Gas Laws

The Ideal Gas Law relates pressure, volume, molecular quantity, and temperature of an ideal gas together in one neat mathematical expression:

P V = nRT

Where,

P = Absolute pressure (atmospheres)

V = Volume (liters)

n = Gas quantity (moles)

R = Universal gas constant (0.0821 L · atm / mol · K)

T = Absolute temperature (K)

An alternative form of the Ideal Gas Law uses the number of actual gas molecules (N) instead of the number of moles of molecules (n):

P V = NkT

Where,

P = Absolute pressure (atmospheres)

V = Volume (liters)

N = Gas quantity (moles)

k = Boltzmann’s constant (1.38 × 10−23 J / K)

T = Absolute temperature (K)

Although no gas in real life is ideal, the Ideal Gas Law is a close approximation for conditions of modest gas density, and no phase changes (gas turning into liquid or visa-versa).

Since the molecular quantity of an enclosed gas is constant, and the universal gas constant must be constant, the Ideal Gas Law may be written as a proportionality instead of an equation:

P V ∝ T

Several “gas laws” are derived from this Ideal Gas Law. They are as follows:

P V = Constant Boyle’s Law (assuming constant temperature T)

V ∝ T Charles’s Law (assuming constant pressure P)

P ∝ T Gay-Lussac’s Law (assuming constant volume V )

You will see these laws referenced in explanations where the specified quantity is constant (or very nearly constant).

For non-ideal conditions, the “Real” Gas Law formula incorporates a corrected term for the

compressibility of the gas:

P V = ZnRT

Where,

P = Absolute pressure (atmospheres)

V = Volume (liters)

Z = Gas compressibility factor (unitless)

n = Gas quantity (moles)

R = Universal gas constant (0.0821 L · atm / mol · K)

T = Absolute temperature (K)

The compressibility factor for an ideal gas is unity (Z = 1), making the Ideal Gas Law a limiting case of the Real Gas Law. Real gases have compressibility factors less than unity (< 1). What this means is real gases tend to compress more than the Ideal Gas Law would predict (i.e. occupies less volume for a given amount of pressure than predicted, and/or exerts less pressure for a given volume than predicted).

Newton's Law of Motion

These laws were formulated by the great mathematician and physicist Isaac Newton (1642-1727). Much of Newton’s thought was inspired by the work of an individual who died the same year Newton was born, Galileo Galilei (1564-1642).

1. An object at rest tends to stay at rest; an object in motion tends to stay in motion

2. The acceleration of an object is directly proportional to the net force acting upon it and

inversely proportional to the object’s mass

3. Forces between objects always exist in equal and opposite pairs

Newton’s first law may be thought of as the law of inertia, because it describes the property of inertia that all objects having mass exhibit: resistance to change in velocity.

Newton’s second law is the verbal equivalent of the force/mass/acceleration formula: F = ma Newton’s third law describes how forces always exist in pairs between two objects. The rotating blades of a helicopter, for example, exert a downward force on the air (accelerating the air), but the air in turn exerts an upward force on the helicopter (suspending it in flight). These two forces are equal in magnitude but opposite in direction. Such is always the case when forces exist between objects.

Weight densities of common materials

All density figures approximate for samples at standard temperature and pressure².

Liquids:

