Physics Part 1MECHANICSPhysics 1700

Gasses (& Hydrodynamics)

W. PezzagliaUpdated: 2013Jul23

Gasses & Hydrodynamics

A. Gas Laws

B. Atmospheric Pressure

C. Hydrodynamics (Bernoulli’s Law)

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A. Gasses

1. Boyle’s Law

2. Kinetic Model of Pressure

3. Temperature of a Gas

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1. Boyle’s Law• Boyle’s Law (1662) at constant temperature,

pressure is inversely proportional to volume, or

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Robert Boyle(1627 – 1691)

constantPV

https://en.wikipedia.org/wiki/File:Boyles_Law_animated.gif

Demo: http://phet.colorado.edu/en/simulation/gas-properties

2. Kinetic Theory of Pressure• 1738 Daniel Bernoulli derives Boyle’s law by assuming gasses

consist of moving molecules, and the impacts with wall causes pressure.

• Impulse by collision

• Time between collisions for box of width “L” knowing x-velocity

• Average Force on wall for 1 molecule

• Average Kinetic Energy in 3D

• Pressure is related to KE

• Boyle’s Law for N molecules

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Daniel Bernoulli (1700 – 1782)

KENPV

KELAAFPV

mvvvvmmvKELmv

tpF

vLt

mvp

xzyx

x

x

32

32

223222

212

21

2

)(

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3. Kinetic Theory of Temperature• Equate Kinetic pressure law

with ideal gas law and we find average kinetic energy of a mole of gas is

• 1900(?) Planck writes that the average Kinetic Energy of a single monoatomic gas atom is given by:

• Hence “temperature” is a measure of average kinetic energy of molecules.

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KJ

avg

k

kTK

23-

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10×1.38Boltzmann Constant=R/Na

RTK avg 23

B. Atmospheric Pressure

1. Barometer

2. Pressure & Altitude

3. Helium Balloons

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1. Barometer

1643 Torricelli invents Mercury Barometer

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2. Pressure and Altitude• Pascal’s law of depth assumes constant density

• Boyle’s law shows that density of gas decreases in proportion to decrease in pressure

• Modified law of depth,Pressure “P” at altitude “h”:P0 is pressure at sea levelScale height H=8500 meters

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HhePhP 0)(

Pressure is 0.33 atm at top of Mt. Everest

3. Helium Balloons• Molecular weight of Helium is about 1/7 that of air

• Hence density of Helium is always about 1/7 that of air at the same pressure.

• Thus Helium balloons have buoyant force which theoretically could take them to the top of the atmosphere.

• Record is only about 16.2 km (the balloon expands and bursts)

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C. Hydrodynamics

1. Torricelli's law (1643)

2. Bernoulli effect

3. Bernoulli equation (1738)

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1. Torricelli's law (1643)

In brief, the velocity of a fluid exiting at the bottom of a tank of depth “h” is independent of the fluid’s density (i.e. the fluid analogy of Galileo’s law that all bodies fall at same rate independent of mass)

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ghv 2

2. Bernoulli Effect

1738, Daniel Bernoulli notes that pressure decreases when a fluid’s velocity increases.

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21222

1 vvP

2b. Bernoulli Equation

1738, at any point in the fluid, the sum of the pressure, kinetic energy density and potential energy density is a constant

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constant221 ghvP

From this can derive Pascal’s law of depth, Torricelli’s law and Bernoulli effect

3a. Continuity Equation

Based upon conservation of mass, when a fluid (liquid or gas) is forced through a smaller pipe, the speed must increase.

If the density is unchanged (“incompressible”) the velocity increases inversely proportional to cross section area

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222111 AvAv

3b. Venturi TubeCan be used to measure flow rate v1 of a liquid (or gas) from the observed pressure

difference (inferred from “h”) when cross section area is decreased.

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2

112

21

21

222

121

AAvv

ghPPvvPP

Notes• Demo PHET Bernoulli:

http://phet.colorado.edu/en/simulation/fluid-pressure-and-flow

• Demo PHET for Gas Law: http://phet.colorado.edu/en/simulation/gas-properties

• Video Mechanical Universe #45 Temperature & Gas Laws

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