Laws Related to Gases

Introduction to Gases

• Gases are one of the fundamental states of matter with no fixed shape or volume.
• Their behavior is governed by several scientific laws that describe the relationship between pressure, volume, temperature, and the number of particles in a gas.

Properties of Gases

1. Compressibility: Gases can be compressed due to large spaces between particles.
2. Expandability: Gases expand to fill the volume of their container.
3. Low Density: Gases are less dense compared to solids and liquids.
4. Diffusibility: Gases mix uniformly without external force.

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Boyle’s Law

• Statement: At constant temperature, the pressure ($P$) of a gas is inversely proportional to its volume ($V$). $P \propto \frac{1}{V} \quad \text{or} \quad PV = \text{constant}$

• Mathematical Expression: $P_1 V_1 = P_2 V_2$ Where:

  • $P_1, V_1$ = Initial pressure and volume,
  • $P_2, V_2$ = Final pressure and volume.

• Graphical Representation:

  • Plot of $P$ vs. $V$: Hyperbolic curve.
  • Plot of $P$ vs. $\frac{1}{V}$: Straight line.

• Example:

  • A balloon expands as it rises in the atmosphere because the external pressure decreases.

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Charles’ Law

• Statement: At constant pressure, the volume ($V$) of a gas is directly proportional to its absolute temperature ($T$). $V \propto T \quad \text{or} \quad \frac{V}{T} = \text{constant}$

• Mathematical Expression: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ Where:

  • $V_1, T_1$ = Initial volume and temperature,
  • $V_2, T_2$ = Final volume and temperature.

• Graphical Representation:

  • Plot of $V$ vs. $T$: Straight line passing through the origin (in Kelvin scale).

• Key Concept:

  • Absolute zero ($0 \, \text{K}$ or $-273.15^\circ \text{C}$): Theoretical temperature where the volume of a gas becomes zero.

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Gay-Lussac’s Law

• Statement: At constant volume, the pressure ($P$) of a gas is directly proportional to its absolute temperature ($T$). $P \propto T \quad \text{or} \quad \frac{P}{T} = \text{constant}$

• Mathematical Expression: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ Where:

  • $P_1, T_1$ = Initial pressure and temperature,
  • $P_2, T_2$ = Final pressure and temperature.

• Graphical Representation:

  • Plot of $P$ vs. $T$: Straight line passing through the origin (in Kelvin scale).

• Example:

  • Pressure inside a sealed container increases when it is heated.

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Combined Gas Law

• Statement: The relationship between pressure, volume, and temperature of a gas is expressed as: $\frac{PV}{T} = \text{constant}$
• Mathematical Expression: $\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$
• Applications:
  • Predicting the behavior of gases in changing conditions.
  • Calculations involving compressed gas cylinders.

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Practical Examples of Gas Laws

1. Boyle’s Law:
   • Using a syringe: Pulling the plunger reduces pressure and draws in liquid or gas.
2. Charles’ Law:
   • Hot air balloons rise when heated due to the expansion of gas.
3. Gay-Lussac’s Law:
   • Pressure cooker: The pressure inside increases as the temperature rises.

Avogadro’s Law

• Statement: At constant temperature and pressure, the volume ($V$) of a gas is directly proportional to the number of moles ($n$) of the gas. $V \propto n \quad \text{or} \quad \frac{V}{n} = \text{constant}$

• Mathematical Expression: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$ Where:

  • $V_1, n_1$ = Initial volume and number of moles,
  • $V_2, n_2$ = Final volume and number of moles.

• Applications:

  • Explains why equal volumes of all gases contain the same number of molecules under identical conditions (known as Avogadro's hypothesis).
  • Forms the basis for the molar volume of gases:
    • At standard temperature and pressure (STP: $0^\circ \text{C}$ and $1 \, \text{atm}$):
      • Molar volume = $22.4 \, \text{L}$.

