Table 12.1 A Macroscopic Comparison of Gases, Liquids, and Solids State Shape and Volume Compressibility Ability to Flow Gas Conforms to shape and volume of container high high Liquid Conforms to shape of container; volume limited by surface very low moderate Solid Maintains its own shape and volume almost none almost none
Figure 12.3 A cooling curve for the conversion of gaseous water to ice.
GASES (e.g. Air: 78% N2, 21% O2, vapors (N2, H2, F2, Cl2, He, Ne, Ar, Kr, Xe)) coined by Jan van Helmont in 1624 from the Greek word chaos which means “confusion” or “empty space” have the complete freedom to fill its container thus they assume the volume and shape of the vessel at which they are contained. are highly compressible forms homogeneous mixtures molecules are far apart: ~0.1% of total volume is occupied by gas molecules compared to 70% for lquids
*Substances that are liquids or solids under ordinary conditions can usually also exist in the gaseous state and they are often referred to as vapors Variables that affect gases: 1. volume, V 3. pressure, P 2. temperature, T 4. number of moles, n *Pressure – a force that tends to move something in a given direction P = F/A F = force; A = area *gases exert pressure on walls with which they are in contact e.g. balloons, tires SI Unit: 1 kg/m.s2 = 1 N/m2 = 1 Pa 1 atm = 760 mmHg = 760 Torr = 1.01325 x 105 Pa or 101.325kPa * standard atmospheric pressure = typical pressure at sea level; pressure exerted by the atmosphere on the Earth’s surface * Standard Temperature and Pressure ( STP ): T = 273 K, P = 1 atm
barometer – device used for atmospheric pressure measurement – made from an inverted glass tubing of more than 760 mm in length. This tube is filled with Hg and the height of the mercury column is proportional to the atmospheric pressure. manometer – device used to measure the pressure of enclosed gases whose pressures are near the atmospheric pressure – if the levels in the two arms of the tube are the same, the enclosed gas has the same pressure as the atmospheric pressure. – if the pressure of the enclosed gas is less than the atmospheric pressure, the mercury is forced toward the direction of the enclosed gas. – if the pressure of the enclosed gas is greater than the atmospheric pressure, the mercury is push towards the atmosphere.
GAS LAWS - equations that express the relationships among P, T, V, and n of gases 1. Boyle’s Law (Robert Boyle) : P-V Relationship “ At constant temperature, the volume of a quantity of gas is inversely proportional to the pressure.” at constant T: V α (1/P) ↔ PV = constant pressure, ↓ volume 2. Charles’ Law (Jacques Charles): temperature-volume relationship “ At constant pressure, the volume occupied by a fixed amount of gas is directly proportional to the absolute temperature.” at constant P : T α V ↔ V/T = constant ↑ temperature, ↑ volume *Note: T = absolute temperature (Kelvin, K) TK = 273 + oC
3. Avogadro’s Law (Amadeo Avogadro): quantity – volume relationship Avogadro’s Hypothesis: “Equal volume of gases at the same temperature and pressure contain equal number of particles (molecules).” Avogadro’s Law : “The volume of gas maintained at constant temperature and pressure is directly proportional to the number of moles of the gas.” at constant T and P: V α n ↔ V = constant x n same volume of any gas, say 10 mL, be it H2, He, O2, NH3, and so on, at the same temperature and pressure, contain the same number of moles (or molecules) at STP, V = 22.4 L 4. Gay-Lusaac’s Law/Laws of Combining Volumes (Joseph Louis Gay-Lusaac) “ At a given pressure and temperature, the volume of gases that react with one another are in ratios of small whole numbers.” e.g. 2H 2 + O 2 -> 2H 2 O 1omL 5 mL 10 mL
Amonton’s Law : P 1 T 1 = P 2 T 2 Boyle’s Law: P 1 V 1 = P 2 V 2 Charles’ Law : V 1 T 1 V 2 T 2 = Combined Gas Laws : P 1 V 1 = P 2 V 2 T 1 T 2 A B C
The Ideal Gas Equation: Combined Gas Laws *ideal gas – hypothetical gas whose P, v, and T behavior are completely described by the ideal gas equation Ideal Gas Equation: PV = nRT where R = ideal gas constant = 0.08206 L · atm / mol · K P = pressure (atm) T = temperature (K) n = mole of gas (mol) V = volume (L)
