States of Matter I: Gases

Gas laws describe the macroscopic behavior of gases through relationships between pressure (P), volume (V), temperature (T), and moles (n).

Boyle’s Law: The Pressure-Volume Relationship in Gases

Definition

Boyle’s Law states that the volume of a fixed amount of gas is inversely proportional to its pressure when temperature is held constant.

Mathematical Expression

P ∝ 1/V

PV = k (where k is a constant)

or

P₁V₁ = P₂V₂ = k

Conditions

• Temperature (T) must remain constant
• The amount of gas (n) must not change
• Applies to ideal gases (no intermolecular forces, negligible particle volume)

Experimental Verification

Robert Boyle verified this law using a J-shaped tube with mercury. Modern verification uses a cylinder with a movable piston:

Pressure (atm)	Volume (dm³)	P × V (dm³·atm)	Observation
2.0	1.0	2.0	Initial state
4.0	0.5	2.0	Pressure doubled → volume halved
6.0	0.33	2.0	Pressure tripled → volume reduced to 1/3

Conclusion: The product PV remains constant (k = 2.0 dm³·atm in this experiment), verifying Boyle’s Law.

Example Calculation

Problem: A gas occupies 10.0 dm³ at 2.5 atm and 0°C. What is its volume at 2.0 atm (temperature constant)?

Solution

Given:

• P₁ = 2.5 atm
• V₁ = 10.0 dm³
• P₂ = 2.0 atm
• T = constant (0°C = 273 K)

Apply Boyle’s Law:

P₁V₁ = P₂V₂

V₂ = (P₁V₁)/P₂ = (2.5 atm × 10.0 dm³)/2.0 atm = 12.5 dm³

Answer: The new volume is 12.5 dm³.

Graphical Representation

A graph with pressure on the x-axis and volume on the y-axis creates a curve called an isotherm, where “iso” means the same and “therm” means heat.

When the gas temperature is increased and held constant while varying pressure and volume, the isotherm moves away from both axes due to the increase in gas volume with higher temperatures. Further temperature increases will result in additional isotherms that are positioned even further from the axes.

Kinetic Molecular Theory Explanation

Boyle’s Law can be explained at the molecular level:

1. Gas particles are in constant random motion, colliding with container walls (creating pressure).
2. When volume decreases (compression):
   • Particles are confined to a smaller space
   • Collisions with walls become more frequent
   • Result: Higher pressure
3. When volume increases (expansion):
   • Particles spread out
   • Collisions with walls become less frequent
   • Result: Lower pressure
4. Temperature remains constant: Average kinetic energy of particles doesn’t change

Practical Applications

1. Syringes:
   • Pulling the plunger increases volume → decreases pressure → draws fluid in
   • Pushing the plunger decreases volume → increases pressure → expels fluid
2. Scuba Diving Tanks:
   • Air is compressed to a high pressure (small volume) for storage
   • As a diver uses air, pressure decreases, and volume remains constant
3. Breathing:
   • Diaphragm expands chest cavity (↑V) → ↓P in lungs → air flows in
   • Diaphragm contracts (↓V) → ↑P in lungs → air flows out

Limitations of Boyle’s Law

Boyle’s Law applies perfectly only to ideal gases. Real gases show deviations:

• At high pressures:
  • • Gas particles occupy a significant volume
    • Intermolecular forces become noticeable
  • The actual volume is greater than predicted
• At low temperatures:
  • Intermolecular attractions become significant
  • Actual pressure is less than predicted
• Near condensation points:
  • Gas begins to liquefy
  • The law completely breaks down

These deviations are accounted for in the van der Waals equation.

Historical Context

Robert Boyle (1627-1691) published this law in 1662 after experiments with air trapped in a J-shaped glass tube sealed with mercury. He observed volume changes by adding mercury to increase pressure, establishing the inverse relationship.

What is Charles’s Law?

Charles’s Law is a fundamental principle in chemistry that describes how gases expand or contract when their temperature changes, provided the pressure remains constant.

Statement:

At constant pressure, the volume of a given mass of gas is directly proportional to its absolute temperature (in Kelvin).

V ∝ T

V = kT (where k is a constant)

V/T = k

or

= k

Direct proportionality means that if the temperature doubles, the volume will also double, provided that the pressure remains constant.

