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Great physics discussion questions!

Great physics discussion questions! These were written by Physicist Dr. Matt Caplan, who used to run the QuarksAndCoffee blog. That blog no longer exists, links to archived copies exist.

If everyone was Kung fu fighting, and their kicks really were fast as lightning, what would happen?

How many calories do superheroes burn using their powers?

What would happen if a 10 meter plasma sphere was transported from the Sun to Earth?

Dyson Sphere: Is there enough material in the solar system to build a shell to enclose the sun?

If it was cold enough for the atmosphere to condense, how deep would the ‘liquid air’ be?

Does my phone weigh more when the battery is charged?

How loud would a literal ‘shot heard round the world’ be?

Why are there seven colors in the rainbow?

Math

What are the odds of solving a Rubik’s cube by making random moves?

Why are some moons spherical while others are shaped like potatoes?

Why are some moons spherical while others are shaped like potatoes?

This blog post was written by Physicist Dr. Matt Caplan, who used to run the QuarksAndCoffee blog. That blog no longer exists, but I’m showing this archived copy of one of his posts for my students.
__________________________________

 

Why are some moons spherical while others are shaped like potatoes?

Short answer: Gravity likes to pull things together, which makes spheres. If the body is small enough gravity isn’t strong enough to deform it, which makes potatoes.

Long answer: Put a ball on top of a hill. What happens? It rolls down to the bottom. Why? Because gravity said so. This isn’t just how it works on the earth, but everywhere in the universe. Clearly, gravity is trying to make spheres. If you tried to dig a super deep hole stuff would fall in from the edges to fill it up. And what happen if we start to pile up rocks? Eventually, the pile of rocks reaches the point where it will all crumble down under its own weight. A sphere is the only shape that has no holes to fill or hills to crush. This is why every planet and star in the universe is round.

Of course, the earth and moon and planets aren’t perfect spheres. They’re lumpy. They’ve got hills and valleys and although none of them are that big compared to the planet, they’re still there. This is because gravity is strong enough to destroy (or prevent the formation) of a really big mountain, but not a small mountain. A small mountain’s own rigidness is enough to support its weight against gravity [1].

Mountain failure How High

This image shows two failure modes for mountains. The mountain on the left experiences shear failure, with the stress from the weight above the diagonal line exceeding the breaking point of the material. The mountain on the right fails due to compression of the base material.

Because materials have some intrinsic rigidity there must be bodies whose gravity isn’t strong enough to pull them into a sphere. Rather, the material is stiff enough to keep an oblong shape. After all, satellites and astronauts and cows don’t collapse into spheres in space.

The limit where gravity is strong enough to overcome the material properties of a body and pull it into a sphere is called the Potato Radius, and it effectively marks the transition from asteroid to dwarf planet [2]. It’s about 200-300 km, with rocky bodies having a slightly larger Potato Radius than icy bodies.

You can use some complicated math with material elasticity, density, and gravity to calculate the Potato Radius from scratch, or you could just look at Mt Everest. It turns out that the same physics determining the maximum height of mountains can be used to determine the Potato Radius – after all, they’re both just the behavior of rocks under gravity.

Check this out. The heights of the tallest mountains on Earth and Mars obey an interesting relation:

height gravity mountains equation

If you know the height of Everest and that Mars surface gravity is 2/5ths of Earth, then you know that Olympus Mons (tallest mountain on Mars) is about 5/2× taller than Everest! This relation also works with Maxwell Montes, the tallest mountain on Venus, but not for Mercury. Planetary science is a lot like medicine in this sense- there are always exceptions because everything is completely dependent on the body you’re looking at.

This is more than a curiosity. It tells us something important. The height of the tallest mountain a planet can support, multiplied by that planet’s surface gravity, is a constant.

For this sake of this piece I’ll call it the Rock Constant because that sounds cool. So why am I spending so long on a tangent about mountains in a piece about potato moons? It’s because the Potato Radius and Rock Constant are determined by the same things – gravity and the elasticity of rock! We can use the Rock Constant to estimate the Potato Radius!