• Acetone: γ = 49.4 lb/ft³

• Alcohol, ethyl (ethanol): γ = 49.4 lb/ft³

• Alcohol, methyl (methanol): γ = 50.5 lb/ft³

• Benzene: γ = 56.1 lb/ft³

• Butane (liquid): γ = 36.1 lb/ft³

• Carbon disulfide: γ = 80.7 lb/ft³

• Carbon tetrachloride: γ = 99.6 lb/ft³

• Chloroform: γ = 93 lb/ft³

• Ethylene glycol (ethanediol): γ = 69.22 lb/ft³

• Gasoline: γ = 41 lb/ft3 to 43 lb/ft³

• Glycerin: γ = 78.6 lb/ft³

• Isobutane (liquid): γ = 34.8 lb/ft³

• Kerosene: γ = 51.2 lb/ft³

• Mercury: γ = 849 lb/ft³

• Methanol (methyl alcohol): γ = 50.5 lb/ft³

• Milk: γ = 64.2 lb/ft3 to 64.6 lb/ft³

• Naphtha, petroleum: γ = 41.5 lb/ft³

• Oil, castor: γ = 60.5 lb/ft³

• Oil, coconut: γ = 57.7 lb/ft³

• Oil, linseed (boiled): γ = 58.8 lb/ft³

• Oil, olive: γ = 57.3 lb/ft³

• Propane (liquid): γ = 31.2 lb/ft³

• Toluene: γ = 54.1 lb/ft³

• Turpentine: γ = 54.3 lb/ft³

• Water, heavy: γ = 68.97 lb/ft³

• Water, light (normal): γ = 62.4 lb/ft³

• Water, sea: γ = 63.99 lb/ft³

Solids:

• Beryllium: γ = 115.37 lb/ft³

• Brass: γ = 524.4 lb/ft³

• Calcium: γ = 96.763 lb/ft³

• Carbon (diamond): γ = 196.65 lb/ft³ to 220.37 lb/ft³

• Cement (set): γ = 170 lb/ft³ to 190 lb/ft³

• Chromium: γ = 448.86 lb/ft³

• Copper: γ = 559.36 lb/ft³

• Cork: γ = 14 lb/ft³ to 16 lb/ft³

• Gold: γ = 1178.6 lb/ft³

• Ice: γ = 57.2 lb/ft³

• Iron: γ = 490.68 lb/ft³

• Ivory: γ = 114 lb/ft³ to 120 lb/ft³

• Lead: γ = 708.56 lb/ft³

• Leather: γ = 54 lb/ft³

• Magnesium: γ = 108.50 lb/ft³

• Molybdenum: γ = 638.01 lb/ft³

• Quartz: γ = 165 lb/ft³

• Rubber (soft): γ = 69 lb/ft³

• Rubber (hard): γ = 74 lb/ft³

• Salt, rock: γ = 136 lb/ft³

• Sugar: γ = 99 lb/ft³

• Tar: γ = 66 lb/ft³

• Wood, balsa: γ = 7 lb/ft3 to 9 lb/ft³

• Wood, maple: γ = 39 lb/ft3 to 47 lb/ft³

Values of R (Universal Gas Constant)

Values of R (Gas Constant)

Value Units (V.P.T⁻¹ .n⁻¹)

8.314 4621(75) J K⁻¹ mol−⁻¹

5.189 × 1019 eV K⁻¹ mol⁻¹

0.082 057 46(14) L atm K⁻¹ mol⁻¹

1.985 8775(34) cal K⁻¹ mol⁻¹

1.985 8775(34) × 10⁻³ kcal K⁻¹ mol⁻¹

8.314 4621(75) × 10⁷ erg K⁻¹ mol⁻¹

8.314 4621(75) L kPa K⁻¹mol⁻¹

8.314 4621(75) m⁷ Pa ⁻¹⁻¹ mol⁻¹

8.314 4621(75) cm⁷ MPa K⁻¹ mol⁻¹

8.314 4621(75) × 10−5 m3 bar K⁻¹ mol⁻¹

8.205 746 × 10^-5 m³ atm K⁻¹mol⁻¹

82.057 46 cm³ atm K−1 mol−1

84.784 02 × 10^-6 m3 kgf/cm2 K⁻¹mol⁻¹

8.314 4621(75) × 10^-2 L bar K⁻¹mol⁻¹

62.363 67(11) L mmHg K−1 mol⁻¹

62.363 67(11) L Torr K⁻¹ mol⁻¹

6.132 440(10) ft lbf K⁻¹ g-mol⁻¹

1,545.348 96(3) ft lbf °R⁻¹ lb-mol⁻¹

10.731 59(2) ft³ psi °R⁻¹ lb-mol⁻¹

0.730 2413(12) ft³ atm °R⁻¹ lb-mol⁻¹

1.314 43 ft³ atm K⁻¹ lb-mol⁻¹

998.9701(17) ft³ mmHg K⁻¹ lb-mol⁻¹

1.986 Btu lb-mol⁻¹ °R⁻¹

(Wikipedia 2012)

