U17- Particle Model: Ideal Gas Unit 17: Ideal...

U17- Particle Model: Ideal Gas

Supported by NSF DUE 1044154

Unit 17: Ideal Gas By the time you finish all lab and class activities, you should be able to understand that: 1) A gas is a fundamental state of matter where the individual atoms or molecules do not have fixed

positions relative to each other and typical are far apart from each other. 2) The ratio of number of particles, N (typically a freakin' huge number) to the volume in which the

gas is contained, V (m3), is defined as the particle density,

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N :

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ParticlesVolume[m3]

≡ Particle Density[m−3]→NV≡ N

Particle number is referenced to Avagadro's number NA = 6.022x1023 pls. If you have this number of avacados it is called one guaca-mole.

3) The ratio of a force exerted over an area, or the work done F∆x to compress the volume A∆x is defined as Pressure. Pressure is a scalar property:

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Force[N ]Area[m2]

=Energy[J ]

Volume[m3]≡ Pressure[Pa]→

FA

=EV≡ P

Consequently pressure is an example of an energy density, reflecting the internal energy of the gas due to the kinetic energy of all the atoms or molecules in the gas. Internal energy of a gas is the result of translational, rotational, and vibrational motion of the particles. The latter two forms of motion can only happen for molecules (minimum of two atoms).

4) In a confined volume, the finite size of the particles means an enormous number of ELASTIC collisions take place between gas particles and the container or between gas particles at any finite interval of time. Gas pressure reflects conservation of linear momentum.

5) INELASTIC collisions involve special kinds of physics or (God forbid) chemistry. 6) Internal energy is proportional to the temperature "T" of the gas, the number of particles "N" and a

proportionality constant called the Boltzmann constant "kB" = 1.381x10-23 J/K, reflecting the statistical description of average particle collision behavior. Units of temperature: a) Absolute scale = Kelvin (K). 0K is the classical ceasing of all particle motion, and the triple

point of water is 273.15K. Note that no degree sign (°) is used with Kelvin. b) Centigrade scale = Celsius (°C). Water freezes at 0°C and water boils at 100°C. The range

between the two is 100C° (note the order, a temperature is °C, a temperature difference is C°). c) Fahrenheit scale = Fahrenheit (°F). Water freezes at 32°F and water boils at 212°F. The story

behind this scale is interesting, based on the freezing point of "brine" and body temperature. 7) The ideal gas law in physics relates the work done by/on an expanding/compressing gas "PV" to the

internal energy in "N" particles at a temperature "T" and proportionality constant "kB": PV = NkBT

8) 1Pa = 1pascal = 1N/m2. Other units for pressure and their relation to one atmosphere: a) 1 atm = 14.7 lbs/in2 = 1.013 bar = 29.92 in Hg = 760 mm Hg = 34 ft H2O b) Absolute Pressure: Pressure measured from zero = vacuum. c) Gauge Pressure: Pressure measured from zero = local atmospheric pressure. d) Atmospheric pressure at sea level is 101,300 Pa ≅ 105 Pa.

Theory – Ideal Gas

The particle theory of matter is one of humankinds greatest achievements. Why? Because empirical evidence, such as the kind you are about to collect in lab, was used to infer the presence of things humans could not directly see, namely atoms undergoing random collisions. In the 1850s French physicist Émile Clapeyron combined results from English physicist Robert Boyle and French baloonist



Jacques Charles into what we now call the Ideal Gas Law (PV=nRT)1. n=N/NA is the number of moles of "N" particles and "R" was the experimentally determined "ideal gas constant", 8.314 J/(moleK). Later this emperical result would be firmly rooted in "kinetic theory" developed by Austrian Physicist Ludwig Boltzmann2 (Figure 1) who defined the ratio of the ideal gas constant to Avagadros number as a fundamental statistical mechanical parameter that would later bear his name, the Boltzman Constant:

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RN A

= kB =1.381x10−23 JK

The importance of this result is that humans had figured out how to measure pressure and temperature, the result of gazillions of colliding particles, that could not be seen! Physicists are more partial to the form of the ideal gas law PV=NkBT because it emphasis the collective result of gazillions of particles elastically colliding with each other and with the walls of container of volume "V". "NA" is an inconceivably huge number even for objects as small as a grain of sand. One NA of beach sand will fill a sandbox approximately 1km wide, 1km long and 1km deep. Reasonable volumes and masses of a mole of particles are obtained only when dealing with objects as small as atoms.

