==> PV=nRT(1+Pnb/nRT-an/RTV+an^3b/nRTV^2) ==> PV=nRT(1+Pnb/PV-an/RTV+an^3b/nRTV^2)==> PV=nRT(1+nb/(V/n)-(a/RT)/(V/n)+(ab/RT)V^2/n^2). Therefore the coefficients of B and C are:

B(T)=(b-a/RT) and C(T)=ab/RT right?

pentazoid said:

## Homework Statement

Even at low density, real gases don't quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion,

PV=nRT(1+B(T)/(V/n) + C(T)/(V/n)^2+...)

where functions B(T), C(T) and so on are called the virial coefficients.

Any proposed relation between P,V, and T , like the ideal gas law or the virial equationm is called an equation of state. Another famous equation of state , which is qualitative accurate even for dense fluids, is the van der waals equations

(P+an^2/V^2)(V-nb)=nRT

where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients(B and C) for a gas obeying the van der waals equation , and terms of a and b. (hint: The binomial expansion says that (1+x)^p=1+px+1/2p(p-1)x^2, provided that abs(px)<<1. Apply this approximation to the quantity [1-nb/V]^-1.)

## The Attempt at a Solution

(P+an^2/V^2)(V-nb)=nRT==> PV-Pnb-an^3b/V^2+an^2/V=nRT==> PV=nRT+Pnb+an^3/V^2-an^2/V ==> PV=nRT(1+Pnb/nRT-an/RTV+an^3b/nRTV^2) ==> PV=nRT(1+Pnb/PV-an/RTV+an^3b/nRTV^2)==> PV=nRT(1+nb/(V/n)-(a/RT)/(V/n)+(ab/RT)V^2/n^2). Therefore the coefficients of B and C are:

B(T)=(b-a/RT) and C(T)=ab/RT right?

Instead ofmultiplying out like you I wrote (as suggested)
$$(V-nb) = V(1-\frac{nb}{V} )\\$$
and took the bracket to the other side, then:
$$nRT(1-\frac{nb}{V} )^-^1 = nRT(1 + \frac{nb}{V} +\frac{n^2b^2}{V^2} + ...) = PV +\frac {an^2}{V} = PV + \frac{n}{V}. \frac {a}{RT}.nRT$$
Giving
$$B = b - \frac{a}{RT} , C = b^2$$

## What is the ideal gas law?

The ideal gas law is a mathematical equation that describes the relationship between the pressure, volume, and temperature of an ideal gas. It is written as PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature in Kelvin.

## What is an approximation of the ideal gas law for real gases?

The approximation of the ideal gas law for real gases is known as the Van der Waals equation. It takes into account the volume occupied by gas molecules and the attractive forces between them, which are not considered in the ideal gas law.

## How is the Van der Waals equation different from the ideal gas law?

The ideal gas law assumes that gas molecules have no volume and do not interact with each other, while the Van der Waals equation takes into account the volume of gas molecules and the attractive forces between them. This makes the Van der Waals equation more accurate for real gases.

## Why do we need an approximation of the ideal gas law for real gases?

The ideal gas law is only accurate for ideal gases, which do not exist in real life. Real gases have volume and interact with each other, so the ideal gas law is not sufficient to describe their behavior. The Van der Waals equation provides a more accurate representation of real gases.

## What are the limitations of the Van der Waals equation?

The Van der Waals equation is still an approximation and has its own limitations. It does not take into account the compressibility of gases at high pressures and low temperatures. It also does not consider the effects of intermolecular forces other than van der Waals forces. Other more advanced equations of state have been developed to address these limitations.

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