Consider an oblong asteroid. Let’s pretend this asteroid is actually a sphere with a large mountain whose height is equal to the radius of that sphere.

potato shaped asteroid or moon

As the radius of a body increases the maximum height of a mountain decreases. If the radius was any bigger the mountain would have to be shorter and our asteroid would be entering ‘sphere’ territory. Let’s check if the radius of this imaginary asteroid is close to the Potato Radius using our relation for the Rock Constant:

height asteroid gravity equation

Height asteroid 3

And now we have everything we need:

 

height asteroid 4

This works out to about 240 km [1], right in the middle of the 200-300 km range of the more rigorous calculation!

(1) How High Can A Mountain Be? P. A. G. Scheuer, Journal of Astrophysics and Astronomy, vol. 2, June 1981, p. 165-169.

Web.archive.org copy
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How high can a mountain grow?

There is some kind of process that builds mountains, but there also must be something limiting that process. After all, we don’t see mountains 20 or 30 miles tall, right? So we must ask, how high can a mountain grow?

We start by asking, what are the highest mountains on Earth?

Which then brings up the next question, what do we mean by “highest”? The answer isn’t obvious because there are three different ways to think about “highest” – see this diagram.

Highest Mountain on the Earth

Given this, we next notice that most mountains on Earth are nowhere near this height. For instance, the highest mountain in New England is Mount Washington New Hampshire 1,900 m (6,300 ft.). The highest mountain in the Rocky Mountains in Mount Elbert in Colorado 4400 m (14,000 ft.)

In general, almost everywhere on our planet, the highest that a mountain can be is about half the height of Everest.  This is as tall as a mountain can grow on a lithospheric tectonic plate.

So our next question is, “why is there one set of rules for the highest that a mountain can be almost everywhere on Earth, and why do some locations have exceptions?”

What factors control the height of a mountain?

There is a balance of the forces:

Tectonic plate forces pushes the Earth’s crust upward.

Gravity pulls the mountain downward.

And, when the mountain is high & big enough, the weight of the mountain can crack and shatter the rock inside of it. This causes the mountain to crumble, and settle down to a lower height.

Don’t believe me? Even rock has a maximum amount of strength. Here is a GIF of what happens to solid rock when you put enough pressure on it! 🙂

Source: Unconfined compressive strength test of rock

Thus, if the weight of mountain > yield strength of the base rock then the mountain’s base will crumble.

Then he mountain will compress down to the maximum allowable height.

Of course, when this happens depends on what the mountain is made of. SiO2 is the most common molecule. But there are many minerals that are lighter, or stronger, or both, that can also be found in a mountain.

By the way, this gives us a neat relation – the surface gravity X maximum height of a mountain should be a constant.

Formula lets us relate height of Mt Everest on earth and Olympus Mons on Mars. Or find max deformation of asteroid before gravity pulls it into a sphere.

All the other downward forces on a mountain

Erosion wears the mountain down

How well does the mountain resist weathering/erosion? This depends on what kind of chemicals it is made out of.

Does being in the ocean affect how high a mountain can be?

Consider Mauna Kea, in Hawaii.

Much of Mauna Kea is underwater. It’s base can support more pressure since it’s underwater. Underwater, there is a buoyant force on the object that counteracts the force of gravity. Since nothing counteracts the gravity on Mount Everest, the mountain’s base can only support so much pressure.

Roy Blitz

What else makes mountains rise or grow?

Even while a mountain is eroding, the underlying plate activity may be forcing the mountain to grow higher.

A tectonic plate pushing more directly against another plate will create higher mountains than a plate moving less directly (say, at an angle) against another plate.

How strong are the crustal roots of the mountain?

As a mountain range grows in height, this root grows in depth, and thus the pressure and temperature experienced by the bottom of this root increases.

At a certain point, rocks in the base of this crustal root metamorphose into a rock called eclogite. At that point this rock will be denser than the material supporting the crustal root.

This causes delamination to occur. Depending on the amount of material removed, the rate of new material added, and erosion, scenarios with net increases or decreases in elevation are possible after a delamination event. This sets another limit on how thick a crustal root can get (and thus how high a mountain range grow on the long term).