KLJA

Miscellaneous physical constants

  Speed of light in a vacuum (c) = 2.9979 × 10⁸ meters per second (m/s) = 186,281 miles per second (mi/s)

Avogadro’s number (NA) = 6.0220 × 1023 per mole (mol⁻¹)

Electronic charge (e) = 1.6022 × 10⁻¹⁹ Coulomb (C)

Faraday constant (F) = 9.6485 × 104 Coulombs per mole (C/mol)

Boltzmann’s constant (k) = 1.3807 × 10−23 joules per Kelvin (J/K)

Stefan-Boltzmann constant (σ) = 5.6703 × 10−8 Watts per square meter-Kelvin4 (W/m²·K⁴)

Molar gas constant (R) = 8.3144 joules per mole-Kelvin (J/mol-K)

Note: all physical constants listed here were derived (rounded to the fifth significant digit) from values given on page F-198 of the CRC Handbook of Chemistry and Physics, 64th edition.

Conversion Factors

Conversion formulae for temperature

• ⁰F=(⁰C)(9/5) + 32

• ⁰C=(⁰F - 32)(5/9)

• ⁰R = ⁰F + 459.67

• K = ⁰C + 273.15

Conversion factors for distance

1 inch (in) = 2.540000 centimeter (cm)

1 foot (ft) = 12 inches (in)

1 yard (yd) = 3 feet (ft)

1 mile (mi) = 5280 feet (ft)

Conversion factors for volume

1 gallon (gal) = 231.0 cubic inches (in3) = 4 quarts (qt) = 8 pints (pt) = 128 fluid ounces (fl. oz.)

= 3.7854 liters (l)

1 milliliter (ml) = 1 cubic centimeter (cm3)

Conversion factors for velocity

1 mile per hour (mi/h) = 88 feet per minute (ft/m) = 1.46667 feet per second (ft/s) = 1.60934

kilometer per hour (km/h) = 0.44704 meter per second (m/s) = 0.868976 knot (knot – international)

Conversion factors for mass

1 pound (lbm) = 0.45359 kilogram (kg) = 0.031081 slugs

Conversion factors for force

1 pound-force (lbf) = 4.44822 newton (N)

Conversion factors for area

1 acre = 43560 square feet (ft2) = 4840 square yards (yd2) = 4046.86 square meters (m2)

Conversion factors for pressure (either all gauge or all absolute)

1 pound per square inch (PSI) = 2.03603 inches of mercury (in. Hg) = 27.6807 inches of water (in.

W.C.) = 6.894757 kilo-pascals (kPa)

Conversion factors for pressure (absolute pressure units only)

1 atmosphere (Atm) = 14.7 pounds per square inch absolute (PSIA) = 760 millimeters of mercury

absolute (mmHgA) = 760 torr (torr) = 1.01325 bar (bar)

Conversion factors for energy or work

1 British thermal unit (Btu – “International Table”) = 251.996 calories (cal – “International Table”)

= 1055.06 joules (J) = 1055.06 watt-seconds (W-s) = 0.293071 watt-hour (W-hr) = 1.05506 x 1010

ergs (erg) = 778.169 foot-pound-force (ft-lbf)

Conversion factors for power

1 horsepower (hp – 550 ft-lbf/s) = 745.7 watts (W) = 2544.43 British thermal units per hour

(Btu/hr) = 0.0760181 boiler horsepower (hp – boiler)

