10 Gases
CHAPTER OBJECTIVES
• To be able to describe the characteristics of a gas
• To understand the relationships between the pressure, temperature, amount, and volume of a gas
• To be able to use the ideal gas law to describe the behavior of a gas
• To know how to determine the contribution of each
component gas to the total pressure of a gaseous mixture
• To understand the significance of the kinetic molecular theory of gases
• To recognize the differences between the behaviors of an
Chemistry: Principles, Patterns, and Applications, 1e
10.1 Gaseous Elements and
Compounds
10.1 Gaseous Elements and Compounds
• The three common phases (or states) of matter are gas, liquid, and solid
1. Gases
a. Have the lowest density of the three states of matter b. Are highly compressible
c. Completely fill any container in which they are placed d. Their intermolecular forces are weak
e. Molecules are constantly moving independently of the other molecules present
2. Solids
a. Dense b. Rigid
c. Incompressible
d. Intermolecular forces are strong e. Molecules locked in place
10.1 Gaseous Elements and Compounds
3. Liquids
a. Dense
b. Incompressible
c. Flow readily to adapt to the shape of the container
d. Sum of the intermolecular forces are between those of gases and solids
• The state of a given substance depends strongly on conditions
10.1 Gaseous Elements and Compounds
• Geometric structure and the physical and chemical properties of atoms, ions, and molecules do not depend on the physical state
• Macroscopic properties of a substance depend strongly on its physical state, which is determined by intermolecular forces and conditions such as temperature and pressure
• The figure of the periodic table shows the locations in the periodic table of those elements that are commonly found in the gaseous, liquid, and solid states
• Elements that occur as gases are on the right side of the periodic table except for hydrogen
• All the noble gases (Group 18) are monatomic gases
10.1 Gaseous Elements and Compounds
• Many of the elements and compounds are typically found as gases
• Gaseous substances include:
1. Many binary hydrides, such as the hydrogen halides 2. Hydrides of the chalcogens
3. Hydrides of the Group-15 elements N, P, and As 4. Hydrides of the Group-14 elements C, Si, and Ge 5. Diborane
6. Many of the simple covalent oxides of the nonmetals such as CO, CO2, NO, NO2, SO2, SO3, and ClO2
7. Many low-molecular-mass organic compounds 8. Most of the commonly used refrigerants
10.1 Gaseous Elements and Compounds
• All of these gaseous substances (other than the monatomic noble gases) contain covalent or polar covalent bonds and are nonpolar or slightly polar molecules.
• The lightest members of any given family of compounds are most likely to be gases.
• Boiling points of polar compounds are higher than those of nonpolar compounds of similar molecular mass.
• In a given series of compounds, the lightest and
least-polar members are the ones most likely to be
gases.
Chemistry: Principles, Patterns, and Applications, 1e
10.2 Gas Pressure
10.2 Gas Pressure
• At the macroscopic level, a complete physical description of a sample of a gas requires four quantities:
1. Temperature (expressed in K) 2. Volume (expressed in liters) 3. Amount (expressed in moles) 4. Pressure (given in atmospheres)
• These variables are not independent — if the values of any three of these quantities are known, the fourth can be
calculated.
• Any object exerts a force on any surface with which it comes in contact.
• Pressure is dependent on both the force exerted and the size of the area to which the force is applied:
P = F
A
• Units of pressure are derived from the units used to measure force and area
1. In the English system, the units of force are pounds and the units of area are square inches, so pressure is expressed in pounds per square inch, lb/in2 (or psi).
2. For scientific measurements, the S unit for pressure, derived from the S
units for force (newtons) and area (square meters), is the newton per square meter, N/m2, called the paschal (Pa):
Units of Pressure
• Every point on Earth’s surface experiences a net pressure called atmospheric pressure.
• Pressure exerted by the atmosphere is considerable.
• A 1.0-m
2column, measured from sea level to the top of the atmosphere, has a mass of about 10,000 kg, which gives a pressure of 100 kPa:
pressure = (1.0 x 10
4kg) (9.807 m/s
2) = 1.0 x10
5Pa = 100 kPa
1.0 m
2• In English units, this is 15 lb/in
2.Atmospheric Pressure
• Atmospheric pressure can be measured using a
barometer, a closed, inverted tube filled with mercury.
