*An experimental balloon takes its inaugural flight in August 2020. This particular balloon can change altitude by shortening or lengthening a cord attached the top and bottom of the balloon. Shortening the cord compresses the balloon which makes it descend while lengthening the cord expands the balloon allowing it to ascend. Photo courtesy of Thin Red Line Aerospace, used with permission.*

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*For this post I invited back fellow thermodynamics enthusiast Mike Pauken, principal engineer at NASA’s Jet Propulsion Laboratory and author of Thermodynamics for Dummies, to continue this series related to his work on developing balloons for Venus. His first post covered the developmental history of balloons. This second post dives into the fundamental reasons why balloons float to begin with. For his third and final post, to be published early next year, Mike will provide a general discussion of balloon flight on other planets. Please extend a warm return welcome to Mike! – Bob Hanlon*

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**Why do Balloons Float? **It almost seems like a question a child would ask. In fact, many children have asked this question. The short answer, which may not quite satisfy a child, is: balloons float because there is a force pushing them upwards. In 1687 Isaac Newton published his first two laws of motion in *Principia Mathematica* which showed us that things, like balloons, move in response to forces applied to them. For a balloon to move at all, some force must act on it. A balloon ascends or descends in response to two basic forces: the lift force, F_{L}, and the mass gravitational force on the balloon, F_{B}. You can imagine that a balloon also responds to other forces, such as those from wind, but we shall ignore these complexities for now.

The lift force on a balloon is created when the gas inside the balloon has a lower density than the gas outside the balloon. But density alone isn’t sufficient to make a lift force for a balloon floating in the air. Gravity is needed as well. Gravity has a way of sorting things out with lighter gases working their way above heavier gases. This was known to the Montgolfier brothers and Jacques Charles in 1783 as I described in my previous post in which I showed the lift force of a balloon can be calculated from gas density, gravity, and volume using this equation:

F_{L} = V·g·(ρ_{a} – ρ_{b})

Where F_{L} is the lift force, V is the balloon gas volume, g is the gravitational acceleration, ρa is the density of the atmospheric gas, and ρb is the density of the balloon gas.

The other force acting on a balloon is the mass gravitational force, F_{B}, which is just a fancy way of saying the weight of the balloon plus its payload. We calculate the balloon mass gravitational force, F_{B}, using the following equation:

F_{B} = g·(m_{b} + m_{p})

where g, is the gravitational acceleration, m_{b} is the mass of the balloon envelop and m_{p} is the mass of the payload the balloon carries.

How a balloon responds to both the lift force, F_{L}, and the mass gravitational force, F_{B}, depends on which force is greater. In the lifetime of a balloon, it rises if the lift force exceeds the mass gravitational force, stops rising when they are equal, and descends when the lift force is smaller than the mass gravitational force. Mathematically these are written as:

F_{L} > F_{B} – rising balloon

F_{L} = F_{B} – stable balloon altitude

F_{L} < F_{B} – descending balloon.

Briefly, this is why and how balloons float, but there is so much more to the story, so read on.

As you can see, both gravity and density show up in the lift force equation:

F_{L} = V·g·(ρ_{a} – ρ_{b})

Density is one of the variables found in the famous Ideal Gas Law which took centuries to figure out through diligent efforts by brilliant people; also briefly described in my previous post. The Ideal Gas Law shows us that there are a couple ways in which one gas can have a lower density than another and demonstrates how both helium and hydrogen balloons and hot air balloons work.

Let’s take a look at the Ideal Gas Law: P·V = *n*·R·T, to understand what properties of a gas affect its density. In the ideal gas law, the variables in the equation are defined as:

P = absolute pressure

V = volume

n = number of moles of gas

R = universal gas constant

T = absolute temperature

The number of moles, *n*, is defined by:

Then making a substitution for *n*:

Density is defined by mass per unit volume:

Finally, we can make substitutions for m and V and rearrange the equation to solve for density:

What this equation tells us is that the density of a gas inside a balloon is based on three different properties: pressure, molar mass, and absolute temperature that are connected together through the universal gas constant. Let’s go over the contents of the Ideal Gas Law to get a better understanding of how it applies to our atmosphere and flying balloons.

**The Universal Gas Constant**

The Universal Gas Constant of the Ideal Gas Law. As its name implies, is universally applied to all gases. It allows us to take the proportional relationship between density and the other properties, specifically pressure, temperature and molar mass, and turn it into an equation to quantify density in absolute terms. Thus, the gas constant does not affect density as pressure, temperature or molar mass change. It is simply an important anchor! So, I’ll leave it there for now and go to the other variables.

