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Avogadro’s law

Previously in Chemistry one has learned about Avogadro’s hypothesis:

Equal volumes of any gas, at the same temperature and pressure, contain the same number of molecules.

Avogardo's Hypothesis gas

Reasoning 

(from Modern Chemistry, Davis, HRW)

In 1811, Avogadro found a way to explain Gay-Lussac’s simple ratios of combining volumes without violating Dalton’s idea of indivisible atoms. He did this by rejecting Dalton’s idea that reactant elements are always in monatomic form when they combine to form products. He reasoned that these molecules could contain more than one atom.

Avogadro also put forth an idea known today as Avogadro’s law: equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

It follows that at the same temperature and pressure, the volume of any given gas varies directly with the number of molecules.

Avogadro’s law also indicates that gas volume is directly proportional to the amount of gas, at a given temperature and pressure.

Note the equation for this relationship.

   V = kn

Here, n is the amount of gas, in moles, and k is a constant.

Avogadro’s reasoning applies to the combining volumes for the reaction of hydrogen and oxygen to form water vapor.

Dalton had guessed that the formula of water was HO, because this formula seemed to be the most likely formula for such a common compound.

But Avogadro’s reasoning established that water must contain twice as many H atoms as O atoms, consistent with the formula H2O.

As shown below, the coefficients in a chemical reaction involving gases indicate the relative numbers of molecules, the relative numbers of moles, and the relative volumes.

Avogadro gas reaction

The simplest hypothetical formula for oxygen indicated 2 oxygen atoms, which turns out to be correct. The simplest possible molecule of water indicated 2 hydrogen atoms and 1 oxygen atom per molecule, which is also correct.

Experiments eventually showed that all elements that are gases near room temperature, except the noble gases, normally exist as diatomic molecules.

As an equation

Avogadro’s Law – also known as Avogadro–Ampère law

when temperature and pressure are held constant:

volume of a gas is directly proportional to the # moles (or # particles) of gas

n1 / V1 = n2 / V2

or

Avogadro's Law gas

What does this imply?

As # of moles of gas increases, the volume of the gas also increases.

As # of moles of gas is decreased, the volume also decreases.

Thus, # of molecules (or atoms) in a specific volume of ideal gas is independent of their size (or molar mass) of the gas.

 

Example problems

These problems are from The Chem Team, Kinetic Molecular Theory and Gas Laws

Example #1: 5.00 L of a gas is known to contain 0.965 mol. If the amount of gas is increased to 1.80 mol, what new volume will result (at an unchanged temperature and pressure)?

Solution:

I’ll use V1n2 = V2n1

(5.00 L) (1.80 mol) = (x) (0.965 mol)

x = 9.33 L (to three sig figs)


Example #2: A cylinder with a movable piston contains 2.00 g of helium, He, at room temperature. More helium was added to the cylinder and the volume was adjusted so that the gas pressure remained the same. How many grams of helium were added to the cylinder if the volume was changed from 2.00 L to 2.70 L? (The temperature was held constant.)

Solution:

1) Convert grams of He to moles:

2.00 g / 4.00 g/mol = 0.500 mol

2) Use Avogadro’s Law:

V1 / n1 = V2 / n2

2.00 L / 0.500 mol = 2.70 L / x

x = 0.675 mol

3) Compute grams of He added:

0.675 mol – 0.500 mol = 0.175 mol

0.175 mol x 4.00 g/mol = 0.7 grams of He added


Example #3: A balloon contains a certain mass of neon gas. The temperature is kept constant, and the same mass of argon gas is added to the balloon. What happens?

(a) The balloon doubles in volume.
(b) The volume of the balloon expands by more than two times.
(c) The volume of the balloon expands by less than two times.
(d) The balloon stays the same size but the pressure increases.
(e) None of the above.

Solution:

We can perform a calculation using Avogadro’s Law:

V1 / n1 = V2 / n2

Let’s assign V1 to be 1 L and V2 will be our unknown.

Let us assign 1 mole for the amount of neon gas and assign it to be n1.

