One Shot 1st Summative - Class 9 West Bengal Board Science

Behaviour of Gases

States of Matter and Molecular Characteristics

Matter exists in three main states: solid, liquid, and gas. The arrangement and interaction of molecules differ in each state, affecting their properties and behavior. Solids have tightly packed molecules with strong intermolecular forces, liquids have moderate forces allowing movement, and gases have minimal forces with molecules far apart and in constant motion.

• Solid: Fixed shape and volume, strong intermolecular attraction, minimal distance between molecules.
• Liquid: No fixed shape, fixed volume, moderate intermolecular attraction, molecules move freely.
• Gas: No fixed shape or volume, minimal intermolecular attraction,…

Behaviour of Gases

States of Matter and Molecular Characteristics

Matter exists in three main states: solid, liquid, and gas. The arrangement and interaction of molecules differ in each state, affecting their properties and behavior. Solids have tightly packed molecules with strong intermolecular forces, liquids have moderate forces allowing movement, and gases have minimal forces with molecules far apart and in constant motion.

• Solid: Fixed shape and volume, strong intermolecular attraction, minimal distance between molecules.
• Liquid: No fixed shape, fixed volume, moderate intermolecular attraction, molecules move freely.
• Gas: No fixed shape or volume, minimal intermolecular attraction, maximum distance between molecules, molecules exhibit Brownian motion.

Properties and Behavior of Gases

Gases display unique properties due to their molecular arrangement and weak intermolecular forces. They are highly compressible, expansible, and diffuse rapidly. Their molecules are in constant random motion, colliding with each other and container walls.

• Compressibility: Gases can be compressed by applying external pressure, reducing their volume.
• Expansibility: Heating increases kinetic energy, causing gas molecules to occupy more volume.
• Diffusion: Gases spread from regions of high concentration to low concentration (e.g., fragrance spreading in a room).
• Brownian Motion: Random movement of gas molecules due to minimal attraction forces.
• Elastic Collisions: Gas molecules collide without loss of energy or deformation.

Measurable Quantities and Pressure

The main measurable quantities for gases are pressure, volume, mass, and temperature. Pressure is defined as the force applied per unit area, and its units and relationships are crucial for calculations.

• Pressure Formula: $P = \frac{F}{A}$ (Force per unit area)
• SI Unit: Pascal ($\text{N/m}^2$)
• CGS Unit: Dyne/cm$^2$
• Conversion: $1\,\text{Pascal} = 10\,\text{Dyne/cm}^2$
• Atmospheric Pressure: Measured using a barometer; standard value is $76\,\text{cm Hg}$ or $1\,\text{atm}$.
• Barometer Principle: Height of mercury column corresponds to atmospheric pressure; $P = h \rho g$.

Boyle’s Law: Pressure-Volume Relationship

Boyle’s Law states that for a fixed mass of gas at constant temperature, the pressure and volume are inversely proportional. Increasing pressure decreases volume and vice versa.

• Statement: At constant mass and temperature, $P \propto \frac{1}{V}$.
• Mathematical Form: $PV = k$ (constant)
• Multiple States: $P_1V_1 = P_2V_2 = P_3V_3 = \dots$
• Graphical Representation:
  • $P$ vs $V$: Rectangular hyperbola (inverse relationship)
  • $PV$ vs $V$ or $PV$ vs $P$: Straight line parallel to axes
  • $P$ vs $1/V$: Straight line through origin
• Conditions: Valid only if mass and temperature are constant; not applicable when gas mass changes (e.g., inflating a balloon).

Charles’s Law: Volume-Temperature Relationship

Charles’s Law describes how the volume of a fixed mass of gas changes with temperature at constant pressure. Volume increases proportionally with temperature.

• Statement: At constant mass and pressure, $V \propto T$.
• Mathematical Form (Celsius): $V_t = V_0 (1 + \frac{t}{273})$
• Mathematical Form (Kelvin): $V \propto T$ or $\frac{V_1}{T_1} = \frac{V_2}{T_2}$
• Graphical Representation: $V$ vs $T$ (Celsius) is a straight line not passing through origin; $V$ vs $T$ (Kelvin) is a straight line through origin.
• Absolute Zero: Extrapolation shows volume becomes zero at $-273.15^\circ$C ($0\,\text{K}$), called absolute zero.
• Conversions:
  • Celsius to Kelvin: $T(\text{K}) = t(\text{C}) + 273$
  • Absolute zero in Fahrenheit: $-459.4^\circ$F

Avogadro’s Law and Molecular Interpretation

Avogadro’s Law states that equal volumes of all gases, at the same temperature and pressure, contain equal numbers of molecules. This forms the basis for mole concept in gases.

• Statement: At constant temperature and pressure, $V \propto n$ (number of moles).
• Mathematical Form: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$
• One mole: Contains $6.022 \times 10^{23}$ molecules (Avogadro number).
• Application: Equal volumes of different gases (e.g., O$_2$, N$_2$, CO$_2$) at same conditions have equal number of molecules.

