States of Matter and Eudiometry

Matter exists in three broad states, influenced by two opposing tendencies among its particles:

• Intermolecular attractive forces.
• Molecular motion or random motion.

Bulk properties like boiling point and wetting properties result from the collective behaviour of numerous atoms, ions, or molecules. Despite consistent chemical composition across states, physical properties differ due to varying molecular energies and aggregation. Chemical properties stay constant, but reaction rates can vary.

1.0The Gaseous State

Below are few important general Properties of Gases:

• Gases lack a definite shape and volume.
• Gases expand to fill the entire space available to them.
• Gases exhibit high compressibility and unlimited dispensability.
• Due to negligible intermolecular forces, gases have very low densities.
• Gases exert pressure on the walls of their container through perfectly elastic collisions.
• Gases diffuse rapidly through each other to form homogeneous mixtures, irrespective of electric and gravitational fields.

The characteristics of gases are typically described using the following four parameters:

• Mass
• Volume
• Pressure
• Temperature

2.0The Gas laws

Today's gas laws stem from centuries of research on gas properties. Robert Boyle's 1662 measurements led to Boyle's Law. Inspired by hot air balloon flights, Jacques Charles and Joseph Louis Gay-Lussac discovered further gas laws. Contributions from Avogadro and others expanded our understanding of gases.

Boyle's Law

In 1662, Robert Boyle established a relationship between the pressure and Volume of a fixed amount of gas at a constant temperature. He observed that the product of pressure and Volume of a fixed amount of gas remained approximately constant. Therefore, Boyle's Law states:

"At constant temperature, the pressure of a fixed amount of gas varies inversely with its volume."

Mathematically, Boyle's Law is expressed as:

$P\propto \frac{1}{V}$

or $P=k_1\frac{1}{V}$ ; where k_1​​ is a proportionality Constant.

This equation implies that at constant temperature, the product of pressure (P) and volume (V) of a fixed amount of gas remains constant. In other words:

PV = k₁ = Constant

If a set quantity of gas, maintained at a constant temperature T, occupies volume V₁ at pressure P₁ and then undergoes expansion to volume V₂​ with pressure P₂​, in accordance with Boyle's Law

𝑃₁𝑉₁ = 𝑃₂𝑉₂ = Constant

Charles' Law

A fixed amount of gas at a constant pressure volume occupied by the gas is directly proportional to its temperature on the absolute temperature scale.  

V ∝ T  or  V = kT 

$\frac{V_1}{N_1}=\frac{V_2}{N_2}\text{ or }\frac{V_1}{n_1}=\frac{V_2}{n_2}$

$\mathrm{V}={\mathrm{V}}_0+\mathrm{bt}and\mathrm{V}={\mathrm{b}}^{\prime }\mathrm{T}$

$\frac{V}{T}=\text{ constant }$; where ‘k’ is a proportionality constant and is dependent on the amount of gas and pressure. 

And, t = temperature on centigrade scale.                

T = absolute temperature (K)

V₀ = volume of gas at 0°C.                                        

b and b' = constants

Important Points:

• Volume is directly proportional to absolute temperature. At absolute zero, the gas volume should be zero.
• No substance exists as a gas near zero, but volume can be extrapolated to zero. Absolute zero is theoretically unreachable but approachable.
• Kelvin devised the Kelvin scale, setting -273.15°C as the lowest approachable limit.

Gay-Lussac's law :

When the volume of a gas remains constant, its pressure exhibits a direct proportionality with its temperature on the absolute scale, such as the Kelvin or ideal gas scale.

𝑃 ∝ 𝑇

This relationship can be expressed as:

𝑃 = 𝑘⋅𝑇 ; where k is a constant.

Note: Originally, the law was developed using the Celsius scale, where it was observed that pressure is a linear function of temperature. This relationship is expressed as:

𝑃 = 𝑃₀ + 𝑏𝑡  ; where 'b' is a constant and 𝑃₀ is the pressure at zero degrees Celsius.

Avogadro’s Law

In 1812, Amadeo Avogadro proposed that "Samples of different gases containing the same number of molecules, regardless of complexity, size, or shape, occupy the same volume at the same temperature and pressure."

Avogadro’s Law states that the volume of a gas is directly proportional to the number of moles of gas when temperature and pressure are kept constant.

Mathematically, this relationship is expressed as:

𝑉 ∝ 𝑛   ; where 'V' is the volume of the gas and 'n' is the number of moles of gas

3.0Ideal Gas Equation 

The Ideal Gas Equation combines Boyle's, Charles', and Avogadro's Laws into a single equation. It describes the simultaneous effects of temperature and pressure changes on the volume of a given amount of gas.