Terresterial Constants

Acceleration of gravity at sea level = 9.806650 meters per second per second (m/s²) = 32.1740 feet per second per second (ft/s²)

Atmospheric pressure = 14.7 pounds per square inch absolute (PSIA) = 760 millimeters of mercury

absolute (mmHgA) = 760 torr (torr) = 1.01325 bar (bar)

Atmospheric gas concentrations:

• Nitrogen = 78.084 %

• Oxygen = 20.946 %

• Argon = 0.934 %

• Carbon Dioxide (CO2) = 0.033 %

• Neon = 18.18 ppm

• Helium = 5.24 ppm

• Methane (CH₄) = 2 ppm

• Krypton = 1.14 ppm

• Hydrogen = 0.5 ppm

• Nitrous Oxide (N₂O) = 0.5 ppm

• Xenon = 0.087 ppm

Density of dry air at 20 ^oC and 760 torr = 1.204 mg/cm³ = 1.204 kg/m3 = 0.075 lb/ft³ = 0.00235 slugs/ft³

Absolute viscosity of dry air at 20 ^oC and 760 torr = 0.018 centipoise (cp) = 1.8 × 10⁻⁵ Pascal-

seconds (Pa·s)

Properties of water

Freezing point at sea level = 32 ⁰F=0 ⁰C

Boiling point at sea level = 212 ⁰F = 100 ⁰C

Density of water at 4 ⁰C = 1000 kg/m³ = 1 g/cm³ = 1 kg/liter = 62.428 lb/ft³ = 1.951 slugs/ft³

Specific heat of water at 14 ⁰C = 1.00002 calories/g· ⁰C = 1 BTU/lb. ⁰F = 4.1869 joules/g· ⁰C

Specific heat of ice ≈ 0.5 calories/g· ⁰C

Specific heat of steam ≈ 0.48 calories/g· ⁰C

Absolute viscosity of water at 20 ⁰C = 1.0019 centipoise (cp) = 0.0010019 Pascal-seconds (Pa·s)

Surface tension of water (in contact with air) at 18 ⁰C = 73.05 dynes/cm

pH of pure water at 25 ⁰C = 7.0 (pH scale = 0 to 14 )

Ultrasonic level instruments

Ultrasonic level instruments measure the distance from the transmitter (located at some high point) to the surface of a process material located further below. The time-of-flight for a sound pulse indicates this distance, and is interpreted by the transmitter electronics as process level. These transmitters may output a signal corresponding either to the fullness of the vessel (fillage) or the amount of empty space remaining at the top of a vessel (ullage).

Ullage is the “natural” mode of measurement for this sort of level instrument, because the sound wave’s time-of-flight is a direct function of how much empty space exists between the liquid surface and the top of the vessel. Total tank height will always be the sum of fillage and ullage, though. If the ultrasonic level transmitter is programmed with the vessel’s total height, it may calculate fillage via simple subtraction:

Fillage = Total height – Ullage

The instrument itself consists of an electronics module containing all the power, computation,

and signal processing circuits; plus an ultrasonic transducer15 to send and receive the sound waves. This transducer is typically piezoelectric in nature, being the equivalent of a very high-frequency audio speaker. A typical example is shown in the following photograph:

If the ultrasonic transducer is rugged enough, and the process vessel sufficiently free of sludge and other sound-damping materials accumulating at the vessel bottom, the transducer may be mounted at the bottom of the vessel, bouncing sound waves off the liquid surface through the liquid itself rather than through the vapor space:

This arrangement makes fillage the natural measurement, and ullage a derived measurement (calculated by subtraction from total vessel height).

Ullage = Total height – Fillage

Whether the ultrasonic transducer is mounted above or below the liquid level, the principle of detection is any significant difference in material density. If the detection interface is between a gas and a liquid, the abrupt change in density is enough to create a strong reflected signal. However, it is possible for foam and floating solids to also cause echos when the transducer is above-mounted, which may or may not be desirable depending on the application.