• The height of the mercury column is proportional to the atmospheric pressure, which is reported in units of
millimeters of mercury (mmHg), also called torr.
Atmospheric Pressure
• Standard atmospheric pressure is the atmospheric pressure required to support a column of mercury
exactly 760 mm tall; this pressure is also referred to as 1 atmosphere (atm).
• A pressure of 1 atm equals 760 mmHg exactly and is approximately equal to 100 kPa:
1 atm = 760 mmHg = 760 torr = 1.01325 x 10
5Pa = 101.325 kPa
Atmospheric Pressure
• Manometers measure the pressures of samples of gases contained in an apparatus.
• A key feature of a manometer is a U-shaped tube containing mercury.
• In a closed-end manometer, the space above the mercury column on the left (the reference arm) is a vacuum (P 0), and the difference in the heights of the two columns gives the pressure of the gas contained in the bulb directly.
• In an open-end manometer, the left (reference) arm is open to the
atmosphere here (P = 1 atm), and the difference in the heights of the two columns gives the difference between atmospheric pressure and the
pressure of the gas in the bulb.
Manometers
Chemistry: Principles, Patterns, and Applications, 1e
10.3 Relationships Between Pressure,
Temperature, Amount,
and Volume
• As the pressure on a gas increases, the volume of the gas
decreases because the gas particles are forced closer together.
• As the pressure on a gas decreases, the gas volume increases because the gas particles can now move farther apart.
• Boyle carried out some experiments that determined the quantitative relationship between the pressure and volume of a gas.
• Plots of Boyle’s data showed that a simple plot of V versus P is a hyperbola and reveals an inverse relationship between pressure and volume; as the pressure is doubled, the volume decreases by a factor of two.
The Relationship between Pressure
and Volume
• Relationship between the two quantities is described by the equation PV = constant.
• Dividing both sides by P gives an equation that illustrates the inverse relationship between P and V:
V = constant = constant(1/P) or V 1/P P
• A plot of V versus 1/P is a straight line whose slope is equal to the constant.
• Numerical value of the constant depends on the amount of gas used in the experiment and on the temperature at which the experiments are carried out.
• This relationship between pressure and volume is known as Boyle’s law which states that at constant temperature, the volume of a fixed amount of a gas is inversely proportional to its pressure.
The Relationship between Pressure
and Volume
• Hot air rises and gases expand when heated.
• Charles carried out experiments to quantify the relationship between the temperature and volume of a gas and showed that a plot of the volume of a given sample of gas versus temperature (in ºC) at
constant pressure is a straight line.
• Gay-Lussac showed that a plot of V versus T was a straight line that could be extrapolated to –273.15ºC at zero volume, a theoretical state.
• The slope of the plot of V versus T varies for the same gas at
different pressures, but the intercept remains constant at –273.15ºC.
• Plots of V versus T for different amounts of varied gases are straight lines with different slopes but the same intercept on the T axis.
The Relationship between Temperature
and Volume
• Significance of the invariant T intercept in plots of V
versus T was recognized by Thomson (Lord Kelvin), who postulated that –273.15ºC was the lowest possible
temperature that could theoretically be achieved, and he called it absolute zero (0 K).
• Charles’s and Gay-Lussac’s findings can be stated as:
At constant pressure, the volume of a fixed amount of a gas is directly proportional to its absolute temperature (in K).
• This relationship is referred to as Charles’s law and is stated mathematically as
V = (constant) [T (in K)] or V T (in K, at constant P).
The Relationship between Temperature
and Volume
• Avogadro postulated that, at the same temperature and pressure, equal volumes of gases contain the same
number of gaseous particles.
• Avogadro’s law describes the relationship between
volume and amount of gas: At constant temperature and pressure, the volume of a sample of gas is directly
proportional to the number of moles of gas in the sample.
• Stated mathematically:
V = (constant) (n) or V n (at constant T and P)
The Relationship between Amount
and Volume
• The relationships between the volume of a gas and its pressure, temperature, and amount are summarized in the figure below.
• The volume increases with increasing temperature or amount but decreases with increasing pressure.
