Physics Part 1 MECHANICS Physics 1700 Gasses (& Hydrodynamics) W. Pezzaglia Updated: 2013Jul23

Gasses & Hydrodynamics A. Gas Laws B. Atmospheric Pressure C. Hydrodynamics (Bernoulli’s Law) 2

A. Gasses 1.Boyle’s Law 2.Kinetic Model of Pressure 3.Temperature of a Gas 3

1. Boyle’s Law Boyle’s Law (1662) at constant temperature, pressure is inversely proportional to volume, or 4 Robert Boyle (1627 – 1691) Demo:

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 5 Daniel Bernoulli (1700 – 1782)

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. 6 Boltzmann Constant=R/N a

B. Atmospheric Pressure 1.Barometer 2.Pressure & Altitude 3.Helium Balloons 7

1. Barometer 1643 Torricelli invents Mercury Barometer 8

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”: P 0 is pressure at sea level Scale height H=8500 meters 9 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) 10

C. Hydrodynamics 1.Torricelli's law (1643) 2.Bernoulli effect 3.Bernoulli equation (1738) 11

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) 12

2. Bernoulli Effect 1738, Daniel Bernoulli notes that pressure decreases when a fluid’s velocity increases. 13

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 14 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 15

3b. Venturi Tube Can be used to measure flow rate v 1 of a liquid (or gas) from the observed pressure difference (inferred from “h”) when cross section area is decreased. 16

Notes Demo PHET Bernoulli: Demo PHET for Gas Law: Video Mechanical Universe #45 Temperature & Gas Laws 17