**A Primer on Molar Mass**

The molar mass describes how many neutrons and protons exist in a molecule. Density is directly proportional to molar mass. The higher the molar mass of a gas molecule, the higher its density will be. A hydrogen atom has only 1 proton, so H_{2} has two protons and no neutrons for a molar mass of 2 grams/mole. Helium has two protons and two neutrons; its molar mass is 4 grams/mole. Air is a mixture of mostly nitrogen (~79%) and oxygen (~21%). We will ignore the contributions of the other gases in air like water vapor, carbon dioxide, methane, etc. to estimate the molar mass of air. Nitrogen has 7 protons and 7 neutrons and combining two atoms together to make N2, gives a molar mass of 28. Oxygen atoms have 8 protons and 8 neutrons, so the oxygen molecule of O_{2} has a molar mass of 32. If we apply the mixture ratio of N_{2} and O_{2} in air to the individual molecular molar masses, we end up with an apparent molar mass of 28.97 grams/mole for air.

The Ideal Gas Law shows that the density ratio of H_{2}/air (holding pressure and temperature constant) is only 2/28.97 = 7% and that of helium/air is 4/28.97 = 14%. We can use the density of hydrogen and helium to determine how much lift they have in a balloon using the balloon lift force equation. Leaving the math to the reader as a homework exercise, the lift force of H_{2} is 10.9 N/m^{3} while that of helium is 10.1 N/m^{3} assuming a pressure of 100 kilopascals and a temperature of 20°C or 293K. So even though hydrogen has half the density of helium it can only lift about 8% more than helium.

**Talk about Pressure!**

Pressure and density go hand-in-hand. If pressure increases, so does density. If pressure decreases, density follows suit. Here I will discuss the effects of pressure on hot air balloons and helium/hydrogen balloons.

**Hot Air Balloons:** The pressure inside hot air balloons is the same as the atmospheric pressure outside the balloon. You probably already know that the pressure of the atmosphere decreases with altitude. (I’ll discuss this in more detail below.) You can conclude then that the air density also decreases with altitude because of pressure decreasing. As a balloon ascends, the pressure inside the balloon decreases to match the atmospheric pressure. For a hot air balloon, lets assume the burner keeps the hot air at roughly a constant temperature. In reality though, the temperature inside the balloon rises when the burner is on, and cools when the burner is off indicating that the temperature actually oscillates around an average value. Furthermore, we can assume the balloon volume remains the same as it rises or descends. The balloon stays the same size as it goes up or down.

We can use the Ideal Gas Law, P·V = *n*·R·T, to show that if the pressure, P, is decreasing with altitude, and the volume, V and the absolute temperature, T, are roughly constant then the only variable left to reduce pressure is the number of moles of gas, n, inside the balloon. If the number of moles of gas inside the balloon decreases with altitude, then this means that some gas inside the balloon leaves through the opening at the bottom of the balloon as the altitude rises. Removing gas from the balloon means the density of the gas inside has decreased as it rises in the atmosphere. As a hot air balloon descends, air enters the opening, thus increasing the number of moles of gas, and thus the gas density, inside the balloon.

**Helium Balloons:** In a helium (or hydrogen) filled balloon, the balloon is closed at the bottom, sealing it off from the atmosphere. There are several different scenarios possible for how pressure affects helium balloons. If the balloon is made from a non-stretchable material, it is usually only partially inflated on the ground as shown in Figure 1. As the balloon rises, the balloon volume grows until it reaches its maximum volume and the balloon will stop rising any further as shown in Figure 2. This means that the gas density inside the balloon decreases as it rises up. The mass of helium or hydrogen inside the balloon doesn’t change, but the volume grows which decreases the density.

If the balloon is filled to its maximum volume on the ground, as Jacques Charles did with his first hydrogen balloon, the pressure inside the balloon will change as the air temperature around the balloon changes. But the density of the helium or hydrogen remains constant because the mass inside the balloon and the volume of the balloon are constant. The density of the gas inside the balloon cannot increase or decrease with altitude or temperature changes. If the balloon is anchored to the ground and the temperature rises during the day, the pressure inside the balloon will rise. If the balloon is set free and rises up in the atmosphere, its pressure will drop as the temperature decreases with altitude. The tricky part is that the pressure difference between the inside and outside of the balloon will grow as the altitude increases. Unless the balloon is very strong, it can burst, which is what happened to Jacques Charles’ first balloon. When the balloon pressure is more than the atmospheric pressure, the balloon is known as a super-pressure balloon. I will talk about these kinds of balloons in my next post on balloon flights on other planets.