Clapeyron Boyle Charles Boltzmann

Figure 1 That's enough of history. What does kinetic theory mean and why should you care? It turns out that understanding fluids requires a grasp of the microscopic picture. If you can understand why gazillions of interacting particles give rise to pressure, then you are more likely to understand more challenging aspects of fluids, such as hydrostatic pressure, dynamic pressure due to moving particles, and real physiological problems such as high blood pressure. One of the important leaps of faith we make when dealing with gasses is that temperature is a measure of kinetic energy corresponding to approximately the average speed of a gas particle of mass "m". The result of kinetic theory is that:

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m < v2 >3

= kBT

<v2> is the "root mean square" speed of the particles in the container, a way of describing an average speed of all particles in the volume. The 1/3 comes about because on average 1/3 of the particles are going in any given coordinate direction, x, y or z. The term to the left then reflects a portion of the average kinetic energy of a gas particle, which is proportional to the temperature of all the particles in the container – a truly impressive result. Only within the past 10 years have humans been able to control the temperature of an individual atom3, so kinetic theory was amazing, exceeding humans abilities to control individual atoms by over 150 years! Because the ideal gas law has four state variables (P, V, N, T) we will need to describe processes in which some of the variables are kept constant. For example, constant pressure processes are called "isobaric". Constant volume processes are called "isochoric" or "isovolumetric". A constant number of particles will be a "constant number of particles" (duh) or isomolar particle number. Lastly a constant temperature is called an "isothermal" process. To help you visualize the features of the ideal gas we will introduce a set of powerful diagrammatic tools (Figure 2) that will attempt to make sense of 1 http://en.wikipedia.org/wiki/Ideal_gas_law 2 http://en.wikipedia.org/wiki/Kinetic_theory 3 http://physics.aps.org/articles/v5/121



inconceivably huge numbers of particle interactions. The table below summarizes how to describe each term in the ideal gas relationship. Particle number and volume are pretty simple to diagram. Temperature and pressure require a bit more thought, especially when those variables are changing.

Term SI units Diagram Particle # "N"

Later on we will find it advantageous for quick drawing purposes to simply associate one arrow with one particle, without a "dot".

Pure #

Volume "V"

Please note that the volume could be a square or any other easy shape to draw. The size difference is the key feature.

Cubic meters "m3"

N.B.: The "cgs" unit is

the cubic centimeter

which can be written as:

cc = cm3 = mL

Temperature "T" For drawing

purposes we will dispense with color coding and simply infer longer arrows

implies more energy and thus

higher temperature.

Kelvin "K" N.B.: This is

the only correct unit of temperature to use with the

ideal gas law.

Pressure "P"

Key:

A "v" represents a collision with the

wall. An "X" represents two "v"s,

i.e. two particles colliding.

Pascal "Pa" N.B.: The number of

particles is 14 in both low

and high pressure – can you see why?

Piscatorial "F" Considered to be a

hidden fishy variable in the ideal

gas law.

Seussian "S"

Figure 2

Small volume: Large Volume:

Low temperature: High temperature:

Low pressure: High pressure:

Few particles: Many particles:

