Expressions Of Pj Problems

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

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Chemical Bonds

Energy Conversion

Chemical Reactions

Electromagnetism

Continuity

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Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

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Sensors Sensings

Beauty

Faith, Love, Charity

Photosynthesis

Weather

Systems

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Tools

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Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

Energy In A Mole Of A Crystalline Solid

The above photo is of a sample of crystalline scandium (Sc, atomic number 21).

The atoms and molecules of a solid are in constant motion at ordinary temperatures eventhough there is equilibrium spacing between them. Thermal agitation due to heat causes the atoms and molecules to oscillate about their equilibrium positions oftentimes at very high frequency. The resulting kinetic energy acquired by the atoms and molecules constitute the thermal energy of the substance.

Consequently, both kinetic energy (thermal) and potential energy (structure) consitute the total energy in solids and liquids (this potential energy is not in gases).

(a) Suppose an individual atom of a crystalline solid behaves as a point mass. Calculate the total energy of the atom at 25^{o}C.

(b) Calculate the total energy in one mole of the crystalline solid 25^{o}C.

**The strings**:

S_{7}P_{3}A_{32} (Force - Push)
**The math**:

Pj Problem of Interest is of type *force*. Energy is the capacity for work. It is force that is the doer of the work. So energy and work problems are of type *force*. In this instance, the total energy consists of kinetic energy and potential energy about 50-50. The kinetic energy is due to atom collissions (more of a *force-push*), the potential energy is more of a *force-pull*. So either categorization is appropriate. *force-push* is selected here because of the influence of temperature.

(a) On average, the total energy of each atom of a crystalline solid = kT for each degree of freedom.

Where, T is the absolute temperature in degree Kelvin (K)

and k is Boltzmann's Constant = 3.30 x 10^{-24} (calorie/^{o}K)/atom.

Since the atom is assumed to be a point mass, it has 3 degrees of freedom in translation and none in rotation.

So, average total energy of atom = 3kT = 3 x 3.30 x 10^{-24} x 298

= 2.77 x 10^{-22} cal.

(b) A *mole* of a crystalline substance contains N atoms.

Where N is Avogadro's Number = 6.02 x 10^{-22}

So, total energy in one mole of crystalline solid = 3NkT

The product, Nk is called the *universal gas constant*, R

R = 1.987 (cal/^{o}K)/mole

So, total energy in one mole of crystalline solid at 25^{o}C = 1788 cal.

Energy = 3RT is approximately correct for many materials at ordinary temperatures. It is not correct at low temperatures.

Math

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 = x^{2} 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