Boyle's Law:

$\mathrm{P}\propto \frac{1}{V}$

$\mathrm{V}\propto \frac{1}{P}$ ; (at constant T)

Charles's Law:

V ∝ 𝑇 (at constant P)

Combining Boyle's and Charles's Laws:

$\mathrm{V}\propto \frac{T}{P}$

$\frac{PV}{T}=\mathrm{constant}(\mathrm{R})$

For 1 mole of gas:

PV = RT

For 'n' moles of gas:

PV = nRT

Universal Gas Constant (R):

R = 0.0821 atm litre mol⁻¹ K⁻¹

R = 8.314 J mol⁻¹ K⁻¹

Note: The plot of pressure versus temperature (in Kelvin) for a fixed mass of gas at constant volume is a straight line.

The volume occupied by 1 mole of an ideal gas under standard temperature and pressure (STP) conditions is 22.7 L . 

4.0Dalton’s Law of Partial Pressures

Partial pressure is the pressure exerted by an individual gas in a mixture. Dalton's law states that the total pressure of non-reacting gases equals the sum of their partial pressures. 

Mathematically, it's expressed as

• P_total = p₁ + p₂ + p₃+…….      

When dealing with a gas mixture in contact with water, subtracting the aqueous tension from the total pressure gives the pressure of the dry gas:

• p_dry gas = p_total − aqueous tension

5.0Kinetic Molecular Theory of Gases 

This Theory states that:

• Gases consist of numerous identical particles (atoms or molecules) that are extremely small and act as hard spheres.
• The actual volume of gas particles is negligible compared to the space between them, so they are treated as point masses.
• Interactions between gas particles are negligible.
• Gas particles are in constant and random motion, with collisions between them being perfectly elastic.
• The average kinetic energy of gas particles rises in direct correlation with the absolute temperature.
• Gas pressure arises from collisions between gas particles and the walls of the container.

6.0Behaviour of Real Gases

Gas liquefaction occurs when gases are subjected to very low temperatures and high pressures. Under these conditions, gas molecules are forced to come close together, leading to significant molecular interactions.

In real gases, molecules are so close at high pressures that intermolecular forces become significant. Due to the attraction forces between them, gas molecules do not hit the container walls with full force. Consequently, the pressure exerted by the gas is lower than expected based on ideal gas behaviour.

This departure from ideal behavior is quantified using the compressibility factor Z, where $Z=\frac{pV}{nRT}$

For Ideal Gases:

Z = 1 at all temperatures and pressures. 

However, for real gases, Z deviates from unity. At very low pressures, most gases behave ideally (Z = 1), but at high pressures, Z exceeds 1, making compression more challenging.

7.0Liquefaction Of Gases

At high temperatures, gases behave like ideal gases and cannot be liquefied even at very high pressures, as noted by Andrews. As temperature drops, the isotherms deviate significantly from ideal behaviour.

Carbon dioxide remains in a gaseous state until it reaches 73 atmospheres at 30.98°C (point E), representing its critical temperature. At this specific pressure and temperature, liquid carbon dioxide begins to form. Critical volume and pressure represent the volume occupied by a single mole of a gas at its critical temperature.

• Critical temperature (Tc): The temperature at which a gas liquefies, given by 

${\mathrm{T}}_{\mathrm{c}}=\frac{8a}{27}\mathrm{Rb}$

• Critical volume (V_c): The volume of one mole of a gas at its critical temperature, calculated as V_c = 3b.
• Critical pressure (P_c): The pressure of a gas at its critical temperature, determined by $P_c=\frac{a}{27}{b}^2$.

For monatomic gases, C_P = 5 cal and C_V = 3 cal, resulting in γ = 5/3 = 1.67.

For diatomic gases, C_P = 7 cal and C_V = 5 cal, yielding γ = 7/5 = 1.4.

For polyatomic gases, C_P = 8 cal and C_V = 6 cal, giving γ = 8/6 = 1.33.

8.0Liquid State

When a partially filled container is evacuated and a liquid is introduced, some evaporates to fill the remaining space with vapour. Initially, the pressure inside the container rises as vapour forms and exerts pressure on the walls. Eventually, equilibrium is reached between liquid and vapour phases, known as saturated vapour pressure or equilibrium vapour pressure.

9.0Vapour Pressure

When a partially filled container is evacuated, and a liquid is introduced, some of it evaporates to fill the remaining space with vapor. Initially, the pressure inside the container rises as vapor forms and exerts pressure on the walls. Eventually, equilibrium is reached between the  liquid and vapor phases, known as saturated vapor pressure or equilibrium vapor pressure.

10.0Surface Tension 

Surface tension refers to the inclination of liquid surfaces to minimize their area while at rest.

