Comprehensive Study Guide on Chemical Quantities, Rotational Kinematics, and Mechanical Work and Power

Physical Quantities Related to Amount of Substance

• Fundamental Conversions and Reference Standards:

  • Mass Conversions:

    • $1\,\text{kg} = 10^3\,\text{g}$

    • $1\,\text{mg} = 10^{-3}\,\text{g}$

  • Volume Conversions:

    • $1\,\text{L} = 10^{-3}\,\text{m}^3$

    • $1\,\text{L} = 1\,\text{dm}^3$

    • $1\,\text{mL} = 10^{-3}\,\text{L}$

    • $1\,\text{cm}^3 = 1\,\text{mL} = 10^{-3}\,\text{L}$

    • Conversion between molar volume units: to convert from $\text{m}^3\cdot\text{mol}^{-1}$ to $\text{L}\cdot\text{mol}^{-1}$, multiply by $10^{-3}$.

  • Volumetric Mass (Density) Conversions:

    • $1\,\text{g/cm}^3 = 10^3\,\text{kg/m}^3$

    • $1\,\text{g/cm}^3 = 1\,\text{g/mL}$

  • Pressure Conversions:

    • $1\,\text{atm} = 1.013 \times 10^5\,\text{Pa}$

    • $1\,\text{bar} = 10^5\,\text{Pa}$

    • $1\,\text{hPa} = 10^2\,\text{Pa}$

    • $1\,\text{kPa} = 10^3\,\text{Pa}$

    • $1\,\text{atm} = 76\,\text{cmHg} = 760\,\text{mmHg}$

  • Absolute Temperature Conversion:

    • $T(\text{K}) = \theta(^\circ\text{C}) + 273.15$

  • Substance Quantity Conversions:

    • $1\,\text{mmol} = 10^{-3}\,\text{mol}$

• Definition and Relations of the Amount of Substance ($n$):

  • The amount of substance $n(X)$ represents the chemical quantity of an elementary entity $X$, expressed in moles ($\text{mol}$).

  • General Master Relationship:         $n(X) = \frac{N}{N_A} = \frac{m(X)}{M(X)} = \frac{V(X)}{V_m} = C \times V$

  • Definition of Variables and Units:

    • $N$: Total number of microscopic entities or constituents (dimensionless).

    • $N_A$: Avogadro's constant, defined as $N_A = 6.03 \times 10^{23}\,\text{mol}^{-1}$.

    • $m(X)$: Mass of sample $X$, expressed in grams ($\text{g}$).

    • $M(X)$: Molar mass of entity $X$, expressed in grams per mole ($\text{g}\cdot\text{mol}^{-1}$).

    • $V(X)$: Volume occupied by gaseous species $X$, expressed in liters ($\text{L}$).

    • $V_m$: Molar volume of a gas under given temperature and pressure conditions, expressed in liters per mole ($\text{L}\cdot\text{mol}^{-1}$).

    • $C$: Molar concentration of solute in solution, expressed in moles per liter ($\text{mol}\cdot\text{L}^{-1}$).

    • $V$: Volume of the solution or gas sample, expressed in liters ($\text{L}$).

• Molar Mass Determinations:

  • Standard Atomic Molar Masses ($\text{g}\cdot\text{mol}^{-1}$):