=X =v



Pre-lab demonstration activities: The instructors will have you examine two introductory activities to get you thinking about how ideal gasses behave. The first will be "Handboilers", sealed glass tubes filled with small amounts of dyed alcohol. These are delicate instruments so please be very careful as they break easily. As you hold the bottom or top of each boiler what do you observe? What do you think is going on? You will be asked to explain the behavior in terms of the diagrammatic tools shown on the previous page. The next activity will be to carefully observe the opening of a fresh bottle of seltzer. The instructor may use a camera to help magnify the activity. What do you observe? Why? Again, as part of the pre-lab discussion use the diagrammatic tools to explain what is going on. Our first lab activity is to characterize a relationship between the various ideal gas law terms: Pressure, Volume, Particle number and Temperature. As an example we will present one result that will NOT be examined in lab, in part because it is hard to do. But we can describe the experiment in easy to understand terms and even model some parts of the relationship with a PHET simulation on ideal gasses (http://phet.colorado.edu/en/simulation/gas-properties). The experiment involves a glass bottle fitted with a stopper that has a small hole in it to let gas in and out, so the volume is constant. The bottle can be immersed in a bath of liquid that can be cooled to the boiling point of nitrogen (77K), cooled to the freezing point of water (273K) and then heated to the boiling point of water (373K). Because there is a small hole in the bottle, dry gas (free of moisture) can move in and out of the bottle as necessary to keep the atmospheric pressure constant in the bottle. The remaining parameters that can be changed are clearly the number of particles (dependent) and temperature (independent). We will first diagram what we expect of the gasses at low and high temperature, and then plot graphically how particle number depends on temperature from a virtual experiment. Notice the diagram on the left side involving two same size circles (constant volumes). As the temperature increases the number of the interactions with the chamber and other particles (i.e. the pressure) stays constant by pushing out particles through the hole, such that the number of lines (particles) decreases in the bottle. The plot at right, from ideal data, is an inverse. This experiment is difficult to accomplish because measuring such a large number of dry gas particles must be inferred and cannot be measured directly, plus it is essential to have a wide range of temperatures so that you can see the inverse nature of the plot. Linearization of this data would yield the following math model:

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N (# pls) =7.3x1025(# pls * K )

T ( K )=

PVkBT

Check this result from the known constants to see if the slope is correct. If we could measure the number of particles and temperature accurately, we could directly extract the Boltzmann's Constant from the slope of the linear plot and the known constants of pressure and volume.

T(K) Constants: Volume = 1L = 0.01m3 Atmospheric Pressure = 101300Pa

0 500

N(#)

7.4x1023


Constant Volume and Pressure



Worksheet Concepts "C": Ideal Gasses

1) The volume (squares below) and number of gas particles (16) in a container are kept constant.

Diagram the situation at low and high temperature and sketch what a graph of pressure versus temperature should look like, labeling all axes appropriately and explaining what the slope would equal.

2) The volume (squares below) and temperature in a container are kept constant. Diagram the

situation at small and large particle number and sketch what a graph of pressure versus particle number should look like, labeling all axes appropriately and explaining what the slope would equal.

3) The pressure and number of gas particles (16) in a container are kept constant. You choose which

parameter to control (temperature or volume). Diagram your and graph your results, labeling all axes appropriately and explaining what the slope would equal.


Constant Volume and Particle Number

Small N: Large N:

Constant Volume and Temperature

Constant Pressure and Particle Number



4) The temperature and number of gas particles (16) in a container are kept constant. Diagram the

pressure at low and high volumes and graph what pressure versus volume should look like, labeling all axes appropriately and explaining what the slope of the linearized data would equal.

5) Diagram how small and large particle numbers would demand of the volume such that the pressure

and temperature in a container are kept constant. Sketch what a graph of V versus N should look like, labeling all axes appropriately and explaining what the slope would equal.

6) Between the example and the five problems above, which of the P,V,N,T comparison's have we left

out? Decide which terms are constant and diagram and graph the variables and explaining what the slope would equal.

Small Volume: Large Volume:

Constant Temp and Particle Number

Small N: Large N:

Constant Pressure and Temperature

Constants:



Basic Worksheet "B": Ideal Gas, Buoyancy, and Hydrostatic Pressure 1) Recall the opening of the fresh bottle of soda water from the beginning of this unit.

a) First draw a diagram and graph of the parameters being kept constant and varied in this demonstration:

b) Calculate the number of gas molecules inside a 3mm diameter bubble using reasonable estimates for the remaining constant state variables (P and T).

c) What is the internal energy of the gas molecules inside the bubble? Compare it to the work required to lift a 100g apple 1m up.