If a balloon is made of a stretchy material like latex rubber, you can fill it with as much gas as you want as long as you don’t fill it up to the point where it will burst. The more gas you put in, the more it will lift up, but with one caveat. The more weight you try to lift up, the lower the altitude the balloon will be able to reach. As the balloon rises higher, the volume of the balloon will grow. If the balloon rises too high, it will reach a bursting point and then your balloon will fall down to the ground. The density of the gas inside this kind of balloon decreases as it rises in the atmosphere because the mass of gas inside is constant but the volume grows larger.

**Temperature Effects on Density**

In a hot air balloon, the gas density inside the balloon is less than the air density outside the balloon, even though the pressure inside equals the pressure outside, by virtue of it being hot compared to the atmosphere. For a typical nylon fabric hot air balloon, the average gas temperature is around 100°C. We can calculate the density of the hot air inside the balloon using the ideal gas law. At sea-level pressure of 101.3 kilopascals and 273.15 K temperature (identical to that specified above) the lifting force of a hot air balloon is only about 2.5 N/m^{3} (compare to ~10-11 N/m^{3} for helium or hydrogen). This is why hot air balloons are so large, they can’t lift nearly as much as helium or hydrogen balloons on a volume basis. If you go back and review my last post and you’ll see that Charles’ hydrogen balloon was a lot smaller than the Montgolfier brothers hot air balloon. Fun fact: if we were to replace 100°C hot air with a mythical ideal gas with the same density (hence lifting capacity), it would have a molar mass of around 23 grams/mole.

I would like to point out that the gas temperature inside a hot air balloon is not uniform. It is hotter at the top of the balloon than at the bottom of the balloon, as shown in the picture below in Figure 3. We used the temperature around the middle of the balloon as the average to make our lift force calculation above. I want to take a moment here to explain why there is a temperature gradient inside a balloon or even inside a building if you’ve ever climbed up near the ceiling.

If we heat a fixed quantity of air (*n* is constant) and keep the pressure, P, constant, the air volume, V, expands as predicted by our good buddy, the Ideal Gas Law: P·V = *n*·R·T. If volume increases, the density decreases because recall that density is mass/volume:

Density is affected by both pressure and temperature. You can decrease density by reducing pressure or by increasing temperature. In a hot air balloon, both effects are taking place simultaneously. The pressure at the top of the balloon is less than at the bottom of the balloon, just as the atmospheric pressure is less at the top than at the bottom. The temperature at the top of the balloon is higher than the temperature at the bottom of the balloon. Density gradients are formed in a gravitational field in which density decreases as elevation increases.

**The Density of the Atmosphere**

At sea-level, the Earths’ atmospheric pressure is 101.3 kiloPascals (or 14.7 pounds per square inch). In Figure 4, I’ve made a graph of the air temperature, pressure and density in the Earth’s atmosphere. As you rise higher in altitude, the pressure decreases in half about every 5 kilometers. The absolute temperature slightly decreases with a rise in altitude. You can see in the graph, that pressure decays more rapidly than absolute temperature. This means the density of the air decreases with altitude. The x-axis is a log-scale plot since the temperature, pressure and density have widely different magnitudes making them hard to plot on the same scale.

Since balloons float in the air as a result of a density difference between the balloon gas and the atmosphere, a balloon will reach a maximum altitude where the gravity force acting downward on the balloon equals the buoyancy force acting upward on the balloon. The maximum float altitude doesn’t necessarily occur at the altitude where the density inside the balloon equals the density outside the balloon. Because the balloon has to support its own mass plus the mass of anything hanging under it, the internal density will always be less than the external air density.