In simple words:

• If you heat a gas, it expands.

• If you cool a gas, it shrinks.

This happens because heating causes gas particles to move faster, hitting the container walls more forcefully and pushing outward, increasing the volume.

Charles’s Law Formula

The mathematical form is:

Where:

• V₁​ = Initial Volume

• T₁​ = Initial Temperature (in Kelvin)

• V₂​ = Final Volume

• T₂​ = Final Temperature (in Kelvin)

Important Note:

• Temperature must always be in Kelvin (K).

• To convert Celsius (°C) to Kelvin (K), use:

K = °C + 273K

Kinetic Molecular Explanation

According to the Kinetic Molecular Theory (KMT):

• Gas particles move randomly and collide with each other and the walls of the container.

• When the temperature increases:

  • The particles gain kinetic energy.

  • They move faster.

  • They strike the container walls harder and more frequently.

  • The gas expands to maintain constant pressure.

Thus, volume increases with temperature.

Graphical Representation of Charles’s Law

Volume vs. Temperature Graph

• • A straight line is obtained when Volume (V) is plotted against Temperature (T in Kelvin).

  • This line passes through the origin (0,0) — meaning zero volume at zero Kelvin.

  • If extended backward, the line meets the temperature axis at –273°C.

Concept of Absolute Zero

What is Absolute Zero?

Absolute Zero is the theoretical temperature at which the volume of an ideal gas would become zero if it continued to contract indefinitely as the temperature decreased.

It corresponds to –273.16°C or 0 Kelvin.

At absolute zero:

• Gas particles would have no kinetic energy.
• Their motion would completely stop.
• However, in reality, gases condense into liquids or solids before reaching absolute zero, so a gas never truly reaches zero volume in practice.

Quantitative Definition of Charles’s Law

Charles’s Law can be quantitatively defined as:

“At constant pressure, the volume of a given mass of a gas increases or decreases by 1/273 of its original volume at 0°C for every 1°C rise or fall in temperature, respectively.”

Breaking it down:

• Suppose a gas has a volume V₀ at 0°C.
• When the temperature increases by 1°C, the volume increases by V₀/273.
• When the temperature decreases by 1°C, the volume decreases by V₀/273.

Thus, for a temperature t (in °C), the volume V_t is given by:

V_t = V₀ (1 + t/273)

where:

• V_t = Volume at temperature t°C
• V₀ = Volume at 0°C

Relationship to Absolute Zero

If you continue lowering the temperature, the gas volume will theoretically decrease steadily by 1/273 of its volume at 0°C for each degree Celsius drop.

By extrapolating this behavior, the volume would reach zero at –273°C.

This temperature, –273°C, is known as absolute zero — the point where all molecular motion would theoretically cease.

Thus, Charles’s Law naturally leads to the concept of absolute zero.

Importance of Absolute Zero and the Kelvin Scale

• Absolute Zero is taken as 0 Kelvin (0 K) on the Kelvin temperature scale.
• The Kelvin scale starts from absolute zero, ensuring all temperatures are positive, and is used in scientific measurements.
• Temperatures in Kelvin are directly proportional to the average kinetic energy of gas molecules.

Experimental Evidence for Absolute Zero

Experimental studies of the volume-temperature relationship of gases show that when temperature is plotted against volume:

• The line extrapolates backward and cuts the temperature axis at about –273°C.
• This provides strong support for the concept of absolute zero.

🌟 Thus, Absolute Zero represents the fundamental limit of physical behavior and underpins modern thermodynamics and kinetic theory.

Real-World Examples of Charles’s Law

• Hot Air Balloons: Air inside the balloon is heated, expanding, and making the balloon rise.

• Car Tire Pressure: In hot weather, the air inside expands, increasing tire pressure.

• Syringe Use: When the syringe is heated, the air inside expands, pushing the plunger outward.

• Basketball: A basketball in the cold shrinks slightly because the air inside contracts.

Key Takeaways:

• Charles’s Law links volume and temperature of a gas.

• Directly proportional relationship: as temperature ↑, volume ↑.

• Always use Kelvin temperature.

• Absolute Zero is –273°C, where particle motion theoretically stops.