The mass of argon now added is exactly equal to the neon, but argon has a higher gram-atomic weight (molar mass) than neon. Therefore less than 1 mole of Ar will be added. Let us use 1.5 mol for the total moles in the balloon (which will be n2) after the Ar is added. (I picked 1.5 because neon weighs about 20 g/mol and argon weighs about 40 g/mol.)

1 / 1 = x / 1.5

x = 1.5

answer choice (c).


Example #4: A flexible container at an initial volume of 5.120 L contains 8.500 mol of gas. More gas is then added to the container until it reaches a final volume of 18.10 L. Assuming the pressure and temperature of the gas remain constant, calculate the number of moles of gas added to the container.

Solution:

V1 / n1 = V2 / n2

5.120 L 18.10 L
–––––––– = ––––––
8.500 mol x

x = 30.05 mol <— total moles, not the moles added

30.05 – 8.500 = 21.55 mol (to four sig figs)

Notice the specification in the problem to determine moles of gas added. The Avogadro Law calculation gives you the total moles required for that volume, NOT the moles of gas added. That’s why the subtraction is there.

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Charles’s Law

Charles’s Law – also known as Charles and Gay-Lussac’s Law.

Describes how gases tend to expand when heated.

When the pressure on a sample of a dry gas is held constant, the temperature and the volume will be in direct proportion.

Volume proportional to temperature

(Only true when measuring temperature on an absolute scale)

This relationship can be written as:

Charles's law gas

-> Gas expands as the temperature increases

-> Gas contracts as the temperature decreases

This relationship can be written as:

Charles's law gas alternate

Important! This is not a law of physics! Rather, this is a generally useful rule, which is only valid when gas temperature and pressure is low enough for the atoms to usually be far apart from each other.  As we begin to deal with more extreme cases, this rule doesn’t hold up.

Let’s see this in action!

Origin

Named after Jacques Alexandre César Charles (1746 – 1823)  a French inventor, scientist, mathematician, and balloonist. Just so we’re all clear on this, he was kind of a mad scientist. And I say that with the utmost approval!

first balloon flight by Charles and Robert 1783

Contemporary illustration of the first flight by Prof. Jacques Charles with Nicolas-Louis Robert, December 1, 1783. Viewed from the Place de la Concorde to the Tuileries Palace (destroyed in 1871)

 

.Apps

Charles’s law app

Learning standards

Massachusetts Science and Technology/Engineering Curriculum Framework

8.MS-PS1-4. Develop a model that describes and predicts changes in particle motion, relative spatial arrangement, temperature, and state of a pure substance when thermal energy is added or removed.

Next Generation Science Standards

MS-PS1-4. Develop a model that predicts and describes changes in particle motion, temperature, and state of a pure substance when thermal energy is added or removed.

College Board Standards

Objective C.1.5 States of Matter

C-PE.1.5.2 Explain why gases expand to fill a container of any size, while liquids flow and spread out to fill the bottom of a container and solids hold their own shape. Justification includes a discussion of particle motion and the attractions between the particles.

C-PE.1.5.3 Investigate the behavior of gases. Investigation is performed in terms of volume (V ), pressure (P ), temperature (T ) and amount of gas (n) by using the ideal gas law both conceptually and mathematically.

Common Core Math

Analyze proportional relationships and use them to solve real-world and mathematical problems.

Ratios & Proportional Relationships

Ratios & Proportional Relationships

CCSS.MATH.CONTENT.7.RP.A.2

Recognize and represent proportional relationships between quantities.

CCSS.MATH.CONTENT.7.RP.A.2.A

Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

CCSS.MATH.CONTENT.7.RP.A.2.B

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

Pascal’s Principle

Pressure applied to an enclosed, incompressible, static fluid is transmitted undiminished to all parts of the fluid.

Hydraulic systems operate according to Pascal’s law.

