Combined gas law, ideal gas equation and universal gas constant

By combining Boyle's and Charles' laws, an equation can be derived that gives the simultaneous effect of the changes of pressure and temperature on the volume of the gas. This is known as combined Ideal Gas Equation. A simple and direct method of deriving this equation is as follows:

According to Boyle's law, for a given mass of a gas at constant temperature

According to Charles' law,

Combining (i) and (ii)

or

or

Now, if V₁ is the volume of a gas at temperature T₁ and pressure P_1. V₂ is the volume of same amount of gas at temperature T₂ and pressure P₂, then

The above relationship is very useful for converting the volume of a gas from one set of conditions to another.

The numerical value of the constant of proportionality (K) depends upon the quantity of gas. The volume of a gas is directly proportional to the number of moles of gas at constant temperature and pressure (Avogadro's law). This means that 'K' is directly proportional to the number of moles, 'n', i.e.,

K n

or K = nR,

where 'R' is the universal gas constant of proportionality. The value of 'R' is same for all gases.

However, the numerical value of 'R' varies with the units in which pressure and volume are expressed. Therefore the Ideal gas Equation is derived as:

For one mole (n = 1),

PV = RT (Ideal Gas Equation)

The Ideal Gas Equation is also known as the equation of state for gases as it expresses the quantitative relation ship between the four variables that describe the state of the gas. The word 'ideal' is used here because in reality no gas obeys the above condition and the gases, which deviate from ideality are called as real gases.

Universal gas constant (R)

From the above ideal gas equation:

The universal gas constant is a measure of energy change (work done) per mole of the gas for one degree change in its temperature.

Numerical Value of R

The magnitude and unit of 'R' depends upon the units in which pressure, volume and temperature are expressed.

Conditions and problems

I)When pressure is expressed in atmosphere and volume in litres

Under standard condition of temperature and pressure i.e., when

P = 1 atm

T = 273.15 K

V = 22.414 L mol^-1

This gives,

II

) When pressure is expressed in atmosphere, and volume in mL

Here,

P = 1 atm

T = 273.15 K

V = 22414 mL mol^-1

This gives,

III) R in energy units

When,

If 'R' is to be expressed in CGS units, the unit of pressure should be dyne cm^-2 and volume in cm³mol^-1. Then,

Normal temperature T = 273.15 K

Normal pressure, P = (1 x 76 x 13.6 x 981) dyne cm^-2

Molar volume under normal temperature and pressure

= 22414 cm³mol^-1

Therefore,

= 8.314 x 10⁷ erg deg^-1 mol^-1

Since, 4.182 x 10⁷erg = 1calorie

Hence,

IV) In SI units

In the SI system of units, under NTP conditions,

P = 101325 N m^-2

T = 273.15 K

V = 22.414x10^-3 m³

Therefore,

= 8.314 J mol^-1 K^-1

The value of Gas constant R in different units

Problems

5. Calculate the number of moles of hydrogen (H₂) present in a 500 cm³ sample of hydrogen gas at a pressure of 760 mm of Hg and 27°C.

Solution

According to ideal gas equation, PV = nRT

T = 27 + 273 = 300 K, R=82.1 cm³ atm K^-1 mol^-1

n = 0.0203 mol = 2.03 x 10^-2 mol.

6. About 200 cm³of a gas is confined in a vessel at 20°C and 740 mm Hg pressure. How much volume will it occupy at S.T.P.?

Solution

We are given

P₁ = 740 mm Hg P₂ = 760 mm Hg

T₁ = 20 + 273 = 293K T₂ = 273 K

V₁ = 200 cm³ V₂ = ?

According to gas equation,

Substituting the values, we get

= 181.4 cm³.

7. Calculate the volume occupied by 2 moles of an ideal gas at

2.5 x 10⁵ Nm^-2 pressure and 300 K temperature.

Solution

According to ideal gas equation,

n = 2 mol, T = 300 K, P = 2.50 x 10⁵ Nm^-2

R = 8.314 Nm K^-1 mol^-1

= 19.95 x 10^-3m³=19.95dm³.