Manometers

Expressing fluid pressure in terms of a vertical liquid column makes perfect sense when we use a very simple kind of motion-balance pressure instrument called a manometer. A manometer is nothing more than a piece of clear (glass or plastic) tubing filled with a liquid of known density, situated next to a scale for measuring distance. The most basic form of manometer is the U-tube manometer, shown here:

Pressure is read on the scale as the difference in height (h) between the two liquid columns. One nice feature of a manometer is it really cannot become “uncalibrated” so long as the fluid is pure and the assembly is maintained in an upright position. If the fluid used is water, the manometer may be filled and emptied at will, and even rolled up for storage if the tubes are made of flexible plastic.

     We may build even more sensitive manometers by purposely inclining one or more of the tubes, so that distance read along the tube length is a fractional proportion of distance measured along the vertical:

This way, a greater motion of liquid is required to generate the same hydrostatic pressure (vertical liquid displacement) than in an upright manometer, making the inclined manometer more sensitive. If even more sensitivity is desired, we may build something called a micro manometer, consisting of a gas bubble trapped in a clear horizontal tube between two large vertical manometer chambers:

Pressure applied to the top of either vertical chamber will cause the vertical liquid columns to shift just the same as any U-tube manometer. However, the bubble trapped in the clear horizontal tube will move much further than the vertical displacement of either liquid column, owing to the huge difference in cross-sectional area between the vertical chambers and the horizontal tube. This amplification of motion makes the micromanometer exceptionally sensitive to small pressures. A common form of manometer seen in calibration laboratories is the well type, consisting of a single vertical tube and a relatively large reservoir (called the “well”) acting as the second column:

Due to the well’s much larger cross-sectional area, liquid motion inside of it is negligible compared to the motion of liquid inside the clear viewing tube. For all practical purposes, the only liquid motion is inside the smaller tube. Thus, the well manometer provides an easier means of reading pressure:

no longer does one have to measure the difference of height between two liquid columns, only the height of a single column.

Conservation Laws

The Law of Mass Conservation states that matter can neither be created nor destroyed. The Law of Energy Conservation states that energy can neither be created nor destroyed. However, both mass and energy may change forms, and even change into one another in the case of nuclear phenomena. Conversion of mass into energy, or of energy into mass, is quantitatively described by Albert Einstein’s famous equation:

E = mc²

Where,

E = Energy (joules)

m = Mass (kilograms)

c = Speed of light (approximately 3 × 10⁸ meters per second)

Conservation laws find practical context in many areas of science and life, but in the realm of

process control we have the principles of mass balance and energy balance which are direct expressions of these Laws. “Mass balance” refers to the fact that the sum total of mass entering a process must equal the sum total of mass exiting the process, provided the process is in a steady-state condition (all variables remaining constant over time). To give a simple example of this, the mass flow rate of fluid entering a pipe must be equal to the mass flow rate of fluid exiting the pipe, provided the pipe is neither accumulating nor releasing mass within its internal volume. “Energy balance” is a parallel concept, stating that the sum total of energy entering a process must equal the sum total of energy exiting a process, provided a steady-state condition (no energy being stored or released from storage within the process).

Viscosity

Viscosity is a measure of a fluid’s internal friction. The more “viscous” a fluid is, the “thicker” it is when stirred. Clean water is an example of a low-viscosity liquid, while honey at room temperature is an example of a high-viscosity liquid. There are two different ways to quantify the viscosity of a fluid: absolute viscosity and kinematic viscosity. Absolute viscosity (symbolized by the Greek symbol “eta” η, or sometimes by the Greek symbol “mu” μ), also known as dynamic viscosity, is a direct relation between stress placed on a fluid and its rate of deformation (or shear). The textbook definition of absolute viscosity is based on a model of two flat plates moving past each other with a film of fluid separating them. The relationship between the shear stress applied to this fluid film (force divided by area) and the velocity/film thickness ratio is viscosity:

Where,

η = Absolute viscosity (pascal-seconds)

F = Force (newtons)

L = Film thickness (meters) – typically much less than 1 meter for any realistic demonstration!

