Gas Laws

Ideal Gas Law

The definition of the ideal gas:

  1. Atoms and molecules are idealized as mathematical points (their volume is neglected compared to the total volume of the gas)
  2. All collisions between atoms or molecules are perfectly elastic
  3. There are no intermolecular attractive forces

 

The equilibrium states of a simple, compressible substance can be specified in terms of its pressure (P), volume (v) and temperature (T). If any two of these state variables is specified, the third is determined. This implies that the states of the substance can be represented as a surface in a three dimensional PvT space.

 

Details

 

The ideal gas law

 

Pv = n RT

                         P: pressure = force / surface: 1 Pa = 1 N / 1 m2 = 1 kg / m s2

(101.325 kPa = 1 atm)

                         v: volume  m3

                        n: mol = m / M

                        m: mass g

                        M: molar mass g/mol

                        R: constant = 8.3143 J / mol K

                        T: temperature en Kelvin, 0ºC = 273.1 K   Kelvin temperature scale

The ideal gas law is generally valid at low pressure, large volume and high temperature.

 

All the possible states of an ideal gas can be represented by a PvT surface as illustrated below. The behavior when any one of the three state variables is held constant is also shown. Pi, vi, and Ti are the Pf, vf, and Tf are the final values.

 

 

 

See details in:

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/idegas.html

 

Real Gases

 

The Ideal Gas Law is invalid:

At low temperature, high pressure, and small volume

van der Waals Equation of State

The ideal gas law treats the molecules of a gas as point particles with perfectly elastic collisions. This works well for gases in many experimental circumstances (low P, large v, high T). But gas molecules are not point masses, and there are circumstances (high P, small v, low T) where the properties of the molecules have an experimentally measurable effect. A modification of the ideal gas law was proposed by Johannes D. van Waals in 1873 to take into account molecular volume and molecular interaction forces. It is usually referred to as the van der Waals equation of state:

             [p + a (n/V)2] [V – n b] = n RT                     (1)

 

where nb is approximately the volume occupied by the molecules themselves,

 

The pressure can be expressed as:        p = nRT / [V – n b] - a (n/V)2

 

The pressure exerted by the gas on the walls of the container is related to the number and frequency of the collisions with the wall. These are both reduced by attractive forces between the molecules and this is included in the -a(n/V)2 term.

 

The numeric values of the constants a and b are determined experimentally.

Table 1: Constants de van der Waals

gas

a [dm6 atm / mol2]

b [dm3 / mol]

ammonia (NH3)

4,170

0,0371

nitrogen (N2)

1,345

0,0391

Carbon dioxide (CO2)

3,64

0,0427

Methane (CH4)

2,283

0,0428

 

Another units:

 

van der Waals Coefficients

Gas

a (Pa m6/mol2)

b(m3/mol)

Helium

3.46 x 10-3

23.71 x 10-6

Neon

2.12 x 10-2

17.10 x 10-6

Hydrogen

2.45 x 10-2

26.61 x 10-6

Carbon dioxide

3.96 x 10-1

42.69 x 10-6

Water vapor

5.47 x 10-1

30.52 x 10-6

 

 

 

 

 

 

 

 

 

 

 

Data from Fishbane, et al.

van der Waals calculator

Gaseous Equation of State Calculator

Home work assignment: Calculate and draw the van der Waals p-V curves for a given gas at 300 and 400 K (use Excel).