Within the bulk of a liquid, molecules experience balanced intermolecular forces from all directions, resulting in no net force acting on individual molecules. However, molecules at the liquid's surface encounter a net attractive force directed towards the interior of the liquid.

11.0Viscosity

Viscosity, a key characteristic of liquids, arises from internal friction between fluid layers as they move during flow. It represents the liquid's resistance to flow.

Strong intermolecular forces, like van der Waals and hydrogen bonding, hinder fluid layer movement, leading to higher viscosity. Glass, for instance, exhibits high viscosity due to these forces.

Viscosity determines flow rate; higher viscosity means slower movement. As temperature increases, molecules gain kinetic energy, overcoming intermolecular forces and reducing viscosity.

12.0Eudiometry

Eudiometry, also known as gas analysis, involves calculations based on reactions involving gases, where the amounts of gases are represented by their volumes measured at the same pressure and temperature. Several assumptions aid in these calculations:

Gay Lussac's Law of Volume Combination: This law states that the volumes of gaseous reactants and products bear a simple ratio when measured at the same temperature and pressure. For instance, in the reaction:

N₂(𝑔) + 3H₂(𝑔) → 2NH₃(𝑔)

1 volume of N₂ reacts with 3 volumes of H₂ to produce 2 volumes of NH₃.

Negligible Volume of Solids or Liquids: The volumes of solids or liquids are considered negligible compared to gases due to the vast difference in volume occupied by substances in different states. For example, in the reaction:

2H₂₍𝑔)+O₂(𝑔)→2H₂O(𝑙)

2 volumes of H₂ react with 1 volume of O₂ to produce 0 volumes of H₂O.

Air Composition: Air is treated as a mixture of oxygen and nitrogen since it constitutes about 99% of air volume.

Nitrogen's Non-Reactivity: Nitrogen gas is considered non-reactive under typical laboratory conditions due to its high thermal stability. Hence, it's assumed not to participate in reactions in eudiometry.

Amagat's Law states that the total Volume of a non-reacting gas mixture is the sum of the partial volumes of its components. Each gas's partial Volume is the Volume it would occupy if it alone exerted the entire pressure of the mixture.

Solvent Absorption: Gases produced in reactions are often absorbed by specific solvents.  For example:

• KOH absorbs CO₂, SO₂, Cl₂
• Ammoniacal Cu₂Cl₂ absorbs CO
• Turpentine oil absorbs O₃
• Alkaline pyrogallol absorbs O₂
• Water absorbs NH₃, HCl
• CuSO₄/CaCl₂ absorbs H₂O

Eudiometer: This laboratory device measures changes in gas mixture volume following a physical or chemical change.

Eudiometry is thus a systematic approach to analyze gases in reactions, relying on these assumptions and principles to make accurate calculations.

13.0Solved Problems

Question 1. A 3L gas mixture of propane (C₃H₈) and butane (C₄H₁₀) on complete combustion at 25°C produced 10L CO₂. Assuming constant P and T conditions, what was the Volume of butane present in the initial mixture ?

Ans.  Solution:

$\underset{xL}{{\mathrm{C}}_3{\mathrm{H}}_8(\mathrm{g})}+5{\mathrm{O}}_2\to \underset{3\mathrm{XL}}{3{\mathrm{CO}}_2(\mathrm{g})}+4{\mathrm{H}}_2\mathrm{O}(\mathrm{I})$

$\begin{array}{lc}{\mathrm{C}}_4{\mathrm{H}}_{10}(\mathrm{g})+& \frac{13}{2}{\mathrm{O}}_2\to 4{\mathrm{CO}}_2(\mathrm{g})+5{\mathrm{H}}_2\mathrm{O}(\mathrm{l})\\ (3-\mathrm{x})\mathrm{L}& 4(3-\mathrm{x})\mathrm{L}\end{array}$

from question 3x + 4 (3 –x) = 10 ⇒ x = 2

∴ Volume of butane, C₄H₁₀ = (3 – x)  ⇒ 1L

Question 2. How can the function 𝑃𝑉𝑅𝑇 demonstrate non-ideal gas behavior, particularly at elevated pressures?

Ans. According to the ideal gas equation 𝑃𝑉 = 𝑛𝑅𝑇  the ratio𝑃𝑉𝑅𝑇​equals the number of moles n of an ideal gas. For an ideal gas, this ratio should remain constant regardless of changes in pressure, as the number of moles n is fixed. However, if𝑃𝑉𝑅𝑇changes with pressure, it indicates that the gas is not behaving ideally. This deviation occurs due to the real gas effects, such as intermolecular forces and the finite volume of gas molecules, which become significant at high pressures.