Exercises 1. A sample of a gas occupies 360mL under a pressure of 0.750atm. If the T is held constant, what volume will the sample occupy under a pressure of 1.00atm? 2. At 0 o C and 5.00 atm, a given sample of a gas occupies 75.0 L. The gas is compressed to a final volume of 30.0L at 0oC. What is the final pressure of the gas? 3. A sample of N2 exerts a pressure of 0.988atm and occupies 12.3L when its T 450K. Assuming constant pressure, what volume will the gas occupy at 300K? 4. At what T will a 10.0 L gas at constant pressure be brought to if the same gas originally occupied 3000cm3 at 27 o C? 5. Two identical cylinders of N2 contain the same weight of N2. The T of one cylinder is 20 o C and the other is 100 o C. If the pressure the one at 20 o C is 1520mmHg, what is the pressure of the other cylinder? 6. In a laboratory, 200cm3 of O2 gas is collected at a pressure of 73 cmHg and temperature of 30oC, Compute the volume of the O2 gas under standard conditions. 7. A sample of gas occupies 400mL at STP. What volume will the sample occupy at 71 o C and 2.50 atm? 8. Calculate the pressure of one mole of an ideal gas which occupies 12.0 L at 25 o C. 9. A gas contained in 50L under 8 atm of pressure and at 20 o C. How many moles of gas are there in the container?
GAS MIXTURES AND PARTIAL PRESSURES
PARTIAL PRESSURE AND MOLE FRACTION
FURTHER APPLICATIONS OF THE IDEAL GAS EQUATION Gas Densities and Molar Mass: the higher the molecular mass and pressure of a gas, the denser is the gas. more denser gases will lie below that of less dense gases. hot gases tend to be less dense than cool gases.
Volumes of Gases in Chemical Reaction The coefficients in a balanced chemical equation tell us the relative amounts (in moles) of reactants and products in a reaction. Collecting Gases Over Water *When a gas is collected over water, the total pressure or atmospheric pressure is equal to the sum of the pressures of the gas and the water vapor. Patm = Pgas + P WATER
Exercises 1: 1. From data gathered by Voyager 1, scientist have estimated the composition of the atmosphere of Titan, Saturn’s largest moon. The total pressure on the surface of Titan is 1220 Torr. The atmosphere consist of 82 mole percent N2, 12 mol percent Ar, and 6.0 mol percent CH4. Calculate the partial pressure of these gases. 2. A sample of 5.0 g He gas, 2.0 g H 2 gas and 10.0 g water vapor exerts a pressure of 0.5 atm at 25 o C. Calculate the mole fraction of each gas, the partial pressure of each gas and the volume occupied by the mixture.
What is the density of CCl4 vapor at 714 torr and 125 o C?
2. The safety air bags in automobile are inflated by nitrogen gas generated by the rapid decomposition of sodium azide, NaN3:
NaN 3 (s) 2 Na (s) + 3 N 2 (g)
If an air bag has a volume of 36 L and is to be filled with nitrogen gas at a pressure of 1.15 atm at a temperature of 26.00C, how many grams of NaN3 must be decomposed?
3. A sample of KClO3 is partially decomposed producing O2 that is collected over water. The volume of gas collected is 0.250 L at 26 o C and 765 Torr of total pressure. How many moles of O 2 are collected? How many grams of KClO3 were decomposed?P H2O at 26 o C = 25 torr
KINETIC MOLECULAR THEORY *provides a model to explain the regularity observed in the behavior of ideal gases. *formulated by Rudolf Clausius Summary: *Gases consist of large number of molecules that are in continuous, random motion. *These gaseous molecules are widely separated in space. Actual volume of gas molecules is negligible compared with empty space between them. *Attractive and repulsive forces between the molecules are negligible. *The collisions are perfectly elastic. Energy can be transferred from one molecule to the other during collisions, but the average kinetic energy of the molecules remain constant. *The average kinetic energy of the molecules is proportional to absolute temperature. At any given temperature, the molecules of all gases have the same kinetic energy.
APPLICATION OF KMT
1. Effect of volume increase on pressure at constant temperature
the speed at which molecules travel remains unchanged when temperature is constant (kinetic energy is proportional to temperature)
an increase in volume also increases the distance at which the molecules travel. When this happens, there will be fewer collisions on the container walls per unit time. Fewer collisions result to decrease in pressure.