11.5 Pascal’s Principle

  • Define pressure.
  • State Pascal’s principle.
  • Understand applications of Pascal’s principle.
  • Derive relationships between forces in a hydraulic system.
Pascal's principle hydraulics

image from littlewhitecoats.blogspot.com

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Learning Standards

tba

 

 

Bernoulli’s equation

Bernoulli’s equation

This is the law of conservation of energy as applied to flowing fluids. We can explain to department heads and parents that basics ideas – such as conservation of energy – appear everywhere in life, everywhere in science, so it is important for us to see examples of how they play out, such as in Bernoulli’s equation.

Online textbook

Bernoulli’s Equation

The Most General Applications of Bernoulli’s Equation

Viscosity and Laminar Flow; Poiseuille’s Law

The onset of turbulence

Motion of an Object in a Viscous Fluid

Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

 

Applications of the Bernoulli effect

Air flow over airplane wings

 

Resources

NASA How the Bernoulli equation works in rockets

Khan Academy What-is-Bernoulli’s-equation

Bernoulli’s Equation OpenStax College

4physics The Bernoulli effect

Energyeducation.ca Bernoulli’s equation

Apps

PhET fluid pressure and flow

lmnoeng.com Bernoulli equation calculator

Endmemo.com Bernoulli equation calculator

Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

HS-PS3-1. Use algebraic expressions and the principle of energy conservation to calculate the change in energy of one component of a system when the change in energy of the other component(s) of the system, as well as the total energy of the system including any energy entering or leaving the system, is known.

Disciplinary Core Idea Progression Matrix

PS3.A and 3.B: The total energy within a physical system is conserved. Energy transfer within and between systems can be described and predicted in terms of energy associated with the motion or configuration of particles (objects)

NGSS leaves out critical guidance on importance of teaching about vectors

NGSS Logo

As we all know the NGSS are more about skills than content. Confusingly, though, they ended up also listing core content topics as well – yet they left out kinematics and vectors, the basic tools needed for physics in the first place.

The NGSS also dropped the ball by often ignoring the relationship of math to physics. They should have noted which math skills are needed to master each particular area.

Hypothetically, they could have had offered options: For each subject, note the math skills that would be needed to do problem solving in this area, for

* a standard (“college prep”) level high school class
* a lower level high school class, perhaps along the lines of what we call “Conceptual Physics” (still has math, but less.)
* the highest level of high school class, the AP Physics level. And the AP study guides already offer what kinds of math one needs to do problem solving in each area.

Yes, the NGSS does have a wonderful introduction to this idea, (quoted below) – but when we look at the actual NGSS standards they don’t mention these skills.

In some school districts this has caused confusion, and even led to some administrators demanding that physics be taught without these essential techniques (i.e. kinematic equations, conceptual understanding of 2D motion, kinematic analysis of 2D motion, vectors, etc.)

To help back up teachers in the field I put together these standards for vectors, from both science and mathematics standards.

– Robert Kaiser

Learning Standards

Massachusetts Science Curriculum Framework (pre 2016 standards)

1. Motion and Forces: Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.1 Compare and contrast vector quantities (e.g., displacement, velocity, acceleration force, linear momentum) and scalar quantities (e.g., distance, speed, energy, mass, work).

NGSS

Science and Engineering Practices: Using Mathematics and Computational Thinking

Mathematical and computational thinking in 9–12 builds on K–8 experiences and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions.

  • Apply techniques of algebra and functions to represent and solve scientific and engineering problems.

Although there are differences in how mathematics and computational thinking are applied in science and in engineering, mathematics often brings these two fields together by enabling engineers to apply the mathematical form of scientific theories and by enabling scientists to use powerful information technologies designed by engineers. Both kinds of professionals can thereby accomplish investigations and analyses and build complex models, which might otherwise be out of the question. (NRC Framework, 2012, p. 65)

Students are expected to use mathematics to represent physical variables and their relationships, and to make quantitative predictions. Other applications of mathematics in science and engineering include logic, geometry, and at the highest levels, calculus…. Mathematics is a tool that is key to understanding science. As such, classroom instruction must include critical skills of mathematics. The NGSS displays many of those skills through the performance expectations, but classroom instruction should enhance all of science through the use of quality mathematical and computational thinking.