**Digging Deeper: The Molecular Point of View**

Some readers may stop here and go about their business. Maybe you’re one of them and already satisfied by this point. But if you wish to dive deeper into the topic and get a certificate of deep knowledge, continue on reading…

Thus far, my description of why balloons float focuses only on observations made by measuring properties using mechanical instruments such as thermometers, pressure gauges, graduated cylinders and balances. These observations of physics are merely that – just observations! They describe what we are able to measure mechanically. These kinds of observations do not explain why molecules, in the form of a gas, have the properties we call pressure, temperature and density connected together through the Ideal Gas Law. To understand how molecules can possess pressure, temperature and density, we must become like a molecule ourselves. This is something that the father of the famed physicist Dr. Richard Feynman emphasized to him – look at things from different perspectives to really understand something. (As an aside, Richard mentored his younger sister, Joan, to do the same. Joan became a well-known research scientist at JPL). Melville Feynman told young Richard: Just because you know the name of a bird, doesn’t mean you know anything about it. You need to look at a bird for a long time to know more about it.

**Statistical Mechanics**

In 1975, Dr. David Goodstein, a professor at Cal Tech, published a book called “States of Matter.” He starts off Chapter One the following alarming introduction: “Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously.” With that warning, let me just say, we will only dip our pinky toe into the deep waters of statistical mechanics of a perfect gas.

Daniel Bernoulli published his classic work, written in Latin, “Hydrodynamica” in 1738 in which he suggested the pressure exerted by a gas on the wall of a container is due to the impact of the particles (now known as atoms or molecules) against the sides of the container. Bernoulli showed the illustration in Figure 5 to explain his theory of what happens inside a closed container full of air. The closed cylinder has a moveable piston with a weight on top to maintain a constant pressure inside the container. Bernoulli described his theory on pressure arising from the motion of molecules as follows: “*let the cavity contain very minute corpuscles, which are driven hither and thither with a very rapid motion; so that these corpuscles, when they strike against the piston and sustain it by their repeated impacts, form an elastic fluid which will expand of itself if the weight is removed or diminished…*”

Bernoulli was ahead of his time, and most scientists in those days rejected his idea, holding on to the hypothesis that gas molecules do not move about and repel one another from a distance.

**Over a Century Later**

A number of scientists contributed to the development of the kinetic theory of gases over the next 120 years, including James Joule, who calculated the molecular velocity required to produce the observed pressures for a number of gases in his 1851 paper “Some Remarks on Heat and the Constitution of Elastic Fluids.” This demonstrated that scientists were finally coming around to accepting Bernoulli’s hypothesis on molecules in motion as the source of pressure in a gas.

In the late 1850s, James Clerk Maxwell started to work on Bernoulli’s hypothesis accepting the notion that gas molecules move as perfectly elastic particles and obey Newton’s laws of motion, bouncing of each other and surfaces with straight-line trajectories between collisions. Maxwell did not believe that gas molecules all moved at the same velocity as James Joule’s analysis suggested even if the container holding them is at a uniform temperature. Maxwell held that temperature is simply an indicator of the mean speed of molecules (more accurately, temperature indicates the mean of the square of the speeds of a gas molecule population). Furthermore, he realized that knowledge of the position and velocity of every molecule at every instant in time was not necessary to describe how molecules produce pressure and other properties such as viscosity and heat capacity.

What was needed was a mathematical representation of the distribution of molecular velocities and the bell-shaped curve made the most sense. Some molecules will move slowly, other will move very fast, but most will be somewhere in the middle. Similar speed statistics are observed in marathons, or other races, where only a few, who practically sprint the whole time, cross the finish line first, then a big wave of “average” runners cross the finish, and finally the race tapers down to the folks who walked the distance to complete the course.

The culmination of Maxwell’s work on this topic were published in his 1860 paper “Illustrations of the Dynamical Theory of Gases.” He continued working on the kinetic theory and concluded by 1867 that molecules do not really collide into each other, rather they repel one another with a force whose magnitude is inversely proportional to the fifth power of the distance between them in his paper “On the Dynamical Theory of Gases.”

The kinetic theories of gases were advanced even further in the 1870s through the genius of Ludwig Boltzmann. Boltzmann reworked Maxwell’s analysis to include various degrees of freedom that describe in more detail the way molecules move. Maxwell only looked at the energy from the linear velocity of molecules. Boltzmann included the energy associated with rotation and vibration of molecules.

**The Speed of Gas Molecules**

The speed of gas molecules allows us to define the properties we know as temperature and pressure. From pressure and temperature and molecular mass, we can define density. One important result of Maxwell and Boltzmann’s work on the Kinetic Theory of Gases is an equation, shown below, for describing the probability of gas molecules having a certain speed in terms of the molecular mass and the bulk gas temperature. While this equation looks complicated and it would take a blog all by itself to discuss it, I want to point out that there are three variables that are very important in the equation – velocity, *v*, absolute temperature, *T*, and molar mass, *m*. The other variables you see here are π, which takes care of the circular nature of the geometry space and *k _{B}*, which is the Boltzmann constant.