Increase in volume, decrease in pressure
2. Effect of temperature increase on pressure at constant volume
No change in volume means constant number of collisions against the container walls.
An increase in temperature means an increase in kinetic energy. The greater the kinetic energy, the more collisions occur. Hence, the pressure of the gas increases.
Increase in temperature, increase in pressure
Molecular Effusion and Diffusion root-mean-square-speed (rms speed), *Note: Particles of lighter gases have a higher rms speed than particles of heavier one. μ =
Calculate the rms speed of N 2 molecule at 25 o C.
rms = = (3)(8.3145 J.mol/K)(298.15K) (28.02 g/mol)(1 kg/1000g) √ = 515 or 5.2 x 10 2 m/s Note : 1 J = 1 kg (m 2 /s 2 )
2. Arrange the following in increasing rms speed: N 2 , O 2 , H 2 O, NO at 298 K N 2 = 28.02 g/mol O 2 = 32 g/mol H 2 O = 18.02 g/mol NO = 30.01 g/mol Answer: O 2 < NO< N 2 < H 2 O
Effusion – the escape of gas molecules through a tiny hole. Diffusion – the spread of one substance throughout a space or throughout a second substance. Graham’s Law (Thomas Graham) “ The effusion and diffusion rates of a gas are inversely proportional to the square root of its molar mass.” Rate α
Arrange the following gases in order of increasing rate of diffusion: CO, NO2, HF, H2S, CO2 at the same temperature
CO= 28.01 Answer: NO2 < CO2 < H2S < CO < HF
NO 2 = 46.01
HF = 20.008
H 2 S = 34.086
CO 2 = 44.01
= d 1 d 2
2. An unknown gas composed of homonuclear diatomic molecule effuses at a rate that is only 0.355 times that of O2 at the same temperature. What is the identity of the unknown gas? 3. Two gases, O2 and N2, are introduced into the two ends of a 100.0cm tube. At what distance from the point where O2 is introduced will the two gases meet?
Deviations from Ideal Gas Behavior (Real or nonideal) High Pressures – molecules are more crowded; volume of gas molecules cannot be considered negligible; observed volume of gas > ideal volume Low Temperature – average KE decreases while intermolecular forces remain; cooling a gas deprives the molecules of the energy needed to overcome the intermolecular attractive forces; less number of collisions with walls; observed P < ideal pressure
Intermolecular Forces of Attraction
-are the forces that exist between molecules
-the strengths of intermolecular forces vary for each substance
-they are generally much weaker than ionic or covalent bond (intramolecular)
-properties of liquids, such as boiling points and melting points, depend on the strength of intermolecular forces of attraction (IMFA)
Types of Intermolecular Forces of Attraction ion-dipole force dipole-dipole force Hydrogen bonding London dispersion
Polar molecule (due to difference in electronegativity)
Figure 12.12 Polar molecules and dipole-dipole forces. solid liquid
Polarizability and Charged-Induced Dipole Forces distortion of an electron cloud
Polarizability increases down a group
size increases and the larger electron clouds are further from the nucleus
Polarizability decreases left to right across a period
increasing Z eff shrinks atomic size and holds the electrons more tightly
Cations are less polarizable than their parent atom because they are smaller.
Anions are more polarizable than their parent atom because they are larger.