Avogadro’s Law

Statement of Avogadro’s Law

Avogadro’s Law states that:

“Equal volumes of all ideal gases at the same temperature and pressure contain the same number of molecules (or moles).”

In other words, if you take the same volume of any ideal gas under identical temperature and pressure conditions, they will contain the same number of gas molecules, regardless of the type of gas.

Mathematical Expression

Avogadro’s Law can be mathematically expressed as:

V₁ / n₁ = V₂ / n₂

or simply,

V / n = k (where k is a constant at constant temperature and pressure)

• V = Volume of the gas
• n = Number of moles of gas
• k = Proportionality constant

Connection with Molar Volume at STP

At standard temperature and pressure (STP) — 273.16 K and 1 atm — it has been experimentally determined that:

• One mole of any ideal gas occupies 22.414 dm³ of volume.
• One mole contains Avogadro’s number of molecules, i.e., 6.022 × 10²³ molecules.

This means:

• 22.414 dm³ of any ideal gas at STP contains 6.022 × 10²³ molecules.

For example, if you have 1 dm³ of any gas at STP, the number of molecules will be:

(6.022 × 10²³) ÷ 22.414 ≈ 2.68 × 10²² molecules

This value applies to any gas: whether it’s hydrogen (H₂), helium (He), nitrogen (N₂), oxygen (O₂), or carbon monoxide (CO).

Example

Suppose you have separate containers each containing 1 dm³ of H₂, He, N₂, O₂, and CO gases at STP:

• The number of molecules in each container will be approximately 2.68 × 10²² molecules.

Even though their masses are different (because molecular masses differ), the number of molecules remains the same!

• For instance, 1 dm³ of hydrogen (H₂) at STP weighs approximately 0.0899 g.
• Whereas, 1 dm³ of oxygen (O₂) at STP weighs about 1.4384 g.

The mass differs because oxygen molecules are heavier (O₂ is 16 times heavier than H₂), but the number of molecules per volume is the same!

Why Mass Doesn’t Affect Volume at STP

Even though molecules have different masses, gases at STP behave similarly because their particles are widely separated. The size of the individual molecules becomes negligible compared to the distances between them.

At room temperature, the average distance between two neighboring gas molecules is approximately 300 times the diameter of a single molecule!

Explanation with Kinetic Molecular Theory (KMT)

According to KMT, gas particles are constantly moving and colliding with container walls, creating pressure.

If we add more gas molecules (increase n) into the same volume at constant temperature:

• The number of collisions would increase.
• The pressure would increase.

To maintain the same pressure, the volume must increase so that the number of collisions per unit area remains constant.

This explains why volume is directly proportional to the number of moles at constant temperature and pressure, as stated by Avogadro’s Law.

Key Takeaways:

• Equal volumes of gases at the same temperature and pressure contain the same number of molecules.
• The volume of a gas is directly proportional to the number of moles.
• The nature or type of gas does not affect this relationship.
• The molar volume of any ideal gas at STP is 22.414 dm³ per mole.

🌟 Key Insight: Even though different gases have different masses and molecular sizes, they occupy the same volume at STP if the number of molecules is the same.

Dalton’s Law of Partial Pressures

Definition

Dalton’s Law of Partial Pressures states that:

“The total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of the individual gases.”

Each gas in a mixture behaves independently and contributes to the total pressure according to its pressure, known as its partial pressure.

Mathematical Expression

The law is expressed mathematically as:

P_total = P₁ + P₂ + P₃ + …

• P_total = total pressure of the gas mixture
• P₁, P₂, P₃ = partial pressures of individual gases

The partial pressure of a gas is the pressure that the gas would exert if it were the only gas present in the container at the same temperature.

Explanation with Kinetic Molecular Theory (KMT)

According to the Kinetic Molecular Theory:

• Gas particles move randomly and independently.
• They exert pressure due to collisions with the walls of the container.

Since the particles of different gases do not interfere with each other’s motion in a mixture (assuming no reaction occurs), each gas contributes independently to the total pressure.

Therefore, the total pressure is just the sum of the individual contributions (partial pressures) of all gases present.