In this equation, the first term (in parentheses) scales the equation in terms of molar mass and absolute temperature. It acts like a constant for a given temperature and gas species. The second term accounts for the motion of the molecules through a spherical geometry representing a system of molecules. This term dominates at low velocities. The last term, with the exponent, is a mathematical formulation that shows the number of molecules with a high speed gets smaller as the speed gets higher. This term dominates at high velocities. The Maxwell-Boltzmann speed distributions comparing the speed of air (at 0°C and 100°C), helium, and hydrogen are plotted in Figure 6. What you can see in this graph is that the speed range of small molecules like hydrogen and helium are higher than the speed range of air molecules. You will also see that as the temperature of the gas increases, (comparing the 0°C air to the 100°C air) the speed range moves to the right, that is, the speed range get faster. The speed of the molecules has very important implications for gas pressure and consequently gas density. This factors into the change in pressure and density with altitude in our atmosphere.

Although the Maxwell-Boltzmann speed distribution is a significant achievement over James Joules single speed interpretation of gas molecules, it turns out that there is a single speed in the distribution that is representative of all the gas molecules and is useful for computing properties like kinetic energy and pressure. This reaffirms James Joules’ representation of molecular speeds as a single value. This is known as the Root-Mean-Square (RMS) speed. I marked the location of the RMS speed for each gas in Figure 6. The RMS speed is not the most probable speed, which would be at the top of the curve. Nor is it the average speed, which would be slightly faster than the most probable speed. The RMS speed defines the average kinetic energy of the gas molecule population. The kinetic energy depends on the square of the speed and higher speed molecules have a disproportionately higher kinetic energy than molecules with below average speed. The RMS speed is simply the average of the square of the speeds.

**Microscopic View of Temperature**

Molecules move with a linear speed because they possess a property known as kinetic energy. Kinetic energy connects the molecular mass and the RMS speed of molecules to the property we physically measure as temperature. The kinetic energy of a molecule is dependent on only its absolute temperature, *T*, and is calculated using the Boltzmann constant, *k _{B}*, in this equation: K.E. = 3/2

*k*·

_{B}*T*. All gas molecules at the same temperature have the same kinetic energy, regardless of whether the molecules are helium, hydrogen or air. The kinetic energy of a gas molecule population determines the RMS speed of molecules using the molecular mass in this equation: K.E. = 1/2 m·

*v*

^{2}_{rms}. Therefore, we can define temperature in terms of the molecular mass and RMS speed of molecules with the following equation:

From this equations we can see that for a given temperature, gases with lower a molecular mass, m, will have higher RMS speeds as shown in the Maxwell-Boltzmann speed distribution plots in Figure 6. When helium or hydrogen molecules are in the air, they will speed around much faster than the nitrogen and oxygen molecules even if they are at the same temperature. I would like to point out that molecules not only move in a linear fashion, they also rotate and vibrate. But these additional kinds of molecular motion are not normally accounted for in defining the kinetic energy temperature.

**Microscopic View of Pressure**

Daniel Bernoulli was correct in thinking that pressure is a result of molecules colliding with each other and with surfaces that constrain gases. Pressure depends on the number of molecules per unit volume and their RMS speed. This is clear from the Ideal Gas Law equation written to solve for pressure: P = (n/V)·R·T where (n/V) is the number of molecules per unit volume and we now know that the absolute temperature, T, determines the RMS speed of the molecules. Pressure is define as a force over an area. Forces are a result of a mass experiencing an acceleration. Acceleration is simply a change in velocity which arises from either a change in speed or direction. When a molecule collides with a surface, it will rebound from the surface and change it’s velocity. It will keep the same speed for an elastic collision and just change it’s direction. This change in direction is an acceleration. When a large number of molecules collide with surfaces, we are able to measure the force of these collisions and determine the property we call pressure.

The Ideal Gas Law can be rewritten to determine the pressure of a gas from the RMS speed of its molecules and the number of molecules contained in a unit volume. If we consider only the vertical velocity component of gases, the pressure is calculated with the RMS velocity with the following equation:

In this equation, *N*/*V*, is the number of gas molecules per unit volume, *m*, is the molar mass of the gas molecule and* v _{z}* is the RMS vertical velocity. You will note that I have switched from speed, which does not refer to any direction, to velocity, which indicates direction. I use the z-axis to define the vertical direction. Also, we do not have to use only the vertical velocity component to compute pressure, we can also formulate the problem to use all three axes of velocity. I’m using the vertical velocity to demonstrate the effects of altitude later on.