1. ION-DIPOLE FORCE
=exist between an ion and the partial charge on the end of a polar molecule (e.g. H2O)
=polar molecules have dipoles , that is, they have partial positive and partial negative ends
=specially important for solutions of ionic compounds in polar liquids,
e.g. solution of NaCl in water
2. DIPOLE-DIPOLE FORCE
exist between neutral polar molecule
polar molecules attract each other when the negative end of molecule is in close proximity with the positive end of the other molecule
effective only when polar molecules are close to each other
are generally weaker than ion-dipole force
for molecules with approximately equal mass and size, the strengths of dipole-dipole attraction increase with increasing polarity
3. Hydrogen Bonding (H-bonding)
-the strongest type of IMFA
-a special case of dipole-dipole interaction occurring between molecules in which H is covalently bonded to small, electronegative atom (N, O, F)
-the small size of the H atom makes the F, N, or O atom of one molecule approach the H atom of another molecule closely sufficient to produce an attraction strong enough to be called a bond
Illustration: H ― F ----------- H ― F
THE HYDROGEN BOND a dipole-dipole intermolecular force The elements which are so electronegative are N, O, and F. A hydrogen bond may occur when an H atom in a molecule, bound to small highly electronegative atom with lone pairs of electrons, is attracted to the lone pairs in another molecule. hydrogen bond donor hydrogen bond acceptor hydrogen bond acceptor hydrogen bond donor hydrogen bond donor hydrogen bond acceptor .. F .. .. .. H O .. N .. F H .. .. .. O .. .. .. N H
Figure 12.21 The H-bonding ability of the water molecule. hydrogen bond donor hydrogen bond acceptor
SAMPLE PROBLEM 12.2 Drawing Hydrogen Bonds Between Molecules of a Substance SOLUTION: (a) C 2 H 6 has no H bonding sites. (c) PROBLEM: Which of the following substances exhibits H bonding? For those that do, draw two molecules of the substance with the H bonds between them. (a) (b) (c) PLAN: Find molecules in which H is bonded to N, O or F. Draw H bonds in the format -B: H-A-. (b)
4.LONDON DISPERSION FORCE
-forces formed from instantaneous (momentary) dipole moment created by the motion of electrons in an atom or molecule
-the movement of electrons in a molecule is influenced by repulsion of neighboring electrons
-IMFA that exist in nonpolar molecules ; the weakest of all known intermolecular forces; exist between all molecules but are generally overshadowed when stronger forces are present.
-the greater the polarizability of the electrons in a molecule, the more easily is its electron cloud can be distorted, hence, the stronger the London dispersion force.
-dispersion forces tend to increase in strength as the molecular weight increases (due to increasing size and polarizability)
Figure 12.18 Summary diagram for analyzing the intermolecular forces in a sample. INTERACTING PARTICLES (atoms, molecules, ions) ions only IONIC BONDING (Section 9.2) ion + polar molecule ION-DIPOLE FORCES ions present ions not present polar molecules only DIPOLE-DIPOLE FORCES HYDROGEN BONDING polar + nonpolar molecules DIPOLE- INDUCED DIPOLE FORCES nonpolar molecules only LONDON DISPERSION FORCES only DISPERSION FORCES ALSO PRESENT H bonded to N, O, or F
Figure 13.1 The major types of intermolecular forces in solutions.
Strengths of IMFA H-bonding > ion dipole > dipole-dipole > london dispersion
SAMPLE PROBLEM 12.3 Predicting the Type and Relative Strength of Intermolecular Forces (a) MgCl 2 or PCl 3 (b) CH 3 NH 2 or CH 3 F (c) CH 3 OH or CH 3 CH 2 OH
Bonding forces are stronger than nonbonding(intermolecular) forces.
Hydrogen bonding is a strong type of dipole-dipole force.
Dispersion forces are decisive when the difference is molar mass or molecular shape.
PROBLEM: For each pair of substances, identify the dominant intermolecular forces in each substance, and select the substance with the higher boiling point. (d) Hexane (CH 3 CH 2 CH 2 CH 2 CH 2 CH 3 ) or 2,2-dimethylbutane PLAN: Use the formula, structure and Table 2.2 (button).
SOLUTION: SAMPLE PROBLEM 12.3 Predicting the Type and Relative Strength of Intermolecular Forces continued (a) Mg 2+ and Cl - are held together by ionic bonds while PCl 3 is covalently bonded and the molecules are held together by dipole-dipole interactions. Ionic bonds are stronger than dipole interactions and so MgCl 2 has the higher boiling point. (b) CH 3 NH 2 and CH 3 F are both covalent compounds and have bonds which are polar. The dipole in CH 3 NH 2 can H bond while that in CH 3 F cannot. Therefore CH 3 NH 2 has the stronger interactions and the higher boiling point. (c) Both CH 3 OH and CH 3 CH 2 OH can H bond but CH 3 CH 2 OH has more CH for more dispersion force interaction. Therefore CH 3 CH 2 OH has the higher boiling point. (d) Hexane and 2,2-dimethylbutane are both nonpolar with only dispersion forces to hold the molecules together. Hexane has the larger surface area, thereby the greater dispersion forces and the higher boiling point.