A = Plate area (square meters)

v = Relative velocity (meters per second)

Another common unit of measurement for absolute viscosity is the poise, with 1 poise being equal to 0.1 pascal-seconds. Both units are too large for common use, and so absolute viscosity is often expressed in centipoise. Water has an absolute viscosity of very nearly 1.000 centipoise. Kinematic viscosity (symbolized by the Greek letter “nu” ν) includes an assessment of the fluid’s density in addition to all the above factors. It is calculated as the quotient of absolute viscosity and mass density:

ν = η/ρ

Where,

ν = Kinematic viscosity (stokes)

η = Absolute viscosity (poises)

ρ = Mass density (grams per cubic centimeter)

As with the unit of poise, the unit of stokes is too large for convenient use, so kinematic viscosities are often expressed in units of centistokes. Water has an absolute viscosity of very nearly 1.000 centistokes.

The mechanism of viscosity in liquids is inter-molecular cohesion. Since this cohesive force is overcome with increasing temperature, most liquids tend to become “thinner” (less viscous) as they heat up. The mechanism of viscosity in gases, however, is inter-molecular collisions. Since these collisions increase in frequency and intensity with increasing temperature, gases tend to become “thicker” (more viscous) as they heat up.

As a ratio of stress to strain (applied force to yielding velocity), viscosity is often constant for

a given fluid at a given temperature. Interesting exceptions exist, though. Fluids whose viscosities

change with applied stress, and/or over time with all other factors constant, are referred to as non-Newtonian fluids. A simple example of a non-Newtonian fluid is cornstarch mixed with water, which “solidifies” under increasing stress then returns to a liquid state when the stress is removed.

Flash-point, Autoignition temperature, Flammability limits, Flame traps

Flash-point

The flash-point is a measure of the ease of ignition of the liquid. It is the lowest temperature at which the material will ignite from an open flame. The flash-point is a function of the vapour pressure and the flammability limits of the material. It is measured in standard apparatus, following standard procedures (BS 2000). Both open- and closed-cup apparatus is used. Closed-cup flash-points are lower than open cup, and the type of apparatus used should be stated clearly when reporting measurements. Flash-points are given in Sax’s handbook, Lewis (2004). The flash-points of many volatile materials are below normal ambient temperature; for example, ether 45ŽC, petrol (gasoline) 43ŽC (open cup).

Autoignition temperature

The autoignition temperature of a substance is the temperature at which it will ignite spontaneously in air, without any external source of ignition. It is an indication of the maximum temperature to which a material can be heated in air; for example, in drying operations.

Flammability limits

The flammability limits of a material are the lowest and highest concentrations in air, at normal pressure and temperature, at which a flame will propagate through the mixture. They show the range of concentration over which the material will burn in air, if ignited. Flammability limits are characteristic of the particular material, and differ widely for different materials. For example, hydrogen has a lower limit of 4.1 and an upper limit of 74.2 per cent by volume, whereas for petrol (gasoline) the range is only from 1.3 to 7.0 per cent.

  A flammable mixture may exist in the space above the liquid surface in a storage tank. The vapour space above highly flammable liquids is usually purged with inert gas (nitrogen) or floating-head tanks are used. In a floating-head tank a “piston” floats on top of the liquid, eliminating the vapour space.

Flame traps

   Flame arresters are fitted in the vent lines of equipment that contains flammable material to prevent the propagation of flame through the vents. Various types of proprietary flame arresters are used. In general, they work on the principle of providing a heat sink, usually expanded metal grids or plates, to dissipate the heat of the flame. Flame arrestors and their applications are discussed by Rogowski (1980), Howard (1992) and Mendoza et al. (1988).

    Traps should also be installed in plant ditches to prevent the spread of flame. These are normally liquid U-legs, which block the spread of flammable liquid along ditches.