Ideal gas

 

Gases are conveniently classified into two types, namely, (a) ideal gases, and (b) nonideal or real gases. An ideal gas is one that obeys certain laws which will be dealt with shortly, while a real gas is one that obeys these laws at low pressure and relatively high temperatures. In ideal gases, the volume occupied by the molecules themselves is negligible as compared with total volume at all pressures and temperatures. The intermolecular attraction is almost absent under all conditions. In case of real gases, both these factors are appreciable and the magnitude of each depends on the temperature and pressure of the gas and nature of the gas.

Ideal Gas Behavior

 Boyle’s Law

Robert Boyle in 1660 performed a series of experiment in which he determined the effect of pressure on the volume of a given amount of air at a constant temperature. He found that the volume of any define quantity of a gas at constant temperature varied inversely to the pressure of the gas. Mathematically,

V ∝ 1/P (Temperature and mass of the gas constant)

Or, PV = Constant

Where V is the volume and P is the pressure of the gas. Thus if V₁ is the volume occupied by a given quantity of the gas at pressure P₁ and V₂ is its volume when pressure changes to P₂, then at constant temperature and mass it follows that

P₁V₁ = P₂V₂

P₁/P₂ =V₂/V₁

When the pressure of the gas is plotted against the volume, we obtain a hyperbola curve. These curves are known as isotherm. These curves are known as isotherms. The upper curve corresponds to the higher temperature.

P-V plots of ideal gas in different temperature

Charles’ Law

In 1787, Charles studied the variations in volume of a gas with temperature at constant pressure and observed that the volume of a certain mass of the gas increases or decreases by 1/273.16 of its value at 0^oC. If V_o is the volume of a gas at 0^oC and V_t its volume at t^oC, then mathematically,

V_t =V_o + (t/273.16)V_o

V_t = V_o(1+t/273.16)

V_o = (273.16+t/273.16)

We may define now a new temperature scale such that any temperature‘t’ on this scale will be given by T = 273.16+t, and 0^oC by T₀=273.16. Then,

V_t = V_o(T/T_o)

V_t/V_o =T/T_o

Or in general,

V₂/V₁ = T₂/T₁

This new temperature scale is known as the absolute or Kelvin scale of temperature and is of fundamental importance in all sciences. In terms of this scale predicts that the volume of a definite quantity of a gas at constant pressure is directly proportional to the absolute temperature. So mathematically,

V ∝ T

V/T = Constant

The volume of a gas should be a linear function of absolute temperature at constant pressure. Such a plot of V versus T at two pressure P₁ and P₂ (P₂ > P₁) is drawn as follows.

Each constant pressure line is called an isobar. For every isobar the slope is greater, lower the pressure. If we wish to cool a gas to 0K (-273.16^oC), its volume should become zero. However, no such phenomenon is ever encountered, for usually long before 0K is approached the gas first liquefies and then solidifies.

If the volume of a gas is maintained constant and its temperature is raised the pressure will increase. The increase in pressure per degree rise in temperature relative to its pressure P_o at 0^oC is again found to be (1/273.16)P_o. Hence the pressure P_t of the gas at temperature t is given as

P_t =P_o(1+t/273.16)

    = P_o(273.16+t/273.16) = P_oT/T_o

Or P_t/P_o = T/T_o

It is clear from this equation that P is directly proportional to the absolute temperature at constant volume of the gas. One can therefore write,

P ∝ T  or  P/T = constant

V-T and P-T plots of a ideal gas at different pressure and volume respectively

Avogadro’s Law: Avogadro, in 1811 suggested that ‘equal volume of different gases at the same temperature and pressure contain equal number of molecules’. It will be recalled here that 1 mole of any substance contain the same number of molecules and this number is known as Avogadro constant. It is represented by N_A and has a value of 6.023X10²³.  Hence, we can write

V∝ n (at constant temperature and pressure)

Where n is the number of moles of the gas

Equation of state of an Ideal gas:

Boyle’s law, Charles’ law and Avogadro’s law can be combined to give a general relation between the volume, pressure, temperature and the number of moles of a gas.

PV = Constant (T constant)

V/T = Constant (P constant)

On combining these equations, we get

PV/T = Constant

The combined equation relating to the variables P, V and T of an ideal gas is known as the equation of state. It is clear from equations that the product PV divided by T is always constant for all specified states of the gas. Hence, if we know these values for any state the constant can be computed. In the standard state (STP), the pressure is 1 atm and temperature 273.16 K. The volume occupied by a mole of an ideal gas under these conditions is 22414 cm³ (22.414 litres). According to Avogadro’s Law this volume is the same for all ideal gases. If we consider n moles of an ideal gas at STP the equation becomes

PV/T = nR

Or, PV = nRT

Here R is a universal constant known as the gas constant per mole. This formula is known as ideal gas equation and it connects directly to the volume, temperature, pressure and the number of moles of a gas and permits all types of calculations as soon as the constant R is known. 

 Here is an ideal gas calculator which calculate P, V, n and T with the help of PV = nRT equation, from this calculator one can calculate any unknown value by inserting 3 known value. The values will be determined in SI method, So if anyone has pressure and volume value in different units they can convert it using the converter.

Dalton’s Law of Partial Pressure

The law states, “if two or more than two gases which do not react chemically at constant temperature are enclosed in a vessel, the total pressure exerted by the gaseous mixture is the sum of the individual partial pressure which each gas will exert if present alone in that volume.” If P₁, P₂, P₃…….P_n are the partial pressures of the individual gases in a mixture then according to the Dalton’s Law of partial pressure the total pressure, P is given by

P = P₁ + P₂ +P₃ +..…..+P_n

The ideal gas equation can be applied to each component of the gas

P₁ = n₁RT/V

P₂ = n₂RT/V

P₃ = n₃RT/V

P = n₁RT/V + n₂RT/V + n₃RT/V +……..

P = (n₁+n₂+n₃+….)RT/V

P = n_tRT/V

Where n_t = n₁+ n₂+ n₃…..n_n and is the total number of moles of the gas mixture in volume V.

The above equation shows that the expression PV = n_tRT can be used for mixture of gases as well as for pure gases.

So from this equation we get

P₁ = (n₁/n_t)P

P₂ = (n₂/n_t)P

P₃ = (n₃/n_t)P

These equations are important as they relate the partial pressure of a gas to the total pressure of the gaseous mixture. Since the fractions n₁/n_t, n₂/n_t, n₃/n_t represent the moles of a particular constituent present in the mixture divided by the total number of moles of all the gases present; these quantities are called the mole fractions and are called the mole fractions and are denoted by X₁, X₂, X₃, etc., respectively. A noteworthy property of mole fraction is that the sum of the mole fraction of all the components of the system is unity,

X₁ + X₂ + X₃ + ….. = 1

Reference

1) Peter Atkins, Julio de Paula, Atkins’ physical chemistry book, eighth edition.