The two properties in this equation that can decrease the pressure are the RMS speed of the molecules, and the number of the molecules per volume, N/V. There is a reduction in both the temperature (RMS speed of molecules) and the number of gas particles as altitude rises in our atmosphere. Both factors contribute to the reduction in pressure in the atmosphere. The vertical velocity component is calculated using the following equation:

You can look up the standard atmospheric pressure at sea level and at 0°C and find the pressure is 101,325 Pascals. We can compute this pressure using the number density of molecules and the vertical velocity component as follows:

*k _{B}* is the Boltzmann constant: 1.38065E-23 J/K

*T* is absolute temperature: 273.15K = 0°C

*m* is the mass of an air molecule: 4.80992E-26 kg

*v _{z}* = 280.01 m/s

If you compare this value for the molecular RMS speed in Figure 6 for air at 0°C, you will see that the RMS speed is about 485 m/s which is a lot faster than what we just computed here. This is because we are only considering the vertical velocity component. To compute the molecular speed, you’ll need to account for the velocity components in the x and y directions too. If the speed distribution is independent of direction, then v_{x} = v_{y} = v_{z} = 280 m/s, and adding all the velocity components correctly results in computing the RMS speed as:

RMS = (v_{x}^{2} + v_{y}^{2} + v_{z}^{2})^{0.5} = (3·280^{2})^{0.5} = 485 m/s.

Getting back to computing the standard sea-level pressure of the atmosphere: For air, at standard temperature and pressure, the molecular density is 44.615 moles/m^{3}. Multiplying this by Avogadro’s number, 6.02214E+23, the molecular density is 2.6868E+25 molecules/m^{3}. We confirm that the atmospheric pressure, based on the RMS vertical velocity component of the Maxwell-Boltzmann distribution function is:

p = (2.6868E+25 molecules/m^{3})(4.80992E-26 kg)(280.01 m/s)^{2} = 101,325 Pa.

**Pressure and Altitude**

We observe that the pressure of the atmosphere decreases with altitude. Why is this the case? If you think about gas molecules moving around, they are all affected by the force of gravity. The vertical component of their velocity will be altered by gravity which will slow them down as they move upwards and speed up when traveling downwards. Molecules moving vertically continually exchange kinetic energy for potential energy. As they move up, kinetic energy decreases, velocity decreases and thus so does temperature. Moving down has the opposite effects.

We can demonstrate that if we allow the gas molecules to move upward by 1 m, we can compute the pressure at this altitude and compare it to the hydrostatic formula for pressure. Let’s assume a population of air molecules travel 1 m vertically. The time, t, it takes to travel 1 m is: t = (1m)/(280 m/s) = 0.00357 sec

The velocity of the molecules after traveling upwards for 1 m is: v_{2} = v_{1} – g·t. The acceleration of gravity, g, is 9.81 m/s^{2}.

v_{2} = 280.01 m/s – 9.81 m/s^{2}·0.00357 sec = 279.97 m/s

This is a very small change in velocity, but it has a noticeable affect on pressure! At 1-m altitude, the pressure will be

p = (2.6868E+25 molecules/m^{3})(4.80992E-26 kg)(279.97 m/s)^{2} = 101,312.3 Pa.

The pressure decreased by 12.7 Pa in 1 m altitude change from sea level. We can compare this result to what we can obtain using the hydrostatic formula:

For a 1-m height change, the change in static pressure is:

which agrees with our pressure change we computed using the decrease in molecular vertical velocity. I should point out here that only the temperature was allowed to change here as a result of a change in kinetic energy, the molecule density remained constant. A more accurate formulation would account for both properties decreasing simultaneously.

**Why do Light Gases Rise in the Atmosphere?**

Imagine yourself as a helium molecule in the atmosphere illustrated in Figure 7 as the small red circle. Below you is a crowd of nitrogen and oxygen molecules, above you is a slightly smaller crowd. If you happen to be heading downward, eventually you’ll collide with a nitrogen or oxygen molecule. You may find yourself moving upwards. It’s getting less crowded now, but you’ll still bump into other molecules that send you back down. Overall, you are working your way up, it might be three bumps up, two bumps down. There are fewer and fewer nitrogen and oxygen molecules to push you downward as you rise in the atmosphere. The collisions between molecules of widely differing masses in a gravity field will sort themselves out with lower mass molecules going above higher mass molecules.

