Expressions Of Pj Problems

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

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Energy

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COHN - Natures Engineering Of The Human Body

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Differential Calculus

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Painting

Equilibrium Temperature Of Water Mixed With Ice

40 grams of ice at 0^{o}C is mixed with 100 grams of water at 60^{o}C.

Determine the final temperature of the water after equilibrium has been established.

Heat of fusion of water (H_{2}O) = 80 cal/gram

Heat capacity/Specific Heat of water = 1 cal/gram degrees C.

**The strings**:

S_{7}P_{5}A_{51} (change - physical change)
**The math**:

Pj Problem of Interest is of type *change*. The melting of the ice in the water changed the water temperature from 60^{o}C to an equilibrium temperature. It is this final temperature due to temperature change that is of interest eventhough heat energy is operative.

**Heat Of Vaporization**: the quantity of heat required to vaporize 1 gram of a liquid substance at its boiling point at constant temperature.
**Heat Of Fusion**: the quantity of heat required to liquefy 1 gram of a solid substance at its melting point at constant temperature.
**Specific Heat**: number of calories required to raise the temperature of 1 gram of liquid by 1 degree.

Heat accounting:

(1) Ice needs heat to melt

(2) The cold water from the melted ice needs heat to warm up to the equilibrium temperature.

(3) The sum of the heat from (1) and (2) is total heat absorbed.

(4) The water at 60^{o}C is the source of the heat needed for (3). Therefore it will loose that amount of heat.

Heat required to melt 40 grams of ice = 40 x 80 = 3200 cal.

Amount of heat absorbed by melted water before attaining final temperature, t (equilibrium temperature):

= 40 x specific heat of water x t = 40 x 1 x t = 40t.

So, total heat absorbed = 3200 + 40t.

Temperature change of water from 60^{0}C to t^{o}C = 60 - t

So, amount of heat lost by water at 60^{o}C

= weight of water x specific heat of water x temperature change

= 100 x 1 x (60 - t) = 100(60 - t) = 6000 -100t

Heat absorbed = heat lost

So, 3200 + 40t = 6000 - 100t

So, 2800 = 140t

So, t = 20^{o}C.

So, final temperature (equilibrium temperature) = 20^{o}C.

Temperature of water dropped by 40^{o}C.

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