 # Pressure of an Ideal Gas

## Ideal Gas

Before we learn how to calculate the pressure of an ideal gas let us first know what exactly an ideal gas is. An ideal gas in simple words is a theoretical gas in which the gas particles move randomly and there is no interparticle interaction. An ideal gas doesn't exist in reality. It follows the ideal gas equation which is a simplified equation we will learn further and is susceptible to analysis under statistical mechanics. At standard pressure and temperature condition, most gases are taken to behave as an ideal gas. As defined by IUPAC, 1 mole of an ideal gas has a capacity of 22.71 liters at standard temperature and pressure.

### Failure of Ideal Gas Model

At lower temperatures and high pressure, when intermolecular forces and molecular size becomes important the ideal gas model tends to fail. For most of the heavy gases such as refrigerants and gases with strong intermolecular forces, this model tends to fail. At high pressures, the volume of a real gas is often considerably larger than that of an ideal gas and at low temperatures, the pressure of a real gas is often considerably less than that of an ideal gas. At some point in low temperature and high-pressure real gases undergo phase transition which is not allowed in the ideal gas model. The deviation from the ideal gas model can be explained by a dimensionless quantity, called the compressibility factor (Z).

Ideal Gas Law

Ideal gas law gives an equation known as the ideal gas equation which is followed by an ideal gas. It is a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas equation in empirical form is given as:

PV=nRT

where P= pressure of the gas (pascal)

V= volume of gas (litres)

n= number of moles of gas (moles)

R= universal or ideal gas constant           (=8.314JK-1mol-1)

T= absolute temperature of the gas (Kelvin)

Ideal gas law is an extension of experimentally discovered gas laws. It is derived from Boyle's law, Charles law, Avogadro's law. When these three are combined, we get ideal gas law.

Boyle's law =>  PV = k

Charle's law => V = kT

Avogadro's law => V = kn

Now, when we combine these three laws we use the proportionality constant 'R', which is the universal gas constant and we get ideal gas equation as

V = RTn/P

=> PV = nRT

Various assumptions are made in the ideal gas model. They are as follows:

• Gas molecules are considered as indistinguishably very small and hard spheres.

• All motions are frictionless and the collisions are elastic that is there is no energy loss in motion or collisions.

• All laws of Newton are applicable.

• The size of the molecules is much smaller than the average distance between them.

• There is a constant movement of molecules in random directions with distributed speeds.

• Molecules don't attract or repel each other apart from point-like collisions with the walls.

• No long-range forces exist between molecules of the gas and surroundings.

### The Pressure of an Ideal Gas

Calculation:

For the calculation let us consider an ideal gas filled in a container cubical in shape. One corner of the container is taken as the origin and the edges as x, y, and z axes. Let A1 and A2 be the parallel faces of the cuboid which are perpendicular to the x-axis. Suppose, a molecule is moving with velocity 'v' in the container and the components of velocity along three axes are vx, vy, and vz. As we assume collisions to be elastic so when this molecule collides with face A1  x component of velocity reverses while y and z component remain unchanged.

Change in the momentum of the molecule is

∆P= -mvx -mvx  = -2mvx. ……….     (1)

The change in momentum of the wall is 2mvx as the momentum remains conserved.

After the collision, the molecule travels towards the face A2 with x component of the velocity equal to −vx

Now, the distance traveled by a molecule from A1 to A2 = L

Therefore, time = L/vx

After a collision with A2 it again travels to A1. Hence, the time between two collisions= 2L/vx

So the number of collisions of molecule per unit time n = vx /2L……….(2)

From (1) and (2),

Momentum imparted to the molecule by the wall per unit time

∆F=n∆P

=m/L×vx²

Therefore, the total force on the wall A1 due to all the molecules is

F = Ʃm/L×vx²

F = m/LƩvx²

Ʃvx²=Ʃvy²=Ʃvz² (symmetry)

= 1/3Ʃv²

Therefore, F=1/3×m/LƩv²

Now, the pressure is the force per unit area hence,

P=F/L²

P=1/3×m/L³×Ʃv²/N

P=1/3ρ∑v²/N

Here, M=total mass of the gas

And ρ=density of the gas

Now, Ʃv²/N is written as v² and is called mean square speed.

P= 1/3ρv²

So, this is what is the pressure exerted by gas.