Different gas species at the same temperature have the same kinetic energy, but not the same RMS molecular speed and not the same momentum. When molecules collide with one another they conserve both kinetic energy (m·v^{2}) and momentum (m·v). Kinetic energy and momentum are frequently confused with each other. Momentum is proportional velocity while kinetic energy is proportional to the square of velocity. The mass of “air” molecules (approximately 79% N_{2} and 21%O_{2}) is 7.2 times more than the mass of helium molecules. If helium and air are at the same temperature, the RMS speed of the helium is about 7.2^{0.5} = 2.7 times faster than air molecules. Despite the higher speed of helium molecules, air molecules have about 2.7 times more momentum than helium molecules. In collisions between air and helium molecules, air molecules will impart large changes to the speed of helium molecules.

In Figure 7, I’ve drawn an imaginary line that the molecules are crossing. In general terms the number of molecules per volume (molecular density) is higher below the line. Many of these molecules are moving upwards. The upward momentum of these molecules is a product of the mass of molecules moving up times their velocity. Above the line, the molecular density is lower and many of them are moving downwards. The downward momentum is also a product of the mass of molecules moving down times their velocity. The velocity of molecules moving up is a bit slower than the average molecular speed because of the downward pull of gravity. Likewise, the velocity of the molecules moving down is a bit faster than the average molecular speed. At the line I’ve drawn, the downward momentum of the molecules above the line equals the upward momentum of the molecules below the line. Mathematically this is expressed as:

m_{down}·v_{down} = m_{up}·v_{up}

Since gravity makes the average upward velocity, v_{up}, less than the average downward velocity, v_{down}, this implies that molecular density of molecules moving up, m_{up}, is more than molecular density of molecules moving down, m_{down}. This creates the decreasing density over altitude gradient we observe in the atmosphere.

If we introduce a population of helium molecules as red circles, as shown in Figure 8, there are two situations we must consider. We assume the helium molecules are at the same temperature as the air molecules so they have the same kinetic energy but not the same momentum. First consider the helium molecules below the top line. If the molecular density of molecules is the same on both sides of the line, the air molecules will push the helium molecules downward because the air has more momentum.

Now look at the helium molecules above the lower line. The air molecules will push the helium upward because the air has more upward momentum than the helium has downward momentum. So where is the helium going if the top is being pushed downward while the bottom is pushed upward? Across the interval between the two lines, the pressure of the atmosphere decreases in the upward direction. The decrease in atmospheric pressure means the downward momentum of the air at the upper line is less than the upward momentum of the lower line. The lines need not be separated by a great distance. Even at the molecular level, there is a pressure gradient and a momentum gradient.

In this thought experiment, we’ve shown how the helium molecules will rise in the atmosphere because of the difference in molecular momentum between air and helium molecules.

But what about hot air? Does it work the same way as a low molar mass molecule? If we increase the temperature of a group of nitrogen and oxygen molecules, temperature is proportional to velocity squared. Temperature is proportional to volume, so volume is proportional to velocity squared. Did you follow that argument? Hopefully it didn’t come across as circular reasoning. But momentum is proportional to plain ole velocity. What happens as air is heated is that its volume increases at a much faster rate than its momentum. The larger volume creates a lower particle density that is proportional to velocity squared. The end result is that hot air has less momentum per unit volume than cold air even though it has more kinetic energy per molecule. Less momentum per unit volume for hot air is the same effect observed in the low molar mass molecules like helium and hydrogen. Thus hot air rises because the momentum balance pushes it upwards.

If we collect the hot air or helium molecules inside of a bag, we are merely replacing the imaginary lines we’ve drawn in Figure 8 with a physical barrier. Let’s call this barrier a balloon for the sake of convenience. The air on the bottom of the balloon exerts an upward pressure force while the air on the top of the balloon exerts a downward pressure force. The upward force will out-muscle the downward force until the momentum of the captured helium molecules balances out the momentum of the atmosphere molecules. This happens when the density of the gas inside the balloon is roughly equal to the density of the atmosphere outside the balloon. I say roughly equal, because of course we have to take into account the mass of the balloon itself holding the helium. Now dear children this is the end of our story on why balloons float.

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