Arrange the following types of interactions in order of increasing stability: covalent bond, van der Waals force, hydrogen bonding, dipole interaction
2.Which has the highest boiling point: H 2 , He, Ne, Xe, CH 4
3.Which is expected to have the highest melting point: PH 3 , NH 3 , (CH 3 ) 3 N? Explain why.
Answer : van der Waals < dipole < hydrogen bonding < covalent Answer : Xe, All are nonpolar molecules, but Xe has the greatest van der Waals forces because it has the most electrons Answer : NH3 has the strongest intermolecular forces, thus it is expected to have the highest melting point .
intermolecular forces (IMF) are strong enough to hold the molecules close together; IMF are not strong enough to keep the molecules from moving past one another
liquids flow and assume the shape of their container
liquids are denser than gases; a 70% of the volume of liquids are occupied by liquid molecules
liquids are incompressible; liquids do not expand to fill the container
diffusion with a liquid occurs slowly
Properties of Liquids
1. Viscosity resistance of liquid to flow related to the ease with which individual molecules of the liquid can move with respect to one another it depends on the IMFA that exist in the liquid; the greater the liquid’s viscosity, the more slowly it flows viscosity decreases with increasing temperature; at higher temperature, the average kinetic energy of the molecules is greater, hence, it more easily overcomes the IMFA ↑ T, ↓ viscosity ↑ P, ↑ viscosity ↑ complexity of molecule, ↑ viscosity ↑ IMF, ↑ viscosity e.g. H2O > CH3CH2OH; oil > H2O
Table 12.4 Viscosity of Water at Several Temperatures Temperature( 0 C) Viscosity (N*s/m 2 )* 20 40 60 80 1.00x10 -3 0.65x10 -3 0.47x10 -3 0.35x10 -3 *The units of viscosity are newton-seconds per square meter. viscosity - resistance to flow
2. Surface tension
energy required to increase the surface area of a liquid by a unit amount
measure of the inward force in liquids
the stronger the IMFA, the larger the surface tension
Consequences of Surface Tension:
Cohesive force – IMFA that binds similar molecules
Adhesive force – IMFA that binds different molecules/surface
Cohesion > Adhesion -> meniscus concave down
Adhesion > Cohesion -> meniscus concave up
Capillary Action – the rise of a liquid up a narrow tube. The surface area is increased when H2O rise up a narrow tube. The H2O levels stop rising until it is balanced by the gravitational pull
Figure 12.19 The molecular basis of surface tension. hydrogen bonding occurs in three dimensions hydrogen bonding occurs across the surface and below the surface the net vector for attractive forces is downward
Table 12.3 Surface Tension and Forces Between Particles Substance Formula Surface Tension (J/m 2 ) at 20 0 C Major Force(s) diethyl ether ethanol butanol water mercury dipole-dipole; dispersion H bonding H bonding; dispersion H bonding metallic bonding 1.7x10 -2 2.3x10 -2 2.5x10 -2 7.3x10 -2 48x10 -2 CH 3 CH 2 OCH 2 CH 3 CH 3 CH 2 OH CH 3 CH 2 CH 2 CH 2 OH H 2 O Hg
Figure 12.20 Shape of water or mercury meniscus in glass. H 2 O capillarity Hg adhesive forces stronger cohesive forces
passage of molecules from liquid to a gaseous state
rate of vaporization increases with increased T and decreased IMFA
4. Vapor pressure the pressure exerted by a vapor over the liquid when the liquid and vapor state are in dynamic equilibrium (equal rates of evaporation and condensation) escaping tendency of the liquid the liquids with weak IMFA have high vapor pressure, hence, they evaporate easily and are said to be volatile; vapor pressure increases with temperature. The higher the temperature, the higher also is the vapor pressure -> ↑ T, ↑ VP ↑ IMF, ↓ VP ↑ VP, ↓ boiling point
5. Boiling Point temperature at which the VP of a liquid equals the external pressure acting on its surface during boiling, every heat absorbed is used to convert liquid to gas and T remains constant until all the liquid has been converted normal boiling point – boiling point at which the external pressure is equal to 1atm, e.g. H 2 O 100 o C Tc, critical temperature – the highest temperature at which a liquid can exist Pc, critical pressure – the minimum pressure required to bring about liquefaction; greater IMFA -> more readily a gas is liquefied -> higher Tc
Figure 12.16 Molar mass and boiling point.