Common Core Standards for Mathematics (CCSM)

High School: Number and Quantity » Vector & Matrix Quantities. Represent and model with vector quantities.

Represent and model with vector quantities.

CCSS.MATH.CONTENT.HSN.VM.A.1
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
CCSS.MATH.CONTENT.HSN.VM.A.2
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
CCSS.MATH.CONTENT.HSN.VM.A.3
(+) Solve problems involving velocity and other quantities that can be represented by vectors.

Perform operations on vectors.

CCSS.MATH.CONTENT.HSN.VM.B.4
(+) Add and subtract vectors.
CCSS.MATH.CONTENT.HSN.VM.B.4.A
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
CCSS.MATH.CONTENT.HSN.VM.B.4.B
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
CCSS.MATH.CONTENT.HSN.VM.B.4.C
Understand vector subtraction v – w as v + (-w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
CCSS.MATH.CONTENT.HSN.VM.B.5
(+) Multiply a vector by a scalar.
CCSS.MATH.CONTENT.HSN.VM.B.5.A
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vxvy) = (cvxcvy).
CCSS.MATH.CONTENT.HSN.VM.B.5.B
Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
American Association for the Advancement of Science

Operating with Symbols and Equations

  • Become fluent in generating equivalent expressions for simple algebraic expressions and in solving linear equations and inequalities.
  • Develop fluency operating on polynomials, vectors, and matrices using by-hand operations for the simple cases and using technology for more complex cases.
AAAS Benchmarks (American Association for the Advancement of Science)
9B9-12#2: Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly. Symbolic statements can be combined to look for values of variables that will satisfy all of them at the same time.
9B9-12#5: When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes more than one, and sometimes maybe not at all.
12B9-12#2: Find answers to problems by substituting numerical values in simple algebraic formulas and judge whether the answer is reasonable by reviewing the process and checking against typical values.
12B9-12#3: Make up and write out simple algorithms for solving problems that take several steps.

#vectors #teaching #standards #kinematics #physics #kaiserscience #pedagogy #education #NGSS #Benchmarks #scalors #highschool

Two and three dimensional motion

Most high school physics courses don’t include algebraic analysis of two or three dimensional kinematics and momentum. But these clearly are of great importance.

In a regular, college prep physics high school setting, I can’t imagine skipping 2D physics! Even if we don’t do 2D kinematic equations, we need to cover 2D vectors, and show examples of parabolic motion, and vector components.

https://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Components

and

https://www.physicsclassroom.com/class/vectors/Lesson-2/What-is-a-Projectile

Even if we are not doing the math, I want them to see examples of conservation of momentum in two dimensions, like this:

https://www.physicsclassroom.com/mmedia/momentum/2di.cfm

2D cars colliding momentum

image from physicsclassroom

These seem like great apps for teaching 2D kinematics without all of the detailed calculations.

https://www.physicsclassroom.com/Physics-Interactives/Vectors-and-Projectiles

We won’t be able to do three dimensional collision or momentum problem solving, but we can at least introduce the idea of it, and show them why almost every collision and conservation of momentum in the real world is 3D:

First students need to be introduced to the idea that there are more than just two axes (X and Y,) there is a third dimension, Z!

XYZ axes 3D

Then we see how we can plot points in three dimensions.

This is the GeoGebra app

Now we realize that the size and motion of any object can be plotted in three dimensions.

The physics of dogs and cats colliding GIF

dogs and cats colliding 3d

Two galaxies colliding, and the resulting amazing display

https://media2.giphy.com/media/T2C2efftbhMTC/giphy.webp?cid=790b7611ec6f62c760e4a1059d923e3f271feb87182a154f&rid=giphy.webp

A practical use of 2D kinematics and conservation of momentum: forensic accident reconstruction.

forensic reconstruction traffic accident 2d GIF

 

Boyle’s law (gas laws)

A general relationship between pressure and volume: Boyle’s Law

As the pressure on a gas increases, the volume of the gas decreases because the gas particles are forced closer together.

Conversely, as the pressure on a gas decreases, the gas volume increases because the gas particles can now move farther apart.

