Pj Problems - Overview
Celestial Stars
The Number Line
Geometries
7 Spaces Of Interest - Overview
Triadic Unit Mesh
Creation
The Atom
Survival
Energy
Light
Heat
Sound
Music
Language
Stories
Work
States Of Matter
Buoyancy
Nuclear Reactions
Molecular Shapes
Electron Configurations
Chemical Bonds
Energy Conversion
Chemical Reactions
Electromagnetism
Continuity
Growth
Human-cells
Proteins
Nucleic Acids
COHN - Natures Engineering Of The Human Body
The Human-Body Systems
Vision
Walking
Behaviors
Sensors Sensings
Beauty
Faith, Love, Charity
Photosynthesis
Weather
Systems
Algorithms
Tools
Networks
Search
Differential Calculus
Antiderivative
Integral Calculus
Economies
Inflation
Markets
Money Supply
Painting
The bonding force F, between atoms may be expressed approximately as follows:
F(r) = A/rM - B/rN (N > M) -----------(1)
Where r, is the center-to-center spacing between atoms and A, B, M, and N are constants that vary according to the type of bond.
A/rM represents the attractive force while B/rN represents the repulsive force.
(a) Express the equilibrium spacing r0 in terms of A, B, M, and N.
(b) Derive another form for F(r) in which the only constants are r0, A, M, and N.
(c) From the equation for F(r) derived in (b) calculate the following:
(i) The spacing r1 for which F is maximum
(ii) The value of the maximum force Fmax.
The strings:
(a) S7P7A72 (Dynamic - Equilibrium)
(b) S7P3A31 (Force - Pull)
(c)(i) S7P1A12 (Containership -Length)
(c)(ii) S7P3A31 (Force-Pull)
The math:
Pj Problem of Interest is of type force (electrostatic). Problem is primarily of type force since interatomic force underlies problems (a), (b) and (c).
F(r) = A/rM - B/rN (N > M) -----------(1)
At the equilibrium spacing r0, the resultant interatomic force is zero, that is the attractive force balances the repulsive force.
So, F(r0) = A/(r0)M - B/(r0)N = 0
So, A/(r0)M = B/(r0)N
So,(r0)N-M = B/A -------(2)
So, equilibrium spacing, r0 = (B/A)(1/N-M)-------(3)
From equation (2) B = A(r0)N-M
Let A(r0)N-M replace B in equation (1) :
Then, F(r) = A/rM[1 - ((r0)N-M/rN-M)]-------(4)
(c)(i) Let r1 be the spacing for which F is maximum
Then the derivative of F at r1, dF(r1)/dr = 0.
Using F(r) of equation (4):
dF(r)/dr = -MAr-(M+1) + NA(r0)N-M(r-(N+M))
dF(r1)/dr = 0, implies:
NA(r0)N-M((r1)-(N+M)) = MA(r1)-(M+1)
So, (N/M)(r0)N-M = [(r1)-M]/[(r1)-N]
So,(N/M)(r0)N-M = (r1)N-M
So, The spacing r1, for which F is maximum:
r1 = r0(N/M)(1/N-M).
(c)(ii) Substitute the expression for r1 in equation 4 to get the expression for the value of maximum Force, Fmax.
The attractive forces in interatomic bonds are primarily electrostatic. So, the expression A/rM for the attractive force is usually assumed to be the same as the forces between electric charges. Consequently, Coulomb's Law applies and the force is inversely proportional to the square of the spacing. So M is usually = 2. The value of N is more dependent on the type of bond. For metallic bonds, N ranges from 7 to 10. For ionic and covalent bond, N ranges from 10 to 12.
The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring
Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle
Derivation Of The Area Of An Ellipse
Derivation Of Volume Of A Cylinder
Derivation Of Volume Of A Sphere
Derivation Of Volume Of A Cone
Derivation Of Volume Of A Torus
Derivation Of Volume Of A Paraboloid
Volume Obtained By Revolving The Curve y = x2 About The X Axis
Single Variable Functions
Absolute Value Functions
Conics
Real Numbers
Vector Spaces
Equation Of The Ascent Path Of An Airplane
Calculating Capacity Of A Video Adapter Board Memory
Probability Density Functions
Boolean Algebra - Logic Functions
Ordinary Differential Equations (ODEs)
Infinite Sequences And Series
Introduction To Group Theory
Advanced Calculus - Partial Derivatives
Advanced Calculus - General Charateristics Of Partial Differential Equations
Advanced Calculus - Jacobians
Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables
Advanced Calculus - Fourier Series
Advanced Calculus - Multiple Integrals
Production Schedule That Maximizes Profit Given Constraint Equation
Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation
Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions
Fourier Series
Derivation Of Heat Equation For A One-Dimensional Heat Flow
Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition
The Universe is composed of matter and radiant energy. Matter is any kind of mass-energy that moves with velocities less than the velocity of light. Radiant energy is any kind of mass-energy that moves with the velocity of light.
Periodic Table
Composition And Structure Of Matter
How Matter Gets Composed
How Matter Gets Composed (2)
Molecular Structure Of Matter
Molecular Shapes: Bond Length, Bond Angle
Molecular Shapes: Valence Shell Electron Pair Repulsion
Molecular Shapes: Orbital Hybridization
Molecular Shapes: Sigma Bonds Pi Bonds
Molecular Shapes: Non ABn Molecules
Molecular Orbital Theory
More Pj Problem Strings