Chem 1701 Key Concepts: Vectors, Force and Energy

A. Vector Quantities

• Interdisciplinary nature of science: mathematics, physics, and chemistry share key concepts that help explain chemical phenomena.
• Three key concepts introduced: A) Vector quantities, B) Force, C) Energy. These concepts work together to explain interactions (e.g., between two charged particles) and underpin understanding of atomic orbitals, stability, periodic trends (electronegativity, ionization energy), bond formation, and molecular attractions.
• Coordinate axes (as shown): x and y axes lie in the plane of the paper; z axis points out of the plane toward you. Positive directions:
  • +x to the right, +y up the page, +z out of the plane.
• Vector quantity definition: has both magnitude and direction; represented by an arrow whose length corresponds to magnitude and orientation corresponds to direction.
• In the plane, arrows lie in the plane of the paper (e.g., velocity components in x and y).

B. Vector Quantities (continued) – Examples

• Velocity as a vector: an object moving with a magnitude (speed) of 3 m/s along the positive x-axis has a velocity vector with magnitude |\vec{v}| = 3 \,\mathrm{m\,s^{-1}} and direction along +x.
• If velocity lies in the plane, it can have components along both x and y:
  • The velocity vector can be decomposed into components, e.g., along +x (vx) and along +y (vy), with a resultant blue vector \vec{v} from these components.
• All arrows described (velocity vectors, components, and resultant) lie in the plane of the paper.

B. Force

• Force is a vector quantity: it has magnitude and direction.
• A force can cause a change in motion by accelerating the object in a given direction.
• Relationship between force and motion (Newton's second law): $\vec{F} = m \vec{a}$
  • \vec{F} is the force vector (unit: N).
  • m is the mass (unit: kg).
  • \vec{a} is the acceleration vector (unit: m s^{-2}).
• Key unit definitions:
  • 1 Newton (N) = 1 kg m s^{-2}.
• Directionality:
  • The direction of acceleration is the same as the direction of the force for any object with positive mass.
  • If a force is applied along the +x direction, the object accelerates along +x.
• Gravitational force (an example): on Earth, gravity yields a constant acceleration of approximately $g \approx 9.8 \,\mathrm{m\,s^{-2}}$ toward the Earth's surface.
• Note on orientation: in the examples, gravitational acceleration is downward relative to the plane, aligning with the concept that force dictates acceleration along its direction.

C. Energy

• Energy forms relevant to Chem 1701: kinetic energy (KE), potential energy (PE), plus other forms (electric, thermal, etc.); the course emphasizes KE and PE and their interconversion.
• Kinetic Energy (KE or E_K):
  • Definition for an object of mass m moving with velocity v:
    $KE = \frac{1}{2} m v^{2}$
  • Units: joules (J) where 1 J = 1 kg m^2 s^{-2}.
  • Note: the velocity term is squared; only the magnitude of velocity matters for KE.
• Potential Energy (PE or V):
  • Energy stored due to an object's location relative to a reference point.
  • Gravity is a familiar example: higher height relative to the ground corresponds to higher PE.
  • Multiple forms of PE exist; the course will introduce the expression for the potential energy arising from interactions between two charged particles later.
• Conservation of Energy (definitions):
  • Isolated system: one that cannot exchange matter or energy with its surroundings.
  • Examples in Chem 111/112 involve interactions among charged particles, atoms, and molecules within a system.

The Law of Conservation of Energy

• Law statement: in an isolated system, the total energy remains constant.
• Total energy (Etot) of a system is the sum of its kinetic and potential energies: $E</em>{tot} = KE + PE$
• Implications:
  • As an object's velocity changes, its KE changes.
  • As an object's position changes relative to a reference point, its PE changes.
  • KE and PE can interconvert, but their sum E_tot remains constant in an isolated system.

The Interaction of Two Charged Particles

• Electrical force (F(r)) between two charged particles: the force that one particle exerts on another through space (no direct contact required).
• Charge description:
  • Charge on a particle has both sign (positive or negative) and magnitude.
  • Charge unit: coulomb (C).
  • Example charges: electron charge =
    $q<em>e = -1.602 \times 10^{-19} \ \mathrm{C}, \quad q</em>p = +1.602 \times 10^{-19} \ \mathrm{C}$
• Variables for two interacting charges:
  • q1 for particle 1, q2 for particle 2 (units: C).
  • r for the distance between the two particles (unit: m).
  • ε0 is the permittivity of vacuum, a constant: $\epsilon</em>0 = 8.854 \times 10^{-12} \ \mathrm{C^2\,N^{-1}\,m^{-2}}$
• Electrical force (Coulomb's law): $\vec{F}(r) = \frac{q<em>1 q</em>2}{4 \pi \epsilon_0 r^2} \hat{\mathbf{r}}$
  • Magnitude varies as 1/r^2; vector direction is along the line joining the charges.
  • The sign of q1 q2 determines whether the force is attractive or repulsive:
  • Like charges (q1 q2 > 0): repulsive force, particles accelerate away from each other.
  • Unlike charges (q1 q2 < 0): attractive force, particles accelerate toward each other.
• Coulombic potential energy (V) for two charged particles: $V(r) = \frac{q<em>1 q</em>2}{4 \pi \epsilon_0 r}$
  • The same constants as the force expression; distance r in the denominator.
  • Key distinction in distance dependence:
  • Force varies as 1/r^2.
  • Coulombic potential energy varies as 1/r.
• Putting it together for atomic relevance:
  • These expressions allow quantitative description of the electrical forces and Coulombic potential energies from electron–proton interactions, which are fundamental to the structure and stability of the atom.

Connections and Implications

• These concepts underpin explanations for:
  • The shapes of atomic orbitals and why atoms are stable.
  • Trends in chemical properties (electronegativity, ionization energy) across the periodic table.
  • Why certain chemical bonds form while others do not.
  • Why certain molecular combinations show stronger attractive interactions.
• Practical and philosophical relevance:
  • Provides a unified framework for describing interactions at the atomic scale.
  • Demonstrates how simple, universal laws (F = ma, Coulomb's law, conservation of energy) govern complex chemical behavior.
• Notation recap:
  • Vector quantities: \vec{v}, \vec{F}, components along axes, magnitudes |\vec{v}|, |\vec{F}|.
  • Energies: KE, PE, E_tot.
  • Charges: q1, q2 (in C); distance r (in m); permittivity ε_0.
  • Constants: ε_0 = $8.854 \times 10^{-12}$ C^2 N^{-1} m^{-2}.