    • Hydrogen ($\text{H}$): $M(\text{H}) = 1\,\text{g}\cdot\text{mol}^{-1}$

    • Helium ($\text{He}$): $M(\text{He}) = 4\,\text{g}\cdot\text{mol}^{-1}$

    • Carbon ($\text{C}$): $M(\text{C}) = 12\,\text{g}\cdot\text{mol}^{-1}$

    • Nitrogen ($\text{N}$): $M(\text{N}) = 14\,\text{g}\cdot\text{mol}^{-1}$

    • Oxygen ($\text{O}$): $M(\text{O}) = 16\,\text{g}\cdot\text{mol}^{-1}$

    • Fluorine ($\text{F}$): $M(\text{F}) = 19\,\text{g}\cdot\text{mol}^{-1}$

    • Sulfur ($\text{S}$): $M(\text{S}) = 32\,\text{g}\cdot\text{mol}^{-1}$

    • Iron ($\text{Fe}$): $M(\text{Fe}) = 56\,\text{g}\cdot\text{mol}^{-1}$

    • Copper ($\text{Cu}$): $M(\text{Cu}) = 63.5\,\text{g}\cdot\text{mol}^{-1}$

    • Zinc ($\text{Zn}$): $M(\text{Zn}) = 65.4\,\text{g}\cdot\text{mol}^{-1}$

    • Silver ($\text{Ag}$): $M(\text{Ag}) = 108\,\text{g}\cdot\text{mol}^{-1}$

  • Molecular Molar Mass Calculation Examples:

    • Water ($\text{H}_2\text{O}$):             $M(\text{H}_2\text{O}) = 2 \times M(\text{H}) + M(\text{O}) = 2 \times 1 + 16 = 18\,\text{g}\cdot\text{mol}^{-1}$

    • Glucose ($\text{C}_6\text{H}_{12}\text{O}_6$):             $M(\text{C}_6\text{H}_{12}\text{O}_6) = 6 \times M(\text{C}) + 12 \times M(\text{H}) + 6 \times M(\text{O})$             $M(\text{C}_6\text{H}_{12}\text{O}_6) = 6 \times 12 + 12 \times 1 + 6 \times 16 = 72 + 12 + 96 = 180\,\text{g}\cdot\text{mol}^{-1}$

• Molar Volume of Gases ($V_m$):

  • Definition: The molar volume is the volume occupied by one mole of any gas at a specified pressure $P$ and temperature $T$.

  • Thermodynamic Formula:         $V_m = \frac{R \times T}{P}$         where $R = 8.314\,\text{S.I.}$, $T$ is in Kelvin ($\text{K}$), and $P$ is in Pascals ($\text{Pa}$).

  • Standard Conditions for Temperature and Pressure (CSTP):

    • Temperature: $T = 20\,^\circ\text{C}$

    • Pressure: $P = 1\,\text{atm}$

    • Resulting Molar Volume: $V_m = 24\,\text{L}\cdot\text{mol}^{-1}$

  • Normal Conditions for Temperature and Pressure (CNTP):

    • Temperature: $T = 0\,^\circ\text{C}$

    • Pressure: $P = 1\,\text{atm}$

    • Resulting Molar Volume: $V_m = 22.4\,\text{L}\cdot\text{mol}^{-1}$

• Density, Relative Density, and Concentrations:

  • Volumetric Mass (Density $\rho$):         $\rho = \frac{m}{V}$         where $m$ is the mass in grams ($\text{g}$), $V$ is the volume in cubic centimeters ($\text{cm}^3$), and $\rho$ is expressed in $\text{g}\cdot\text{cm}^{-3}$.

  • Relative Density of a Condensed Body (Liquid or Solid) relative to Water ($d$):         $d = \frac{\rho}{\rho_0}$         where $\rho_0$ is the volumetric mass of water, with a standard value of $\rho_0 = 1\,\text{g}\cdot\text{cm}^{-3} = 1\,\text{g}\cdot\text{mL}^{-1}$.         $\rho = d \times \rho_0$

  • Relative Density of a Gas relative to Air ($d$):         $d = \frac{M(X)}{29}$         where $M(X)$ is the molar mass of the gaseous species expressed in $\text{g}\cdot\text{mol}^{-1}$, and $29\,\text{g}\cdot\text{mol}^{-1}$ represents the equivalent molar mass of atmospheric air.

  • Molar Concentration ($C$):         $C = \frac{n(X)}{V} = \frac{m(X)}{M(X) \times V}$         expressed in $\text{mol}\cdot\text{L}^{-1}$.