Example: Weather balloons get larger as they rise through the atmosphere to regions of lower pressure because the volume of the gas has increased; that is, the atmospheric gas exerts less pressure on the surface of the balloon, so the interior gas expands until the internal and external pressures are equal.

from Libretexts, Chemistry, 5.3: The Simple Gas Laws: Boyle’s Law, Charles’s Law and Avogadro’s Law, CC BY-NC-SA 3.0.

This means that, at constant temperature, the pressure (P) of a gas is inversely proportional to the volume (V).

PV = c

Important! This is not a law of physics! Rather, this is a generally useful rule, which is only valid when gas temperature and pressure is low enough for the atoms to usually be far apart from each other.  As we begin to deal with more extreme cases, this rule doesn’t hold up.

Let’s see the relationship in action, here:

Boyle's law pressure temp

from http://www.grc.nasa.gov/WWW/K-12/airplane/boyle.html

How was this general rule discovered?

Early scientists explored the relationships among the pressure of a gas (P) and its temperature (T), volume (V), and amount (n) by holding two of the four variables constant (amount and temperature, for example), varying a third (such as pressure), and measuring the effect of the change on the fourth (in this case, volume).

The history of their discoveries provides several excellent examples of the scientific method.

The Irish chemist Robert Boyle (1627–1691) carried out some of the earliest experiments that determined the quantitative relationship between the pressure and the volume of a gas. Boyle used a J-shaped tube partially filled with mercury.

In these experiments, a small amount of a gas or air is trapped above the mercury column, and its volume is measured at atmospheric pressure and constant temperature. More mercury is then poured into the open arm to increase the pressure on the gas sample.

The pressure on the gas is atmospheric pressure plus the difference in the heights of the mercury columns, and the resulting volume is measured. This process is repeated until either there is no more room in the open arm or the volume of the gas is too small to be measured accurately.

Boyle's Law pressure temp of a gas

Details: Boyle’s Experiment Using a J-Shaped Tube to Determine the Relationship between Gas Pressure and Volume.

(a) Initially the gas is at a pressure of 1 atm = 760 mmHg (the mercury is at the same height in both the arm containing the sample and the arm open to the atmosphere); its volume is V.

(b) If enough mercury is added to the right side to give a difference in height of 760 mmHg between the two arms, the pressure of the gas is 760 mmHg (atmospheric pressure) + 760 mmHg = 1520 mmHg and the volume is V/2.

(c) If an additional 760 mmHg is added to the column on the right, the total pressure on the gas increases to 2280 mmHg, and the volume of the gas decreases to V/3

(This section from from Libretexts, Chemistry, 5.3: The Simple Gas Laws: Boyle’s Law, Charles’s Law and Avogadro’s Law, CC BY-NC-SA 3.0)

 

Learning standards

Massachusetts Science and Technology/Engineering Curriculum Framework

8.MS-PS1-4. Develop a model that describes and predicts changes in particle motion, relative spatial arrangement, temperature, and state of a pure substance when thermal energy is added or removed.

Next Generation Science Standards

MS-PS1-4. Develop a model that predicts and describes changes in particle motion, temperature, and state of a pure substance when thermal energy is added or removed.

College Board Standards

Objective C.1.5 States of Matter

C-PE.1.5.2 Explain why gases expand to fill a container of any size, while liquids flow and spread out to fill the bottom of a container and solids hold their own shape. Justification includes a discussion of particle motion and the attractions between the particles.

C-PE.1.5.3 Investigate the behavior of gases. Investigation is performed in terms of volume (V ), pressure (P ), temperature (T ) and amount of gas (n) by using the ideal gas law both conceptually and mathematically.

Common Core Math

Analyze proportional relationships and use them to solve real-world and mathematical problems.

Ratios & Proportional Relationships

Ratios & Proportional Relationships

CCSS.MATH.CONTENT.7.RP.A.2

Recognize and represent proportional relationships between quantities.

CCSS.MATH.CONTENT.7.RP.A.2.A

Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

CCSS.MATH.CONTENT.7.RP.A.2.B

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.