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Ideal Gas Equation

• Combines Boyle’s, Charles’, and Avogadro’s laws into a single equation: $PV = nRT$ Where:

  • $P$ = Pressure,
  • $V$ = Volume,
  • $n$ = Number of moles,
  • $R$ = Universal gas constant ($8.314 \, \text{J mol}^{-1} \text{K}^{-1}$ or $0.0821 \, \text{L atm mol}^{-1} \text{K}^{-1}$),
  • $T$ = Temperature in Kelvin.

• Applications:

  • Used for calculations involving gases under various conditions.
  • Example: Determining the amount of gas in a container or the pressure exerted by a gas.

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Dalton’s Law of Partial Pressures

• Statement: The total pressure ($P_\text{total}$) exerted by a mixture of non-reacting gases is equal to the sum of their individual partial pressures. $P_\text{total} = P_1 + P_2 + P_3 + \dots$ Where:

  • $P_1, P_2, P_3, \dots$ = Partial pressures of individual gases.

• Partial Pressure: The pressure a gas would exert if it alone occupied the entire volume of the container. $P_i = \frac{n_iRT}{V}$ Where:

  • $n_i$ = Number of moles of the gas,
  • $V$ = Volume of the container,
  • $T$ = Temperature in Kelvin.

• Applications:

  • Scuba diving: Calculation of oxygen and nitrogen partial pressures in breathing mixtures.
  • Atmospheric pressure: Calculated as the sum of pressures from nitrogen, oxygen, carbon dioxide, and other gases.

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Graham’s Law of Diffusion

• Statement: The rate of diffusion or effusion ($r$) of a gas is inversely proportional to the square root of its molar mass ($M$). $r \propto \frac{1}{\sqrt{M}} \quad \text{or} \quad \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$ Where:

  • $r_1, r_2$ = Rates of diffusion of two gases,
  • $M_1, M_2$ = Molar masses of the gases.

• Key Terms:

  • Diffusion: Mixing of gas molecules due to random motion.
  • Effusion: Passage of gas through a tiny hole without collisions between gas molecules.

• Applications:

  • Separation of isotopes (e.g., $\text{U}^{235}$ and $\text{U}^{238}$).
  • Explains why lighter gases (like hydrogen) diffuse faster than heavier gases (like oxygen).

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Real Gases and Deviations from Ideal Behavior

• Real gases do not always follow the ideal gas equation due to:

  1. Intermolecular Forces: Attraction or repulsion between gas particles.
  2. Finite Volume of Particles: Gas molecules occupy a small but finite volume.

• Van der Waals Equation: Accounts for deviations from ideal behavior: $\left(P + \frac{a}{V^2}\right)(V - b) = nRT$ Where:

  • $a$: Corrects for intermolecular forces,
  • $b$: Corrects for the finite volume of gas molecules.

• At high temperature and low pressure, gases behave ideally because:

  • Intermolecular forces become negligible.
  • Volume of particles becomes insignificant compared to the container volume.

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Key Concept Table

Law	Mathematical Expression	Key Idea
Boyle’s Law	$PV = \text{constant}$	$P \propto \frac{1}{V}$ at $T = \text{constant}$
Charles’ Law	$\frac{V}{T} = \text{constant}$	$V \propto T$ at $P = \text{constant}$
Gay-Lussac’s Law	$\frac{P}{T} = \text{constant}$	$P \propto T$ at $V = \text{constant}$
Avogadro’s Law	$\frac{V}{n} = \text{constant}$	$V \propto n$ at $P, T = \text{constant}$
Ideal Gas Law	$PV = nRT$	Combines all gas laws
Dalton’s Law	$P_\text{total} = P_1 + P_2 + \dots$	Total pressure in a gas mixture
Graham’s Law	$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$	Rate of diffusion/effusion

Kinetic Molecular Theory of Gases

The Kinetic Molecular Theory (KMT) explains the behavior of ideal gases based on the motion and interactions of their particles.