Combined Gas Law and Ideal Gas Equation

The combined gas law integrates Boyle’s, Charles’s, and Avogadro’s laws, relating pressure, volume, temperature, and number of moles.

• Combined Law: $V \propto \frac{nT}{P}$
• Equation: $PV = nRT$
• Alternate Forms:
  • $PV = \frac{W}{M}RT$ (where $W$ = mass, $M$ = molar mass)
  • $PV = \frac{X}{N_A}RT$ (where $X$ = number of molecules, $N_A$ = Avogadro number)
• Gas Constant ($R$):
  • SI: $8.314\,\text{J mol}^{-1}\text{K}^{-1}$
  • CGS: $8.314 \times 10^7\,\text{erg mol}^{-1}\text{K}^{-1}$
  • $0.082\,\text{L atm mol}^{-1}\text{K}^{-1}$
• Dimension: $[\text{ML}^2\text{T}^{-2}\text{N}^{-1}\Theta^{-1}]$
• Significance: $R$ is the work required to increase the temperature of one mole of gas by one Kelvin.

Kinetic Theory and Real vs Ideal Gases

The kinetic theory of gases is based on several assumptions about molecular behavior. Ideal gases strictly follow these assumptions, while real gases deviate due to intermolecular forces and finite molecular volume.

• Assumptions:
  • Gas molecules are numerous, small, and elastic.
  • Volume of molecules is negligible compared to container.
  • No intermolecular attraction or repulsion.
  • Collisions are perfectly elastic; momentum and energy conserved.
  • Molecules move in straight lines between collisions.
• Mean Free Path: Average distance traveled between collisions.
• Deviation: Real gases have finite molecular volume and intermolecular forces, leading to errors in pressure and volume predictions.
• Van der Waals Equation (for real gases):$$\left(P + \frac{a}{V^2}\right)(V - b) = RT$$(where $a$ and $b$ account for intermolecular forces and volume)
• Conditions for Ideal Behavior: Real gases behave ideally at high temperature and low pressure.

Exam-Focused Applications and Calculations

Several types of questions and calculations are directly based on the laws and equations discussed. Understanding the conditions and correct formula application is crucial for solving problems.

• Pressure Calculation: Use $P = \frac{F}{A}$ or barometer readings.
• Volume Change: Apply Boyle’s or Charles’s law depending on which variables are constant.
• Mole and Volume: Use Avogadro’s law for comparing volumes and moles.
• Ideal Gas Equation: $PV = nRT$ for calculations involving mass, volume, pressure, temperature, and moles.
• Density Relation: $d = \frac{PM}{RT}$
• Kinetic Energy: Average kinetic energy per mole: $E = \frac{3}{2}RT$; per molecule: $E = \frac{3}{2}kT$ ($k = R/N_A$).
• Graph Interpretation: Recognize the nature of $P$ vs $V$, $V$ vs $T$, $V$ vs $n$ graphs for different laws.
• Common Pitfalls:
  • Always convert temperature to Kelvin for calculations.
  • Use correct units for pressure and volume.
  • Boyle’s law is not valid if gas mass changes (e.g., inflating a balloon).
  • Real gases deviate from ideal behavior at low temperature and high pressure.

Key Formulas and Units

A summary of essential formulas and units for quick reference in problem-solving.

• Pressure: $P = \frac{F}{A}$; SI: Pascal ($\text{N/m}^2$), CGS: Dyne/cm$^2$

• Barometric Pressure: $P = h \rho g$

• Boyle’s Law: $PV = k$; $P_1V_1 = P_2V_2$

• Charles’s Law: $V_t = V_0 (1 + \frac{t}{273})$; $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Avogadro’s Law: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

• Combined Gas Law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

• Ideal Gas Equation: $PV = nRT$

• Density: $d = \frac{PM}{RT}$

• Gas Constant ($R$): $8.314\,\text{J mol}^{-1}\text{K}^{-1}$; $0.082\,\text{L atm mol}^{-1}\text{K}^{-1}$

• Kinetic Energy: $E = \frac{3}{2}RT$ (per mole), $E = \frac{3}{2}kT$ (per molecule)

• Absolute Zero: $-273.15^\circ$C ($0\,\text{K}$), $-459.4^\circ$F

• Avogadro Number: $6.022 \times 10^{23}$ molecules per mole

• Mean Free Path: Average distance between collisions, denoted as "free path" in kinetic theory

• Van der Waals Equation: $\left(P + \frac{a}{V^2}\right)(V - b) = RT$ for real gases

• Conditions for Ideal Gas: High temperature, low pressure

• Graph Types: Rectangular hyperbola for Boyle’s law, straight line for Charles’s and Avogadro’s laws

• Common Units: Volume in liters or cm$^3$, pressure in atm, mm Hg, or Pascal, temperature in Kelvin

• Key Exam Tip: Always check which variables are held constant to select the correct law and formula.