Atmospheric Pressure and Buoyancy in Gases: A Comparison with Liquids, Slides of Physics

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Finish Chapter 13 (Liquids) from last time

Start Chapter 14 (Gases and Plasmas)

Gases and plasmas: Preliminaries

•^

Will now apply concepts of fluid pressure, buoyancy,flotation of Ch.13, to the atmosphere.

-^

Main difference between a liquid like water and a gas like airis that in the gas, the density can vary hugely; ouratmosphere’s density is depth dependent unlike liquid’s

-^

Gases vs liquids: both are fluids but molecules in gas are farapart and can move much faster, free from cohesive forces.

-^

A gas will expand to fill all space available

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Note! An “empty” cup is not really empty – it’s filled with air.In fact a 1 m

3

“empty” cube of air has a mass of 1.25 kg (at

sea level).

The atmosphere

•^

What determines the thickness of our atmosphere? Balance between:kinetic energy of molecules

vs

gravity

spreads molecules apart

holds molecules near earth

Consider extremes:(i) If very little gravity (eg on moon), then molecules would move, collide, and

eventually disappear into space. So no atmosphere. (ii) If gravity very strong

c.f.

kinetic energy (eg on a remote planet), molecules

move too slowly, and form a liquid or solid, like the planet itself – so againno atmosphere. Earth – balance between the two effects, so we do fortunately have an

atmosphere! (we can breathe!!)

-^ Exactly how tall is the atmosphere?

Not a meaningful question, since it gets thinner and thinner asyou go higher and higher. Even in interplanetary space, haveabout 1 gas molecule (mostly hydrogen) every cubic meter.

Air is least dense uphere… and most dense here

Atmospheric pressure cont.

•^

Unlike water, density of atmosphere varies with height, so pressurerelation in terms of depth is not as simple. Not uniform.

-^

At sea level, 1 m

3 of air has mass of 1.25 kg

At 10km height, 1 m

3 of air has mass of 0.4 kg

(this is why need additional mass of air to pressurize airplanes).^ Recall Pressure = Force/area = weight/area.So to find pressure at sea level, need tocalculate weight of a column of air rising up to“top” of atmosphere, say about 30 km.Find that a 1m

2 area cylinder, 30 km high, has

mass of 10 000kg.i.e. weight of 100 000 N.So pressure = 100 000 N/ (1 m)

2

= 100 kPa

Precisely, sea-level atmospheric pressure = 101.3 kPa

1 Pa = 1 N/m = Pascal

Clicker Question

It would be easier to pull evacuatedMagdeburg hemispheres apart whenthey areA) 20 km above the ocean surface.B) at sea level.C) 20 km beneath the ocean surface.D) held upside down.E) none of these

Answer: AIt’s atmospheric pressure that we have to counter. This isleast higher up in the atmosphere out of the given options.

Barometers

•^

Measure pressure of atmosphere

-^

Simple mercury barometer: Fill tube with mercury and then turnupside down into dish. Mercury runsout into the dish until level in tube is 76cm, as shown.

vacuum

Why 76cm?Because, of pressure balance: barometer balances when weight of liquidin tube exerts same pressure as atmosphere outside.It’s 76cm, regardless of how wide the tube is:

weight of any 76cm

column of mercury equals weight of same width column of 30 km of air. If atmospheric pressure increases, then air pushes down harder on themercury , so column pushed up higher than 76 cm.

Barometers cont.

-^ How about a barometer made of water?Why not – but how tall would the glass tube have to be?

The weight of the water column would need to be the sameweight as 76cm column of mercury, but density of water is 13.6 xless than the density of mercury – hence, water barometer wouldhave to be (at least) 13.6 x 76cm = 10.3 m tall. Again, regardlessof tube’s width.

-^

Just like barometer, when you drink through a straw, it’s the

• atmospheric pressure outside the straw that is pushing the water up.See next slide! This also explains why you can’t get water to be more than 10.3m tall,with a vacuum pump.

Buoyancy of Air

An object surrounded by air is buoyed up by a force equal to theweight of the air displaced

.

c.f.

Archimedes principle for liquids in the previous chapter.

-^ An object will rise in air (ie float upward) if its density is less than air’s density:Why? (c.f. sinking vs floating in previous chapter)Downward grav force (= weight-density x volume) is then less than upwardbuoyant force (= weight-density-of-air x volume). So there is a net upward force.Eg. He-gas filled balloon (or heated air balloon – since hot air is less densethan normal air)

Greater buoyancy if the helium could beevacuated – but not practical since how wouldkeep the balloon sides from collapsing in?Could use stronger material but then weight istoo large, so wouldn’t rise at all…

Clicker Question

The buoyant force on a bird is largest when it fliesA) Closer to the groundB) Higher in the atmosphereC)It is the same at whatever height it is atAnswer: AThe buoyant force is the weight of the air displaced. If weassume the bird’s volume does not change too much, thensince the weight-density of the air decreases at higheraltitudes, the buoyancy force does too.(but note that birds don’t fly due to the buoyant force – flight is morecomplicated, to do with “lift”, part of this is Bernouilli’s effect, see soon)

Differences with buoyancy in air and liquid

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Important differences: (i)

due to the air density becoming less as you go higher (liquid densityremains about the same). So

buoyant force

decreases as you rise

in

atmosphere (but stays same while rise in water). (ii) there is no “top” to the atmosphere (it just keeps thinning out), unlike liquid

surface.

• Consequence: a light balloon released from bottom of ocean will rise all the way towater’s surface; whereas if released from surface of earth, will stop rising at a certainheight.• Why, and how high will a helium balloon rise?When buoyant force on balloon equals its weight, it will stop accelerating upwards.(Buoyant force = displaced-weight-of-air, so for same volume of balloon, thisdecreases as it rises because air is becoming less dense).May continue to rise at the const. speed it reached (but will slow due to airresistance).If balloon material is able to expand, then it will as it rises, as there’s less pressureoutside, so will displace a greater volume of air – net effect is that buoyant forceremains same. If it continues to expand, it will eventually pop…

Consider an air-filled balloon weighted so thatit is on the verge of sinking—that is, its overalldensity just equals that of water.Now if you push it beneath the surface, it will^ 1. sink.2. return to the surface.3. stay at the depth to

which it is pushed.

IMPORTANT NOTE: the balloon is compressible.

Boyle’s Law

•^

When you increase the pressure of a confined gas, how does thevolume change? And vice-versa? This is Boyle’s law: i.e. - If you halve the volume of container, the pressure is doubled,since more collisions (bouncing) between molecules and with walls.

P^1

V^1

= P

V 2

2

for a fixed temperature.

Effectively, the density is doubled.pressure ~ density (at fixed temp)

proportional to

Notes: (i) fixed temperature means fixed average speed of molecules(ii) strictly speaking, Boyle’s law applies to “ideal gases” – i.e. when neglect any“sticky” forces between molecules and treat them as point particles.At normal temps and pressures, air is well-approximated to be an ideal gas.

Moving fluids

•^

So far, talked about stationary fluids (

hydrostatics

). When fluids are

moving, (

hydrodynamics

), have additional effects.

•^

Consider water moving through pipe of varying thickness: The volume passing through any cross-sectionis the same in a given time interval.So, in narrower region, speed must be faster.Eg. Squeeze on end of garden hose, waterspeeds up.Eg. River entering a narrow gorge speeds up.

Water flows faster here

-^ Streamlines –(eg thin lines above) represent paths (trajectories) ofparts of fluid. So are closer together in narrower regions where flow isfaster.