Source, Reddit comment

Why are there some special spots on Earth where mountains can grow twice as high?

George W Hatcher writes

Mauna Kea rests on oceanic crust, which is denser than continental crust and able to support more weight without displacement. Being mostly inundated with seawater precludes some of the erosional processes to which mountains exposed to the upper atmosphere are subjected.

In addition, the very material of which Mauna Kea is composed (basaltic igneous rocks) is stronger than the variety of rocks that make up the continental crust and uplifted limestone seafloor that can be found atop Everest.

The actual lithospheric limit to mountain height averages about half the height of Everest, which is why Fourteeners are so famous in Colorado. Mountains that exceed this limit have local geologic circumstances that make their height possible, e.g. stronger or denser rocks.

In the case of Everest and the Himalayas, you have a geologic situation that is very rare in Earth history. The Indian plate is ramming into the Eurasian plate with such force that instead of just wrinkling the crust on either side into mountain ranges it has actually succeeded in lifting the Eurasian plate up on top.

So the Himalayas have double the thickness of the average continental plate, thus double the mountain height that would be considered “normal”.

George W Hatcher, Planetary Scientist, Aerospace Engineer

References

How high can a mountain possibly get? Earth Science StackExchange

How High can a mountain get? 2 Earth Science Stack Exchange

How tall can a mountain become on Earth? Quora

What is the theoretical limit to how tall mountains can get on Earth? Reddit

Glacial Buzz Saw Hypothesis: New Scientist article

Examples with math details

Why are some moons spherical while others are shaped like potatoes? Quarks & Coffee

How High Can Mountains Be? Talking Physics

How High Could A Mountain Be? Physics World hk-phy.org

How tall can I make a column of stone? Rhett Allain, Wired magazine columnist

Related lab ideas

Play Doh Modeling Folds: Block Diagrams and Structure Contours

Play-Doh Modeling Folds: Block Diagrams and Structure Contours

Play Doh Introduction to Igneous Intrusions

Play Doh Unconformities

Preparing lessons in case schools close for coronavirus

Be Prepared Preparation

Due to the likely imminent arrival of coronavirus outbreaks in America, many school districts are preparing to fight the pandemic with the most effective tool: social distancing. This may include cancelling school, and other public events, for up to 2 weeks in affected areas.

As such, teachers may be asked to prepare 2 weeks worth of lessons for students to do at home as stand alone work. How do we efficiently prepare for this? I have been uploading resources to my website and creating worksheets based on them. In a pinch I could email our students 2 weeks worth of worksheets in PDF format, and print out packets for students with little or no internet access.

Teachers can supplement the work they assign  by creating a YouTube channel with daily mini-lectures. Students could listen & ask questions from their homes.

Teachers of course don’t need their own website. They can use any of their favorite science websites, such as The Physics Classroom.

However, we shouldn’t need something like the coronavirus to begin preparing. It is just good practice for teachers to create a library of ready-to-go, self-contained lesson plans for each topic. Instead of waiting for an emergency, make this part of your weekly schedule: Each week pick your best lesson, and write a self-contained worksheet for it.

As you create handouts (whether destined for PDF or paper) please think about readability, and about students who are slow readers or who have IEPs:

  • Have a brief, clear introduction so the student knows what we are learning and why. See below for details.

  • Use a large enough font.

  • Use double-spacing.

  • Have some space between each section.

  • Break long paragraphs into a smaller paragraphs.

  • Add a color graphic to help explain the concept in each section.

A packed page is a poorly-designed page. Trying to shove a class onto just a page or two is especially irrelevant since we are no longer limited by paper. Students can view as many pages as we need on their PCs, tablets, or phones.

Writing a brief, clear introduction

* Content objective:

Briefly discuss what we are learning, and/or why we are learning this. This may involve teaching new ideas, procedures, or skills.

* Vocabulary objective – What are the critical words in this lesson?

These include not only new terms that you introduce, but supposedly “common” words that one assumes the students “already know.” (The problem is that many students don’t always know what these terms means. Please click the link for more information.)

Tier II vocabulary words: High frequency words used across content areas. They are key to understanding directions, understanding relationships, and for making inferences.