  • Mass Concentration ($C_m$):         $C_m = \frac{m(X)}{V} = \frac{n(X) \times M(X)}{V}$         expressed in $\text{g}\cdot\text{L}^{-1}$.

  • Interrelation between Molar and Mass Concentration:         $C = \frac{C_m}{M(X)}$         $C_m = C \times M(X)$

  • Relationship between Amount of Substance $n(X)$ and Relative Density $d$:         $m(X) = \rho \times V = d \times \rho_0 \times V$         $n(X) = \frac{\rho \times V}{M(X)} = \frac{d \times \rho_0 \times V}{M(X)}$         where $V$ is expressed in $\text{mL}$ when using $\rho_0 = 1\,\text{g}\cdot\text{mL}^{-1}$.

  • Relationship between Molar Concentration $C$ and Relative Density $d$:         $C = \frac{n(X)}{V} = \frac{\rho}{M(X)} = \frac{d \times \rho_0}{M(X)}$

  • Mass Percentage ($P$):

    • The mass percentage $P$ denotes the mass fraction of a pure chemical species $X$ contained within a solution of total mass $m_s$:             $m(X) = P \times m_s = P \times \rho_s \times V_s = P \times d \times \rho_0 \times V_s$

    • The resulting molar concentration $C$ of the solution is given by:             $C = \frac{m(X)}{M(X) \times V_s} = \frac{P \times d \times \rho_0}{M(X)}$

• Gas Laws:

  • Ideal Gas Equation of State:         $P \times V = n(X) \times R \times T$

    • International System of Units (SI):

      • Pressure: $P$ in Pascals ($\text{Pa}$)

      • Volume: $V$ in cubic meters ($\text{m}^3$)

      • Amount of substance: $n(X)$ in moles ($\text{mol}$)

      • Universal gas constant: $R = 8.314\,\text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}$ (or $\text{S.I.}$)

      • Absolute temperature: $T$ in Kelvin ($\text{K}$)

    • Alternative Practical Units:

      • Pressure: $P$ in atmospheres ($\text{atm}$)

      • Volume: $V$ in liters ($\text{L}$)

      • Universal gas constant: $R = 0.082\,\text{L}\cdot\text{atm}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}$

  • Boyle-Mariotte's Law:

    • Applies to an ideal gas undergoing an isothermal transformation (where temperature $T$ remains strictly constant):             $P_1 \times V_1 = P_2 \times V_2 = \text{constant}$

Kinematics of Rotational Motion of a Rigid Solid Around a Fixed Axis

• Positional Kinematic Quantities:

  • Curvilinear Abscissa ($s(t)$):

    • Defined as the algebraic arc length along the circular trajectory traced by point $M$ from an initial reference point $M_0$ at time $t_0$:             $s(t) = \overset{\frown}{M_0 M}$

    • Unit: meters ($\text{m}$).

  • Angular Abscissa ($\theta(t)$):

    • Defined as the oriented angle formed between the reference radius vector $\vec{OM_0}$ and the instantaneous radius vector $\vec{OM}$:             $\theta(t) = (\vec{OM_0}, \vec{OM})$

    • Unit: radians ($\text{rad}$).

  • Geometric Relation:         $s = R \times \theta$         where $R$ is the radius of the circular trajectory traced by point $M$.

• Linear and Angular Velocities:

  • Linear Velocity ($v$):

    • Average Linear Velocity ($v$): Calculated between two positions $M_1$ and $M_2$ over a time interval $\Delta t$:             $v = \frac{M_1 M_2}{\Delta t}$             expressed in $\text{m}\cdot\text{s}^{-1}$.

    • Instantaneous Linear Velocity ($v_i$): Calculated at recorded position $M_i$ using neighboring positions:             $v_i = \frac{M_{i-1} M_{i+1}}{2 \times \tau}$             where $\tau$ is the constant time interval between consecutive recordings ($\Delta t = 2 \times \tau$).

