ALL EQUATION OF STATES AT ONE PAGE!!

The most convincing method of representing a state of a substance is writing it In an equation of (p,v, T)

IDEAL GAS EQUATION

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This concept is just a mixture of the theory developed by Boyles(v directly proportional to 1/p when T is constant) and Charles(v directly proportional to T when p is constant).so the final equation is pv=RT where R is just a proportionality constant. The most important part of this ideal gas concept is that its u(internal energy) does not depend upon p and v of the system depends only on T.i.e. 

$\left( \dfrac{\partial U}{\partial P}\right) _{T}=0$ and $\left( \dfrac{\partial U}{\partial V}\right) _{T}=0$

U=u(T) which is proposed by joules by his famous bathtub experiment

Experiment setup by joules

VANDERWALL'S GAS OR REAL GAS EQUATION OF STATE

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This equation is a modification of ideal gas in terms of gas having volume and second one gases have internal attraction between them.

1. For volume factor

The dotted circle(actually sphere) is the area where no other center of gas can come so this is the excluded volume.

$b=\dfrac{N_{0}}{2}\times \dfrac{4}{3}\times \pi \times \left( 2r \right) ^{3}=\dfrac{16}{3}N_{0}\pi r^{3}$

but again we divide the volume by 2 (that is N/2 in the above image)  as when two gases come closure the excluded volume submerged they both have the common excluded volume so we have to divide it by 2.

2.For attraction factor

$p'=p+a'\left( \dfrac{N_{0}}{v}\right) ^{2}$

Where, 

p'=pressure exerted by the gas in absence of attraction forces.

p=observed pressure 

a= an arbitrary constant.

Here n/v represents the number density of gas in the container and square is taken because attraction forces are like electrostatics forces so that depends upon the square of the number. 

FINAL EQUATION BECOMES

$\left( p+\dfrac{a}{v^{2}}\right) \times \left( v-b\right) =RT$

This equation us for 1 mole you can find for n no. Of moles by multiplying n with a/v²,b, RT. 

BEATTIE-BRIDGEMAN EQUATION OF STATE

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$pv^{2}=RT\left( 1-\dfrac{c}{vT^{3}}\right) \times \left( v+B-\dfrac{bB_{0}}{V}\right) -A_{0}\left( 1-\dfrac{d}{V}\right)$

This equation has 5 constants (Ao, Bo. a. b, and c) which depend upon gas but more accurate than Vanderwall's gas interns of state variables.

REDLICH -KWONG EQUATION OF STATE

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This equation is widely used for engineering calculations due to its accuracy and simplicity. EQUATION IS:

$p=\dfrac{RT}{v-b}-\dfrac{a}{v\left( v+b\right) }$

Its constant values are calculated from critical states of gas like Cinderella gas.

$a=\dfrac{0.42748R^{2}T_{c}^{2.5}}{P_{c}T^{1/2}}$

$b=\dfrac{0.867RT_{c}}{P_{c}}$

BENEDICT-WEBB RUBIN EQUATION

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THIS EQUATION IS MOST COMPLICATED AND MOST ACCURATE.

P = RT/V+$\dfrac{1}{v^{2}}$  ( RT(Bo +b/V)-(Ao+(a/V)-(a$\alpha$/$V^{4}$))-$\dfrac{1}{T^{2}}\left( C_{0}-\dfrac{C}{V}\left( 1+\dfrac{\gamma }{V^{2}}\right) \exp \left( -\dfrac{\gamma }{V^{2}}\right) \right)$

where Ao. Bo, C0, a,b, c, alpha, and gamma are 8 constants.

VIRIAL  EQUATION  OF  STATE

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The word virial means force this equation suggests interaction forces between molecules. 

In 1901 Kamerlingh Onnes suggested this equation. Given by:

$\dfrac{pv}{RT}=1+\dfrac{B}{V}+\dfrac{C}{v^{2}}+\dfrac{D}{V^{2}}\ldots$

here the constant are B, C, D.... and these constant only depends upon temperature. Most importantly more no of constant you know more will be your equations accuracy. "B" is called the second virial coefficient & "C" is called the third virial coefficient ....

VIRIAL EQUATION CAN BE DERIVED FROM REAL GAS EQUATION (VANDERWALLS GAS EQUATION)LIKE THIS:
Now expanding the last derived equation by binomial you will get the above-mentioned equation.
Changing the equation into

$\dfrac{pv}{RT}=1+B'p+C'p^{2}+D'p^{3}+\ldots$

 pressure term we need only change the coefficient.

$B'=\dfrac{B}{RT}\\ c'=\dfrac{C-B^{2}}{\left( RT\right) ^{2}}$

$D'=\dfrac{D-3BC+2B^{3}}{\left( RT\right) ^{3}}$