Tier III vocabulary words: Low frequency, domain specific terms.

* Build on what we already know.

Very few lessons start completely from scratch. Most will include some vocabulary & scientific concepts that were learned in earlier grades. So in this section of your introduction, briefly make connections to prior concepts.

How are you preparing for this?

 

Reliable sources of information

CDC: Centers for Disease Control – Coronavirus Disease 2019 (COVID-19)

Massachusetts Department of Public Health

US FDA Food and Drug Administration Coronavirus Disease 2019

Coronavirus disease: Myth busters – WHO World Health Organization

How Do Airplanes Fly?

After all this time you’d think that we know how an airplane flies. There must be some simple, straightforward explanation. Air hits a plane, air and plane then follow appropriate laws of physics, and voilà, the plane flies, right?

But although the flight of airplanes indeed is in accord with the laws of physics, all of the traditional, straightforward ideas about how this happens are notoriously incomplete, if not incorrect.

What’s the controversy about, and what new ideas today are being proposed?

newtons 3rd law airplane reaction

Bernoulli theorem idea

Intro

and

How Airplanes Fly Bernoulli's Theorem

Newton’s laws of motion

Primary ideas

Reaction force on airplane wing

and

How Airplanes Fly Newton's Third Law

 

New Theories of lift

 

How Airplanes Fly New Ideas of Lift

 

References

No One Can Explain Why Planes Stay in the Air: Do recent explanations solve the mysteries of aerodynamic lift? By Ed Regis

Scientific American, February 2020, Volume 322, Issue 2

Aerodynamic Lift, Part 1: The Science, Doug McLean, The Physics Teacher Vol. 56, issue 8, 516 (2018)

https://doi.org/10.1119/1.5064558

Aerodynamic Lift, Part 2: A Comprehensive Physical Explanation, Doug McLean, The Physics Teacher Vol. 56, 521 (2018)

https://doi.org/10.1119/1.5064559

Understanding Aerodynamics: Arguing from the Real Physics, Doug McLean. Wiley, 2012

You Will Never Understand Lift. Peter Garrison, Flying; June 4, 2012.

Flight Vehicle Aerodynamics. Mark Drela, MIT Press, 2014.

av8n.com – by John S. Denker

#Flight #aerodynamics #Bernoulli #Lift

 

Iridescence and thin film interference

Iridescence is a spectacular optical trick – it is the creation of color without pigment!

Consider surfaces that gradually change color as the angle changes. Soap bubbles, feathers, butterfly wings, some seashells, and certain minerals. Let’s dig in to what causes this phenomenon.

GIF Iridescence Python snake scales

The word iridescence comes from Iris (Ἶρις) the Greek goddess of the rainbow.

There are three ways to get color

Additive color – mixing together light of two or more different colors. Red, green, and blue are the additive primary colors normally used in additive color systems such as smartphones, TVs, projectors and computer displays.

Subtractive color – uses dyes, inks, or pigments to absorb some wavelengths of light and not others. The color that we see comes from the wavelengths of light that are not absorbed by these chemicals.

But iridescence is nature’s special, third way of producing color. In this method, color is created by wave interference with tiny physical structures on the scale of the color’s wavelength.

Iridescence in animals

Iridescence Bird feathers

Iridescence: a functional perspective

Iridescence in minerals

synthetic bismuth

Bismuth is a great example of thin-film interference. The colors come from a thin film of bismuth(III) oxide that forms on the surface if the crystals are formed in air.

Chemistry.stackexchange What causes the iridescent colour in bismuth?

The physics of thin film interference

https://en.wikipedia.org/wiki/Thin-film_interference

http://physics.highpoint.edu/~jregester/potl/Waves/InterferenceColors/interfcolors.html

Videos for thin film interference

Apps

Molecular Expressions Interference Phenomena in Soap Bubbles

https://micro.magnet.fsu.edu/primer/java/interference/soapbubbles/

Optical Interference – Java Tutorial

https://www.olympus-lifescience.com/en/microscope-resource/primer/java/interference/

Molecular Expressions – Interference Between Parallel Light Waves

https://micro.magnet.fsu.edu/primer/java/interference/waveinteractions2/index.html

Physics Hanukkah Fun

During the holiday season many science teachers have a fun day on “The physics of Christmas.” What about “the physics of Hanukkah?” (*) Here’s an idea, would love feedback.