  • Angular Velocity ($\omega$):

    • Average Angular Velocity ($\omega$): Calculated over an angular displacement $\Delta \theta$ across elapsed time $\Delta t$:             $\omega = \frac{\Delta \theta}{\Delta t}$             expressed in $\text{rad}\cdot\text{s}^{-1}$.

    • Instantaneous Angular Velocity ($\omega_i$): Calculated at recording instant $t_i$ using adjacent angular measurements:             $\omega_i = \frac{\theta_{i+1} - \theta_{i-1}}{2 \times \tau}$

  • Kinematic Relationship between Linear and Angular Velocity:         $v = R \times \omega$

• Uniform Rotational Motion:

  • Definition: A rotational motion is defined as uniform when the instantaneous angular velocity remains constant over time:         $\omega_0 = \frac{\Delta \theta}{\Delta t} = \text{constant}$

  • Period ($T$):

    • The time duration required for the solid to execute one complete revolution ($1\,\text{turn}$):             $T = \frac{2 \times \pi}{\omega_0}$

    • Unit: seconds ($\text{s}$).

  • Frequency ($f$ or $N$):

    • The number of complete revolutions executed per unit time (per second):             $f = N = \frac{1}{T} = \frac{\omega_0}{2 \times \pi}$

    • Unit: Hertz ($\text{Hz}$).

  • Angular Displacement and Total Number of Turns ($n$):         $\Delta \theta = \theta_2 - \theta_1$         $\Delta \theta = 2 \times \pi \times n$         where $n$ denotes the total number of full revolutions performed during the interval.

  • Time Equations of Motion (Lois Horaires):

    • Angular Abscissa Time Law:             $\theta(t) = \omega_0 \times t + \theta_0$             where $\theta_0$ is the initial angular position at $t = 0\,\text{s}$. The plot of $\theta(t)$ against $t$ yields a straight line with slope equal to $\omega_0$ and vertical intercept $\theta_0$.

    • Curvilinear Abscissa Time Law:             $s(t) = v \times t + s_0$             where $s_0$ is the initial curvilinear position at $t = 0\,\text{s}$.

Work and Power of Mechanical Forces

• General Concepts and Work Expressions:

  • Mechanical work represents the energy transferred to or from an object via the application of a force along a displacement. Unit: Joules ($\text{J}$).

  • Work of a Constant Force in Rectilinear Translation:

    • For a constant force $\vec{F}$ acting during a straight displacement from point $A$ to point $B$:             $W_{A \rightarrow B}(\vec{F}) = F \times AB \times \cos(\alpha)$             where $F$ is the magnitude of the force in Newtons ($\text{N}$), $AB$ is the displacement distance in meters ($\text{m}$), and $\alpha$ is the angle between force vector $\vec{F}$ and displacement vector $\vec{AB}$.

  • Work of a Force in Rotation Around a Fixed Axis ($\Delta$):

    • For a force exerting a rotational effect about axis $\Delta$ over an angular displacement $\Delta \theta$:             $W_{A \rightarrow B}(\vec{F}) = \mathcal{M}_\Delta(\vec{F}) \times \Delta \theta$             where $\mathcal{M}_\Delta(\vec{F})$ is the moment of force $\vec{F}$ with respect to axis $\Delta$ in $\text{N}\cdot\text{m}$, and $\Delta \theta$ is in radians ($\text{rad}$).

  • Physical Classification of Work Based on Sign:

    • $W > 0$: Motive work (travail moteur), occurring when $\cos(\alpha) > 0$ ($0 \le \alpha < \frac{\pi}{2}$), supplying mechanical energy to the system.

    • $W < 0$: Resistive work (travail résistant), occurring when $\cos(\alpha) < 0$ ($\frac{\pi}{2} < \alpha \le \pi$), removing mechanical energy from the system.

