# Compare ideal and real gases

## Compare between ideal gas and real gas

Compare between the ideal and real gas can be described by the ideal gas law for gas molecules. Ideal gas law for 1-mole gas

PV = nRT

The gas which obeys this law under all conditions of temperature and pressure called ideal gas but the gas which does not obey this law under all conditions of temperature and pressure called real gases.

An ideal gas, really hypothetical one, which follows the ideal gas law rigorously under all circumstances, has been named as ideal gas or perfect gas as distinct properties of gases.

### Assumptions of the ideal gas law

1. An ideal gas can not be liquefied because ideal gas molecules have no inter-molecular attraction. The gas molecules will not condense.
2. The coefficient of thermal expansion(ɑ) depends on the temperature of the gas and does not depends on the nature of the gas.
3. The coefficient of compressibility(β) similarly depends on the pressure of the gas and will be the same for all gases.
4. When pressure is plotted against volume at a constant temperature a rectangular hyperbola curve obtained. The hyperbola curve at each temperature called one isotherm and at a different temperature, we have different isotherms. Two isotherms will never intersect
5. When PV is plotted against pressure at a constant temperature a straight line plot obtained parallel. There will be Different parallel lines obtained at different temperatures.
6. Ideal gas passes through a porous plug from higher pressure to lower pressure within the insulated enclosure, there will be no change in the temperature of the gas. This confirms that the ideal gas has no inter-molecular attraction.

#### Coefficient of thermal expansion

Thermodynamic definition of the coefficient of thermal expansion of a gas

$\alpha&space;=\frac{1}{V}\left&space;(&space;\frac{dV}{dT}&space;\right&space;)_{P}$

For 1-mole ideal gas law for 1-mole gas
PV = RT

$\left&space;(&space;\frac{dV}{dT}&space;\right&space;)_{P}=\frac{R}{P}$

$\therefore&space;\alpha&space;=\frac{1}{V}\times&space;\frac{R}{P}=\frac{R}{PV}=\frac{1}{T}$

Thus thermal expansion will be independent of the nature of the gas and will be a function of temperature only. The values of thermal expansion for different gases are found to be different.

The coefficient of thermal expansion for hydrogen and carbon dioxide 2.78 × 10-7 and 3.49 × 10-7 respectively at 0°C and 500 atmospheres.

#### Compressibility factor of a gas

Coefficient of compressibility defined

$\beta&space;=-\frac{1}{V}\left&space;(&space;\frac{dV}{dP}&space;\right&space;)_{T}$

Ideal gas law for 1-mole gas
PV = RT

$or,&space;\left&space;(&space;\frac{dV}{dP}&space;\right&space;)_{T}=-\frac{RT}{P^{2}}$

$\therefore&space;\beta&space;=\left&space;(&space;-\frac{1}{V}&space;\right&space;)\times&space;\left&space;(-\frac{RT}{P}&space;\right&space;)=\frac{1}{P}$

Thus β should be a function of pressure only and the same for all gases. But experimentally the coefficient of compressibility has been found to be individual property.

### Real gases deviations from ideal behavior

1. Real gas could be liquefied because gas molecules have an intermolecular attraction which helps to coalesce the gas molecules.
2. Thermal expansion (ɑ) found to vary from gas to gas. The coefficient of thermal expansion depends on the nature of the gas.
3. The coefficient of compressibility (β) also is found to depend on the nature of the gas.
4. When pressure plotted against volume a rectangular hyperbola curve obtained only at a high temperature above the critical temperature.
5. But a temperature below the critical temperature(C), the gas can be liquefied after certain pressure depends on temperature. Liquid and gas can be indistinguishable in the critical point of the gases.
When PV is plotted against pressure for real or Van der Waals gases Amagat curve obtained.
6. Real gases pass through porous plug from higher pressure to lower pressure within the insulated enclosure, there occurs a change of temperature.
7. Real gases have inter-molecular attraction and when the gas expands, the molecules have to spend kinetic energy to overcome inter-molecular attraction and so the temperature of the gas drops down.

#### Z vs P graph for real gases

Most gases, the value of Z decreases attains minimum and then increases with the increased pressure of the gas.

Hydrogen and helium gas-only baffle this trend and the curve rise with the increased pressure of the gas from the very beginning.

Carbon dioxide gas can be easily liquified and Z dips sharply below the ideal gas line in the low-pressure region.

TB called Boyle temperature, the initial slope at TB zero. At TB, the Z vs P line of a gas tangent to that of a real gas when pressure approaches zero but latter rises above the ideal gas line only very slowly.

Thus, at TB real gas behaves ideally over a wide range of pressure, because the effect of the size of gas molecules and intermolecular forces roughly compensate each other.

#### Boyle temperature of hydrogen gas

 Gases TB Hydrogen (H2) -1560C Helium (He) -2490C Nitrogen (N2) 590C Methane (CH4) 2240C Ammonia (NH3) 5870C

For hydrogen and helium, TB lowers then 0⁰C temperature so Z values greater than unity.

For nitrogen, methane, and ammonia TB greater then 0⁰C so Z values less than unity in the low-pressure region.

### Compressibility for real gases

An important single parameter called the compressibility used to compare the extent of deviation of the real gas from ideal behavior.

Z = PV/RT

1. Z=1, the gas is ideal gas or there is no deviation from ideal behavior.
2. When Z ≠ 1, the gas is non-ideal and the departure of the value of Z from unity is a measure of the extent of non-ideality of the gas.
3. Zく1, the gas is a more compressible then ideal gas and when Z 〉1, the gas has less compressible then ideal gas.