Postulates of KMT:

1. Gas Particles are in Constant Motion:
   • Particles move in straight lines until they collide with each other or the walls of the container.
2. Negligible Particle Volume:
   • The volume of individual gas particles is negligible compared to the total volume of the gas.
3. No Intermolecular Forces:
   • Gas particles neither attract nor repel each other.
4. Elastic Collisions:
   • Collisions between gas particles or with container walls are perfectly elastic, meaning no kinetic energy is lost.
5. Kinetic Energy and Temperature:
   • The average kinetic energy of gas particles is directly proportional to the absolute temperature of the gas. $KE_\text{avg} = \frac{3}{2} k_B T$ Where:
   • $KE_\text{avg}$ = Average kinetic energy,
   • $k_B$ = Boltzmann constant ($1.38 \times 10^{-23} \, \text{J/K}$),
   • $T$ = Temperature in Kelvin.

Applications of KMT:

• Explains gas laws such as Boyle’s, Charles’, and Gay-Lussac’s laws.
• Describes diffusion and effusion based on particle motion.

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Diffusion and Effusion (Detailed)

1. Diffusion:

   • Movement of gas particles from a region of higher concentration to lower concentration.
   • Factors Affecting Diffusion:
     • Molar mass: Lighter gases diffuse faster (Graham’s law).
     • Temperature: Higher temperatures increase diffusion rates.
   • Example: Smell of perfume spreading in a room.

2. Effusion:

   • Movement of gas particles through a tiny hole without collisions.
   • Governed by Graham’s law: $\text{Rate of Effusion} \propto \frac{1}{\sqrt{\text{Molar Mass}}}$

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Real Gases vs. Ideal Gases

Ideal Gases:

• Follow the ideal gas equation $PV = nRT$ under all conditions.
• Assume no intermolecular forces and negligible particle volume.

Real Gases:

• Deviate from ideal behavior under high pressure and low temperature.
• Reasons for Deviation:
  1. Intermolecular forces become significant.
  2. Volume of gas particles is not negligible.

Van der Waals Equation: To account for deviations: $\left(P + \frac{a}{V^2}\right)(V - b) = nRT$ Where:

• $a$: Corrects for intermolecular attractions,
• $b$: Corrects for finite particle volume.

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Liquefaction of Gases

Liquefaction:

• The process of converting a gas into a liquid by applying high pressure and lowering temperature.

Critical Temperature ($T_c$):

• The highest temperature at which a gas can be liquefied by pressure alone.
• Above $T_c$, gases cannot be liquefied.

Critical Pressure ($P_c$):

• The minimum pressure required to liquefy a gas at its critical temperature.

Applications:

• Liquefied Petroleum Gas (LPG): Used as a domestic and industrial fuel.
• Liquefied Natural Gas (LNG): Used for energy storage and transportation.

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Key Applications of Gas Laws in Daily Life

1. Boyle’s Law:
   • Breathing: During inhalation, lung volume increases, reducing pressure, and air enters.
2. Charles’ Law:
   • Hot air balloons: Gas expands when heated, making the balloon rise.
3. Gay-Lussac’s Law:
   • Pressure cookers: Pressure inside increases with rising temperature.
4. Avogadro’s Law:
   • Balloons: Adding more gas increases volume at constant pressure.
5. Dalton’s Law:
   • Scuba diving: Gas mixtures are calculated to avoid decompression sickness.

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Summary Table of Key Constants and Concepts

Concept	Value/Expression	Significance
Universal Gas Constant ($R$)	$8.314 \, \text{J mol}^{-1} \text{K}^{-1}$ or $0.0821 \, \text{L atm mol}^{-1} \text{K}^{-1}$	Used in ideal gas law
Standard Temperature and Pressure (STP)	$T = 273.15 \, \text{K}$, $P = 1 \, \text{atm}$	Standard conditions for gases
Molar Volume at STP	$22.4 \, \text{L}$	Volume of 1 mole of gas at STP
Graham’s Law	$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$	Rate of diffusion/effusion
Van der Waals Equation	$\left(P + \frac{a}{V^2}\right)(V - b) = nRT$	Accounts for real gas behavior