    • $W = 0$: Null work (travail nul), occurring when $\cos(\alpha) = 0$ ($\alpha = \frac{\pi}{2}$), where the force vector is perpendicular to displacement.

• Work Done by the Gravitational Force (Weight $\vec{P}$):

  • General Path-Independent Formulation:         $W(\vec{P}) = m \times g \times (z_i - z_f)$

    • During downward motion (descent, where $z_i > z_f$):             $W(\vec{P}) = +m \times g \times h$

    • During upward motion (ascent, where $z_i < z_f$):             $W(\vec{P}) = -m \times g \times h$         where $h = |z_i - z_f|$ is the vertical altitude variation.

  • Specific Trajectory Configurations:

    • Horizontal Displacement from $A$ to $B$:             $W_{A \rightarrow B}(\vec{P}) = 0\,\text{J}$             because the weight vector $\vec{P}$ is orthogonal to the horizontal displacement vector $\vec{AB}$.

    • Straight Inclined Plane (Descent from $A$ to $B$ at Angle $\alpha$ to Horizontal):             The vertical drop is $h = AB \times \sin(\alpha)$, yielding:             $W_{A \rightarrow B}(\vec{P}) = +m \times g \times AB \times \sin(\alpha)$

    • Straight Inclined Plane (Ascent from $B$ to $A$ at Angle $\alpha$ to Horizontal):             $W_{B \rightarrow A}(\vec{P}) = -m \times g \times AB \times \sin(\alpha)$

    • Circular Concave Trajectory of Radius $R = OA$ (Descent from $A$ to Equilibrium Position $B$):             For an arc subtending angle $\alpha$ from the vertical equilibrium axis:             $h = R - R \times \cos(\alpha) = R \times (1 - \cos(\alpha))$             $W_{A \rightarrow B}(\vec{P}) = +m \times g \times R \times (1 - \cos(\alpha))$

• Work Done by the Contact Surface Reaction Force ($\vec{R}$):

  • Motion Without Friction (Sans frottement):

    • The reaction force $\vec{R}$ remains strictly normal (perpendicular) to the contact surface ($\vec{R} = \vec{R}_N$ and $\vec{R} \perp \vec{AB}$):             $W(\vec{R}) = 0\,\text{J}$

  • Motion With Friction (Avec frottement):

    • The reaction force tilts opposite the direction of motion and decomposes into a normal component $\vec{R}_N$ and a tangential frictional component $\vec{R}_T$ (or $\vec{f}$):             $\vec{R} = \vec{R}_N + \vec{R}_T$

    • Since $\vec{R}_N \perp \vec{AB}$, its work is zero, leaving exclusively the resistive work of friction:             $W(\vec{R}) = W(\vec{R}_T) = -R_T \times AB$

• Principle of Inertia and Equilibrium Condition:

  • In uniform rectilinear motion, the vector sum of external forces applied to a solid body is zero (Principle of Inertia):         $\sum \vec{F}_i = \vec{0} \implies \sum W_{A \rightarrow B}(\vec{F}_i) = 0$

• Power of a Force ($P$):

  • Mechanical power characterizes the rate at which work is performed per unit time. Unit: Watts ($\text{W}$).

  • Average Power ($P_m$):         $P_m = \frac{W(\vec{F})}{\Delta t}$         where $W(\vec{F})$ is the work performed across time interval $\Delta t$.

  • Instantaneous Power ($P(t)$):

    • In Translatory Motion:             $P(t) = \vec{F} \cdot \vec{v} = F \times v \times \cos(\vec{F}, \vec{v})$             where $\vec{v}$ is the instantaneous linear velocity vector.

    • In Rotational Motion:             $P(t) = \mathcal{M}_\Delta(\vec{F}) \times \omega(t)$             where $\mathcal{M}_\Delta(\vec{F})$ is the moment of the force about axis $\Delta$, and $\omega(t)$ is the instantaneous angular velocity in $\text{rad}\cdot\text{s}^{-1}$.