2) This is a very subtle and tricky problem that you now have enough

information to solve. Consider an air-filled balloon weighted so that is on the verge of sinking, that is, its overall density just equals that of water. a) If you push it beneath the surface it will do what and why?

b) Sketch the molecular interactions and size of the balloon at the surface and at depth in the space below. Explain why your diagram makes sense at depth?

c) At what depth would the volume of the balloon be half of its volume at the surface of the water? What must you assume to solve this problem? You may assume a 1L balloon and STP at surface, but can be solved without this information.

Constants:



3) Calculate the volume of Avagadro's number of gas molecules at STP.

a) How big is this volume? Compare it to ordinary household items.

b) How much internal energy is stored in this much ideal gas? Compare it to the work required to lift a 100g apple 1m up.

4) From the following data complete the following questions:

a) Graph and linearize the data at right as needed.

b) Write down the mathematical model explaining the slope and intercept in terms of ideal gasses:

c) How much internal energy does the system have?

d) If this corresponds to a mole of gas particles, calculate the temperature they are at.

V (m3) P (Pa) .1 40 .5 8 1 4 2 2 4 1 5 .8 8 .5 10 .4



5) Below is a diagram of air molecules at a density of 1.3kg/m3 inside the crew volume (1.0x103m3) of

a submarine when the submarine is at the ocean's surface. Note only two of the molecules colliding with the walls of the ship – that will be a model for atmospheric pressure in the sub.

a) Determine the mass of air in the submarine. How many significant digits here?

b) Determine the number of air molecules (MW = 29g/mole).

c) On August 12, 2000, the Oscar II class Russian submarine Kursk sank in 100m of water near the Finish/Russian border, when the fuel tank of one of its own torpedoes exploded. All hands were lost. What was the pressure of the air in the submarine assuming all the air was trapped by the exposure to seawater at 100m depth?

d) Use the ideal gas relationship to draw the volume the air trapped in the sub at 100m depth, employing the same number of air molecules as in part "a" to explain diagrammatically what happens to pressure in the sub.

e) Explain if air is a good example of an ideal fluid in this situation.



Advance Worksheet "A": Ideal Gas, Buoyancy, and Hydrostatic Pressure

1) Recall the "handboiler" from the beginning of this unit.

a) First draw a diagram and graph of the parameters being kept constant and varied in this demonstration:

b) Using a reasonable guess for the volume of gas inside the handboiler and assuming the number of gas particles in the boiler when it is at room temperature correspond to atmospheric pressure, estimate the pressure of the gas inside the toy when at thermal equilibrium with your hand.

c) What is the internal energy of the gas at that temperature? Compare it to the work required to lift a 100g apple 1m up.

2) A typical hot air balloon can be modeled as a sphere with a radius of about 8.8m. The temperature

of the air inside the balloon is heated to around 210°F. a) Determine the number of ideal gas particles inside the balloon. What assumption did you have

to make to solve the problem?

b) Determine the lift capacity (weight of balloon and passengers) of the balloon at STP.

Constants:



3) Earth's atmospheric pressure does not decrease linearly

with increasing altitude because air is compressible. The temperature changes dramatically with elevation too. a) How do high altitude balloons account for this

difference? Diagram below the "volume" of a high altitude balloon released at the surface of the earth and again when it reaches and altitude of 15km.

b) A weather balloon filled with helium (ρ = 0.164kg/m3) and can lift a payload of 100kg. Calculate the radius of the spherical balloon at the surface of the earth assuming about one atmosphere of pressure and the temperature from the chart above.

c) Calculate the number of helium particles in the balloon.

d) Calculate the volume of the envelope at an altitude of 15km.

e) Calculate the pressure inside the balloon at 15km.



4) A diver 50 m deep in 5°C fresh water exhales a 0.50-cm-diameter bubble. The following steps will

help you determine the bubble's diameter just as it reaches the surface of the lake, where the water temperature is 20°C. Assume that the air bubble is always in thermal equilibrium with the surrounding water. a) Determine the pressure of the air inside the bubble at depth.

b) Determine the volume (and therefore diameter) of the bubble at the surface of the lake using the ideal gas law and pressure at the surface of the lake.

c) Beer bubbles and soda bubbles double or triple in diameter as they float upwards a mere 20cm in a beer stein or soda bottle. Explain if hydrostatic pressure can be used to explain the size change.

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