Figure 12.17 Molecular shape and boiling point. more points for dispersion forces to act fewer points for dispersion forces to act
Figure 12.14 Hydrogen bonding and boiling point.
Figure 12.13 Dipole moment and boiling point.
Problem: Compare the two substances base on the factors listed. Put X to the appropriate box. X X X X X Viscosity Boiling Point Vapor pressure Rate of evaporation IMFA Water, H 2 O Alcohols,CH 3 OH Factors
IMF’s are strong enough to hold the molecules close together and virtually lock them in place
solids retain their shape and have fixed volume
solids are incompressible
diffusion within a solid is extremely slow
solids do not flow
Table 12.5 Characteristics of the Major Types of Crystalline Solids Particles Interparticle Forces Physical Behavior Examples (mp, 0 C) Atomic Molecular Ionic Metallic Network Group 8A(18) [Ne-249 to Rn-71] Molecules Positive & negative ions Atoms Atoms Soft, very low mp, poor thermal & electrical conductors Dispersion Atoms Dispersion, dipole-dipole, H bonds Fairly soft, low to moderate mp, poor thermal & electrical conductors Nonpolar - O 2 [-219], C 4 H 10 [-138], Cl 2 [-101], C 6 H 14 [-95] Polar - SO 2 [-73], CHCl 3 [-64], HNO 3 [-42], H 2 O[0.0] Covalent bond Metallic bond Ion-ion attraction Very hard, very high mp, usually poor thermal and electrical conductors Soft to hard, low to very high mp, excellent thermal and electrical conductors, malleable and ductile Hard & brittle, high mp, good thermal & electrical conductors when molten NaCl  CaF 2  MgO  Na [97.8] Zn  Fe 
Bonding in Solids
Types of Solids Based on Bonding
1. molecular solids
consist of atoms/molecules held together by IMF
soft; low melting point
e.g. Ar, H2O, CH4, C12H22O11, CO2
consist of atoms held together in large networks or chains by covalent bonds
hard; high melting point
e.g. diamond (mp = 3550oC), quartz (SiO2), graphite (interconnected hexagonal planes; used as lubricants), silicon carbide (SiC), boron nitride (BN)
3. ionic solids
ions held together by ionic solids (electrostatic attraction)
structure depends on size and charge of ions
e.g. NaCl, MgO, ZnS, CaF2, Ca(NO3)2
4. metallic solids
consist of metal ions
e.g. Na, Cr
Structure of Solids
e.g. diamond, quartz, NaCl
no specific melting temperature
have specific melting point
lack well defined shapes
have regular shapes
no orderly structure
ions or molecules are in well-defined arrangement
b. amorphous solid a. crystalline solids
PHASE DIAGRAMS – graphical way to summarize the conditions under which equilibria exist between different states of matter. This allows us to predict the phase of a substance that is stable at any given T and P Features of a phase diagram includes: Vapor-pressure curve: generally as temperature increases, vapor pressure increases. Critical Point: critical temperature and pressure for the gas Normal melting point : melting point at 1 atm Triple point : temperature and pressure at which all three phases are in equilibrium Any temperature and pressure combination not on the curve represents a single phase
*Note: Gases may be liquefied by increasing the pressure at a suitable temperature Critical Temperature – the highest temperature at which a substance can exist as a liquid Critical Pressure – the minimum pressure required for liquefaction at this critical temperature The greater the IMF, the easier it is to liquefy a substance, thus, the higher the critical temperature Critical Point – point at which the liquid and vapor becomes indistinguishable Supercritical Fluid – have high densities like liquids and have low viscosity like gases
Phase Diagrams of H2O and CO2 1. Water In general, an increase in pressure favors the more compact phase of the material; this is usually the solid Water is one of the few substances whose solid form is less dense than the liquid form -> the melting point curve for water slopes to the left The triple point occurs at 0.0098 o C and 4.58 Torr The normal melting point is 0 o C The normal boiling point is 100 o C The critical point is 374 o C and 218 atm CO 2
2. Carbon dioxide The triple point occurs at -56.4oC and 5.11 atm The normal sublimation point is -78.5oC. (At 1 atm, CO2 sublimes, it does not melt) The critical point occurs at 31.1oC and 73 atm Freeze drying: Frozen food is placed in a low pressure (<4.58 Torr) chamber -> the ice sublimes H 2 O