The Relationship between Amount
and Volume
Chemistry: Principles, Patterns, and Applications, 1e
10.4 The Ideal Gas Law
Deriving the Ideal Gas Law
• Any set of relationships between a single quantity (such as V) and several other variables (P, T, n) can be combined into a single expression that describes all the relationships simultaneously.
• The following three expressions
V 1/P (at constant n, T)V
T ( at constant n, P)
V n (at constant T, P)can be combined to give
V nT or V = constant (nT/P)
• The proportionality constant is called the gas constant, represented by the letter R.
• Inserting R into an equation gives
V = RnT = nRT P PDeriving the Ideal Gas Law
• Multiplying both sides by P gives the following equation, which is known as the ideal gas law:
PV = nRT
• An ideal gas is defined as a hypothetical gaseous substance whose behavior is independent of attractive and repulsive forces and can be completely described by the ideal gas law.
• The form of the gas constant depends on the units used for the other quantities in the expression — if V is expressed in liters (L), P in
atmospheres (atm), T in kelvins (K), and n in moles (mol), then R = 0.082057 (L•atm)/(K•mol).
• R can also have units of J/(K•mol) or cal/(K•mol).
Deriving the Ideal Gas Law
• A particular set of conditions were chosen to use as a reference;
0ºC (273.15 K) and 1 atm pressure are referred to as standard temperature and pressure (STP).
• The volume of 1 mol of an ideal gas under standard conditions can be calculated using the variant of the ideal gas law:
V = nRT = (1 mol) [0.082057 (L•atm)/(K•mol)] (273.15 K) = 22.41 L P 1 atm
• The volume of 1 mol of an ideal gas at 0ºC and 1 atm pressure is 22.41 L, called the standard molar volume of an ideal gas.
• The relationships described as Boyle’s, Charles’s, and Avogadro’s
laws are simply special cases of the ideal gas law in which two of
the four parameters (P, V, T, n) are held fixed.
Applying the Ideal Gas Law
• The ideal gas law allows the calculation of the fourth variable for a gaseous sample if the values of any three of the four variables (P, V, T, n) are known.
• The ideal gas law predicts the final state of a sample of a gas (that is, its final temperature, pressure, volume, and quantity)
following any changes in conditions if the parameters (P, V, T, n) are specified for an initial state.
• In cases where two of the variables P, V, and T are allowed to
vary for a given sample of gas (n is constant) and the change in
the value of the third variable under the new conditions needs to
be calculated, the ideal gas needs to be arranged.
Applying the Ideal Gas Law
• The ideal gas law is rearranged so that P, V, and T, the quantities that change, are on one side and the constant terms (R and n for a given sample of gas) are on the other:
PV = nR = constant T
• The quantity PV/T is constant if the total amount of gas is constant.
• The relationship between any two sets of parameters for a sample of gas can be written as
P1V1 = P2V2.
T1 T2
• An equation can be solved for any of the quantities P2, V2, or T2 if the initial conditions are known.
Using the Ideal Gas Law to Calculate Gas Densities and Molar Masses
• The ideal gas law can be used to calculate molar masses of gases from experimentally measured gas densities.
• Rearrange the ideal gas law to obtain
n = PV RT
• The left side has the units of moles per unit volume, mol/L.
• The number of moles of a substance equals its mass (in grams) divided by its molar mass (M, in grams per mole):
n (in moles) = m (in grams) M (in grams/mole)
• Substituting this expression for n in the preceding equation gives
m = P
Using the Ideal Gas Law to Calculate Gas Densities and Molar Masses
• Because m/V is the density d of a substance, m/V can be replaced by d and the equation rearranged to give
d = PM RT
• The distance between molecules in gases is large compared to the size of the molecules, so their densities are much lower than the densities of liquids and solids.
• Gas density is usually measured in grams per liter (g/L) rather than
grams per milliliter (g/mL).
Chemistry: Principles, Patterns, and Applications, 1e
10.5 Mixtures of Gases
Partial Pressures
• The ideal gas law assumes that all gases behave identically and that their behavior is independent of attractive and repulsive forces.