Application of Dalton’s Law

Dalton’s Law is useful in many real-world and laboratory scenarios, including:

• Collecting gases over water — The gas collected contains both the gas and water vapor, so the total pressure includes the vapor pressure of water.
• Calculating pressures in gas mixtures — Useful in chemical reactions involving gases.
• Respiratory systems — Oxygen and carbon dioxide behave as components of gas mixtures in the lungs and blood.

Example 1: Calculating Total Pressure

Suppose you have a container with the following gases at a certain temperature:

• O₂: 0.5 atm
• N₂: 0.8 atm
• CO₂: 0.7 atm

Total pressure:

P_total = 0.5 + 0.8 + 0.7 = 2.0 atm

Example 2: Gas Collected Over Water

If a gas is collected over water, the total pressure includes the pressure of the gas and the vapor pressure of water at that temperature.

P_total = P_gas + P_{water vapor}

To find the actual pressure of the gas, subtract the vapor pressure of water from the total pressure:

P_gas = P_total – P_{water vapor}

Important Considerations

• Dalton’s Law applies only to non-reacting gases.
• Temperature and volume must remain constant for accurate application.
• Real gases may slightly deviate due to intermolecular forces, but the law holds well under most conditions.

Summary

• Dalton’s Law helps us understand how pressure works in a gas mixture.
• P_total = P₁ + P₂ + P₃ + … — total pressure is the sum of individual partial pressures.
• The theory is supported by Kinetic Molecular Theory because gas particles act independently in mixtures.
• It’s useful in calculations involving gas mixtures, respiration, and laboratory gas collection methods.

🌟 Key Insight: Each gas in a mixture behaves like it were alone in the container, and all their pressures simply add up to make the total pressure!

Graham’s Law of Diffusion and Effusion

Definition

Graham’s Law states that:

“The rate of diffusion or effusion of a gas is inversely proportional to the square root of its molar mass.”

In simpler terms, lighter gases move faster than heavier ones when temperature and pressure are the same.

Key Terms

• Diffusion: The process by which gas particles spread out to fill the available volume, mixing with other gases.
• Effusion: The process in which gas particles pass through a small hole in a container without collisions between particles.

Mathematical Form

The law is expressed as:

rate₁ / rate₂ = √(M₂ / M₁)

• rate₁ and rate₂ are the diffusion or effusion rates of two gases
• M₁ and M₂ are their respective molar masses

👉 This means that a gas with a lower molar mass will diffuse or effuse faster than a gas with a higher molar mass.

Explanation with Kinetic Molecular Theory (KMT)

According to the Kinetic Molecular Theory:

• At a given temperature, all gas particles have the same average kinetic energy (KE).
• Kinetic energy is given by: KE = ½mv²

For two gases at the same temperature:

½m₁v₁² = ½m₂v₂²

Which leads to:

v₁ / v₂ = √(M₂ / M₁)

👉 Thus, the lighter gas (lower M) has a higher average speed and diffuses/effuses more quickly.

Real-Life Examples

• Smell of perfume spreading: A gas like ethanol (lightweight) diffuses quickly through the air.
• Hydrogen vs. Oxygen: Hydrogen (M = 2 g/mol) effuses about four times faster than oxygen (M = 32 g/mol).
• Leak detection: Helium is often used because it effuses quickly and is safe to detect small leaks in vacuum systems.

Example Calculation

Compare the rates of effusion of hydrogen (M = 2 g/mol) and oxygen (M = 32 g/mol):

rate_H₂ / rate_O₂ = √(32 / 2) = √16 = 4

Conclusion: Hydrogen effuses 4 times faster than oxygen under the same conditions.

Applications of Graham’s Law

• Separating isotopes (e.g., uranium enrichment using UF₆)
• Gas leak detection and control
• Understanding how quickly gases disperse in air

Limitations of Graham’s Law

• It applies best to ideal gases.
• Significant deviations may occur with real gases under high pressure or low temperature.
• Diffusion rates can also be affected by collisions with other gas particles (especially in non-vacuum conditions).

Summary

• Graham’s Law describes how gas diffusion and effusion rates depend on molar mass.
• Lighter gases move faster than heavier ones.
• It is based on kinetic theory and helps explain many natural and industrial gas behaviors.

🌟 Key Insight: At the same temperature, all gases have the same average kinetic energy — so lighter ones must move faster to compensate for having less mass!