**Van der Waals equation** is related to the general gas equation PV=nRT. Both equations are for different natures of gases. Van der Waals equation is for Real gases and the ideal gas equation is for ideal gases. This equation was given by Johannes Diederick van der Waals in 1873.

This equation is very helpful to explain or state the physical state of a non-ideal gas or real gas. Van der Waals equation has a dissimilarity with the volume, molecular size as well as molecular interaction of ideal gas. To understand these two equations, we have to understand the basic difference between ideal gas and real gas.

## Difference between Real and Ideal gas

A real gas is also referred to as non-ideal gas. So, the basic difference between these two gases is as under:

REAL GAS | IDEAL GAS |

1. It does not follow the real gas equation. | 1. It follows the ideal gas equation. |

2. At low pressure and high-temperature Real gas will follow the ideal gas equation. | 2. At all the temperature and pressure, it follows the ideal gas equation…… |

3. In Real gas, molecules are rigid and spheres have the same volume. | 3. Molecules of gas are point masses having negligible volume. |

4. There exists a force of attraction between molecules due to their large size. | 4. There exists no Force of attraction between molecules due to their small size. |

### Compressibility factor “Z”

The compressibility factor is denoted by Z. It is the ratio between the Volume of real gas to the volume of an ideal gas.

Z = V_{real gas} / V_{ideal gas}

- It also gives an idea about the positive, negative, and zero deviation between V
_{real gas}and V_{ideal gas}. - It also tells at which pressure gas beaves as ideal or real gas. If compressibility is equal to one then it will be Ideal gas but if it is greater or less than 1 then it will be a real gas.
- Z = PV/RT; if Z = 1 ⇒ Ideal gas but if Z ≠ 1 ⇒Real gas
- In real gases, two possibilities take place Z > 1 or Z < 1.
- If Z is greater than 1 then the volume of real gas dominates over the volume of an ideal gas which causes or enhances the gas to expand and repulsive forces dominate to follow the +ve deviation.
- But if Z is less than 1 then there is a vice-versa situation of the above explained. This means Videal > Vreal which causes gas to compress and to dominate attractive forces which causes -ve deviation.
- In +ve or -ve deviation measured volume has a deviation from the expected volume.

## Van der Waals Equation & Derivation

Van der Waals equation gives the relationship between the corrected volume and corrected pressure. It also shows an interesting relationship between temperature and the amount of gas. The real gas equation is also called the **Van der Waals equation**. This equation can be written as:

(P – an^{2}/V^{2})(V – nb) = nRT

Whereas,

- P, V, & T are Pressure, Volume, and temperature.
- an
^{2}/V^{2}⇒ Pressure correction. - nb ⇒ Volume correction.
- a, b ⇒ Van der Waals constants.
- n ⇒ No. of moles of gas.

### Derivation of Van der Waals Equation

This equation will follow the Non-ideal gas. This equation is basically based upon correction in volume and pressure. If we consider Ideal gas they follow Kinetic molecular theory and the Kinetic equation of gases. An ideal gas is basically based on three postulates of KMT.

- The volume of molecules is negligible as compared to the volume of the container. (Basically, the volume of gas is available space in a container in which it can move freely).
- As they are considered point masses so didn’t possess the force of attraction between them.
- All the collisions between the molecules of gas are perfectly elastic.

But if we talk about non-ideal gas, they show many deviations from the ideal gas. Van der Waals equation is based on basic two postulates:

- If we talk about volume then molecules of real gas are greater which did not occupy a small volume. So, the volume factor is affected and it is necessary to correct it.
- Due to the large molecular size of the gas, there is a greater force of attraction which causes correct factor pressure.

#### Volume Correction

As it is discussed earlier the cause of volume correction is molecular size. So,

The volume of gas = V_{container} – V_{molecules}

Vg = V – b

*b is the excluded volume. So, for n number of moles. b is also the van der Waals constant and its units are liter per mole*.

Vg = V – nb

If we find the total volume then by using the formula:

Total volume = No. of particles × Volume of single-molecule

Vg = n × 4/3πr^{3}

This equation occurs where molecules are independent or free to move but in the Real gas case, molecules have the force of attraction between them which did not allow any independent motion and have interaction with each other.

If two molecules are interacting then volume correction becomes:

b = n × 4/3πr^{3} × 2

b = 8/3 n π r^{3}

For a single molecule it becomes;

b = 4 × 4/3 n π r^{3}

b = 4n × Volume of a single molecule of gas

For n number of moles;

nb = 4n × Volume

So, ** **Volume = V – nb

A very noticeable factor in volume correction is the significance of b. b is directly proportional to the size of a molecule of gas.

b ∝ Size of a molecule of gas

#### Pressure Correction

In the case of pressure where there is an ideal gas. These gas molecules strike on the walls of the container and perform elastic collisions which exert Pressure “P” on the walls of the container due to no Force of attraction.

But in real gas, there is an exception. No molecular collisions or wall collisions take place due to their greater size and they didn’t strike too hard. So, there is a reduction in the value of pressure.

So, the total pressure of a gas is the sum of pressure and the internal pressure between particles.

P_{gas} = P + P_{internal}

As there are two different molecules of gas with large molecular sizes A & B. So,

Concentration of A-type molecule = n/V

Concentration of B-type molecule = n/V

Concentration is basically molarity and molarity is defined as the number of moles per unit volume. So,

P_{internal} ∝ (A)(B)

P_{internal} ∝ (n/V)(n/V)

P_{internal} = an^{2}/V^{2}

a is van der Waals constant. As we know,

P_{gas} = P + P_{internal}

P_{gas} = P + an^{2}/V^{2}

The significance of a is essential is pressure correction. a is directly proportional to the force of attraction and molar mass. An increase in “a” causes an increase in the intermolecular force of attraction which causes the liquefaction of gases.

But it is important to notice that the small size atom of a gas molecule has zero force of attraction. So, their pressure correction constant is equal to zero.

If we substitute the original pressure and volume with pressure correction and volume correction then the general gas equation becomes:

PV = nRT

**(P – an ^{2}/V^{2})(V – nb) = nRT**

## How Van der Waals equation is related to compressibility factor Z?

By using this equation, we will know a relationship between the ** compressibility factor Z**.

- If the temperature is constant then an increase in volume causes a decrease in pressure. So, correction in volume is negligible as compared to actual volume. So, Z < 1 which gives a negative deviation, and V
_{real}< V_{ideal}. In this case, attractive forces dominate. - If the temperature is constant then an increase in pressure causes a decrease in volume. So, the correction in pressure is negligible as compared to the actual pressure applied. So, Z > 1 which gives a positive deviation, and V
_{real}> V_{ideal}. In this case, repulsive forces dominate.

## Uses of Van der Waals equation

- It allows the prediction of gases betters than the ideal gas equation.
- Volume correction and pressure correction terms are not only for gases but also for fluids.
- It has the ability to calculate some of the critical conditions at which the liquefication of gas occurs.
- The results derived from the Van der Waals equation have accuracy below the critical temperature.
- At high pressure, it allows for the creation of an approximation for real gases and also allows the prediction of a non-ideal gas.

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