During this holiday Jews light a Chanukah menorah  מנורת חנוכה, also called a Ḥanukiyah חַנֻכִּיָּה.

The wick is above the oil, drawing fluid up the wick through capillary action.

Hanukiyah Chanukah oil candle menorah

Capillary action & molecule forces

Oil is drawn up through a wick by a mechanism called capillary action, or wicking. This is a tale of two competing forces:

There is an adhesive force between the molecules of oil and the cotton molecules.

There is an intermolecular/cohesive force between the oil molecules.

Cohesion = ability of like molecules to stick together

Adhesion = ability of dissimilar molecules to stick together

cohesion and adhesion forces

from Bioninja

When the former force > latter force then oil molecules are slowly pulled into the wick.

capillary action and surface tension

From Hyperphysics, Surface tension

Sean Snider, writing on Quora, explains:

A fluid such as heating oil will tend to flow upwards against gravity due to capillary motion. The individual atoms in the oil will interact with the fiber atoms to cause adhesion. The oil atoms will bump into the fiber atoms and move upwards due to intermolecular forces. The difference in charge between the two types of atoms causes them to repel in all directions, including up.

The oil atoms will keep moving up unless the forces between them cause them to clump together so that intermolecular forces weaken and their collective mass is too much to repel the force of gravity.

Typically the density of the fiber itself prevents the oil particles from clumping enough to reach this threshold, so they continue to move upward. This allows the oil to reach the top of the wick and burn. Instead of the fiber burning quickly, the oil burns. Some of the fiber also burns, but much less quickly.

Student activities

Wick lab/game! sciensation.org

Capillary action and diffusion lab

Lights, Camera, (Capillary) Action! Scientific American

Once lit, the heat from the flame would likely warm up the small olive oil vessel below. Those vessels are often transparent.

Arched oil Chanukah menorah

Convection & temperature differentials

That heat would cause a temperature differential, with warmer oil at the top and cooler oil at the bottom. This would cause convection and/or turbulence in the fluid, which might be visible if we study it with a high speed, high-resolution smartphone camera.

(Convection, turbulence, and related topics are usually left out of high school physics curriculum, this might be a fun way to introduce it.)

Heat convection GIF

Experiment: Add drop of coloring into oil. Light the wick. Visually observe convection currents.

Dreidel physics

A dreidel (Yiddish: דרײדל‎) or sevivon (Hebrew: סביבון) is a four-sided spinning top, played by children during the Jewish holiday of Hanukkah.

It is a Jewish variant on the teetotum, a gambling toy found in many European cultures.

Let’s take a look at Extreme High-Speed Dreidel Physics by Alexander R. Klotz:

… a dreidel is an example of a spinning top, a source of extremely difficult homework problems in undergraduate classical mechanics related to torque and angular momentum and rigid body motion and whatnot. I was chatting with a theorist I know who mentioned that it would be fun to calculate some of these spinning-top phenomena for the dreidel’s specific geometry (essentially a square prism with a hyperboloid or paraboloid base), and I suggested trying to compare it to high-speed footage [1000 frames per second] ….

dreidel rotation and precession

Check out the article and videos here.

Related dreidel topics to investigate

rotation

What keeps spinning tops upright? Ask a Mathematician/Physicist

precession

Precession, Wikipedia

conservation of angular momentum

Angular momentum

gryroscopes – Gyroscopes

.

Gyroscope precession GIF

Statistics

Are dreidels fair? In other words, does the average dreidel have an equal chance of turning up any one of its four sides? Dreidel Fairness Study

Ultra High Speed Physics.

