3.1.2 Amount of Substance - The ideal gas equation

Specification

Students should:
  • be able to recall the ideal gas equation pV = nRT and be able to apply it to simple calculations in S.I. units, for ideal gases

The Ideal gas equation

The equation of state refers to a fixed mass of gas. From Avogadro's law we know that the same volume of all gases contain the same number of moles and from this, it follows that the volume is proportional to the number of moles.

Volume ∝ number of moles (n)

These two equations can be combined to obtain an expression involving all the quantities:

After rearrangement, for 'n' moles of gas the proportionality constant is called the Universal Gas Constant and is given the symbol 'R'

This gives the ideal gas equation:

PV = nRT

where:

It is often more convenient to express the pressure in kPa and the volume in litres (dm3). This leaves the value of R the same (see below).

Example: Calculate the number of moles of gas present in 2.6 dm3 at a pressure of 1.01 x 105 Pa and 300 K.

PV = nRT

2.6 dm3 = 0.0026 m3

0.0026 x 1.01 x 105 = n x 8.314 x 300

n = 0.0026 x 1.01 x 105 / 8.314 x 300

n = 0.105 moles

There are several units used for gas volume, gas pressure and temperature. It is important to be consistent with the use of units when carrying out gas law calculations.


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Universal gas constant - R

Although called "Universal", its value depends on the units used for P, V and T.

With the SI units of metres, kilograms, Kelvin and Joules, using P, V and T values at STP gives:

PV=nRT

therefore: R=PV/nT

for 1 mole of gas at STP (using accepted values of P = 1.01325 x 105 Pa, V = 0.022414 m3, T = 273.15 K)

R = (1.01325 x 105) x 22.414)/273.15

R = 8.314 J K-1 mol-1

In chemistry, the units of volume used are the decimetre cubed (dm3) and pressure in kiloPascals (kPa), so one unit is 100x greater and the other 100x smaller than the SI equivalent. Consequently the differences in the product, PV, both cancel out (multiplying AND dividing by 1000), so that the final value for R is the same as in SI units.

The Universal gas constant, R, calculated using atmospheres Pressure and volume in litres, then:

There are, of course, many other values of R as there are many ways of measuring both the volume and the pressure of a gas.


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SI units and 'R'

The SI units of P, V and T give rise to the previously used value for the universal gas constant, R = 8.314 J K-1 mol-1.

How does this happen when chemists do not use these SI units?

Remember:

1 litre = 1 dm3 = 1000cm3

Consequently, if litres are used in the Ideal Gas equation then the pressure units must also be divided by 1000 (as PV = constant). Pressure is measured in Pa or Nm-1, and so the unit of the kPa correct for the difference in volume units.

Atmospheric pressure in Pa = 1.00 x 105 Pa

Atmospheric pressure in kPa = 1.00 x 102 kPa

Provided that you are consistent with the application of units there will be no problem. It is always a good idea when carrying out calculations to look at the value of your answer and ask yourself, "does it seem reasonable?"


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quick test