You’re not a mad scientist unless you ask questions like “Imagine a game of dreidel with a 60-billion-RPM top….” Focus: The Fastest Spinners. APS Physics

(*) How is the holiday spelled? ELA connections

Why write “Hanukkah” instead of “Chanukah” – surely one spelling is right and the other is wrong? The reason for the spelling confusion is the limitations of the English alphabet. Hanukkah is a Hebrew word (חנוכה)

That first Hebrew letter of this word, ח , has a guttural sound. This sound used to exist in ancient English but doesn’t exist in modern English. The modern pronunciation of this letter is a voiceless uvular fricative (/χ/)

As such there is no one correct-and-only way to transliterate this letter. Over the past 2 centuries four ways have developed:

KH – Khanukah (used in old fashioned translations of Yiddish)

CH – Chanukah

H – Hanukkah (the extra ‘k’ is added just to make it 8 letters long.)

H – Ḥanukah (notice the H with a dot under it.)

Each of these transliterations is equally valid.

Learning Standards

Convection & Temperature differential

College Board Standards for College Success in Science

ESM-PE.1.2.1 Describe and contrast the processes of convection, conduction and radiation, and give examples of natural phenomena that demonstrate these processes.

ESM-PE.1.2.1c Use representations and models (e.g., a burning candle or a pot of boiling water) to demonstrate how convection currents drive the motion of fluids. Identify areas of uneven heating, relative temperature and density of fluids, and direction of fluid movement.

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.

Massachusetts Science and Technology/Engineering Curriculum Framework

7.MS-PS3-6 (MA). Use a model to explain how thermal energy is transferred out of hotter regions or objects and into colder ones by convection, conduction, and radiation.

Transliteration

Library of Congress (USA) ALA-LC Romanization Tables

Why Transliteration Matters

 

The nature of time

What is time?

What is time? Where does time come from?

In what way is time really something objective? (something actually out there?)

In what ways is time not objective? (so it would be just a way that humans use to describe our perception of the universe)

What is time?

 

Why does time never go backward?

The answer apparently lies not in the laws of nature, which hardly distinguish between past and future, but in the conditions prevailing in the early universe.

The Arrow of Time, Scientific American article. David Layzer

Is there a relationship between time and the second law of thermodynamics?

Before reading further, understand that these topics require at least some familiarity with the laws of Thermodynamics

“According to many, there might be a link between what we perceive as the arrow of time and a quantity called entropy…. [but] as far as we can tell, the second law of thermodynamics is true: entropy never decreases for any closed system in the Universe, including for the entirety of the observable Universe itself. It’s also true that time always runs in one direction only, forward, for all observers. What many don’t appreciate is that these two types of arrows — the thermodynamic arrow of entropy and the perceptive arrow of time — are not interchangeable.”

No, Thermodynamics Does Not Explain Our Perceived Arrow Of Time, Starts With A Bang, Ethan Siegel, Forbes

No, Thermodynamics Does Not Explain Our Perceived Arrow Of Time

Is time (and perhaps space,) quantized?

Ethan Siegel leads us in a fascination discussion:

The idea that space (or space and time, since they’re inextricably linked by Einstein’s theories of relativity) could be quantized goes way back to Heisenberg himself.

Famous for the Uncertainty Principle, which fundamentally limits how precisely we can measure certain pairs of quantities (like position and momentum), Heisenberg realized that certain quantities diverged, or went to infinity, when you tried to calculate them in quantum field theory….

It’s possible that the problems that we perceive now, on the other hand, aren’t insurmountable problems, but are rather artifacts of having an incomplete theory of the quantum Universe.

It’s possible that space and time are really continuous backgrounds, and even though they’re quantum in nature, they cannot be broken up into fundamental units. It might be a foamy kind of spacetime, with large energy fluctuations on tiny scales, but there might not be a smallest scale. When we do successfully find a quantum theory of gravity, it may have a continuous-but-quantum fabric, after all.

Are Space And Time Quantized? Maybe Not, Says Science

Is time quantized? In other words, is there a fundamental unit of time that could not be divided into a briefer unit?

Even In A Quantum Universe, Space And Time Might Be Continuous, Not Discrete

Time’s Arrow (may be) Traced to Quantum Source: A new theory explains the seemingly irreversible arrow of time while yielding insights into entropy, quantum computers, black holes, and the past-future divide.

Theoretical physics: The origins of space and time

 

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.

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.

At what point does Avogadro’s law not apply?

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.