• If the volume and temperature are held constant, the ideal gas
equation can be arranged to show that the pressure of a sample of gas is directly proportional to the number of moles of gas present:
P = n(RT/V) = n(constant)
• Nothing in the equation depends on the nature of the gas, only on the quantity.
• The total pressure exerted by a mixture of gases at a given
temperature and volume is the sum of the pressures exerted by
each of the gases alone.
Partial Pressures
• If the volume, temperature, and number of moles of each gas in a mixture is known, then the pressure exerted by each gas individually, which is its partial pressure, can be calculated.
• Partial pressure is the pressure the gas would exert if it were the only one present (at the same temperature and volume).
• The total pressure exerted by a mixture of gases is the sum of the partial
pressures of component gases.• This law is known as Dalton’s law of partial pressures and can be written mathematically as
P
t= P
1+ P
2+ P
3 - - - + Piwhere P
tis the total pressure and the other terms are the partial
pressures of the individual gases.
Partial Pressures
• For a mixture of two ideal gases, A and B, the expression for the total pressure can be written as
P
t= P
A+ P
B= n
A(RT/V) + n
B(RT/V) = (n
A+ n
B) (RT/V).
• More generally, for a mixture of i components, the total pressure is given by
P
t= (n
1+ n
2+ n
3+ - - - +n
i) (RT/V).
• The above equation makes it clear that, at constant temperature and volume, the pressure exerted by a gas depends on only the total number of moles of gas present, whether the gas is a single
chemical species or a mixture of gaseous species.
Mole Fractions of Gas Mixtures
• The composition of a gas mixture can be described by the mole fractions of the gases present.
• Mole fraction (
) of any component of a mixture is the ratio of the number of moles of that component to the total number of moles of all the species present in the mixture (nt)mole fraction of A = A = moles A = nA total moles nt
• Mole fraction is a dimensionless quantity between 0 and 1.
• If
A = 1, then the sample is pure A, not a mixture.• If
A = 0, then no A is present in the mixture.• The sum of the mole fractions of all the components present must equal 1.
Mole Fractions of Gas Mixtures
• To see how mole fractions help in understanding the properties of gas mixtures, the ratio of the pressure of a gas A to the total
pressure of the gas mixture that contains A is evaluated.
• Use the ideal gas law to describe the pressures of both gas A and the mixture: P
A= n
ART/V and P
t= n
tRT/V
• The ratio of the two is
PA = nART/V = nA = A Pt ntRT/V nt
• Rearranging the equation gives P
A=
Ap
t.
• The partial pressure of any gas in a mixture is the total pressure
multiplied by the mole fraction of that gas.
Chemistry: Principles, Patterns, and Applications, 1e
10.6 Gas Volumes and
Stoichiometry
10.6 Gas Volumes and Stoichiometry
• The relationship between the amounts of gases (in moles) and their volumes (in liters) in the ideal gas law is used to calculate the stoichiometry of
reactions involving gases, if the pressure and temperature are known.
• Relationship between the amounts of products and reactants in a chemical reaction can be expressed in units of moles or masses of pure substances, of volumes of solutions, or of volumes of gaseous substances.
• The ideal gas law can be used to calculate the volume of gaseous products or reactants as needed.
• In the lab, gases produced in a reaction are collected by the displacement of water from filled vessels — the amount of gas can be calculated from the volume of water displaced and the atmospheric pressure.
Chemistry: Principles, Patterns, and Applications, 1e
10.7 The Kinetic Molecular
Theory of Gases
A Molecular Description
• The kinetic molecular theory of gases explains the laws that describe the behavior of gases and it was
developed during the nineteenth century by Boltzmann, Clausius, and Maxwell
• Kinetic molecular theory of gases provides a molecular explanation for the observations that led to the
development of the ideal gas law
A Molecular Description
• The kinetic molecular theory of gases is based on the following postulates:
1. A gas is composed of a large number of particles called molecules (whether monatomic or polyatomic) that are in constant random motion.
2. Because the distance between gas molecules is much greater than the size of the molecules, the volume of the molecules is negligible.
3. Intermolecular interactions, whether repulsive or attractive, are so weak that they are also negligible.
A Molecular Description
4. Gas molecules collide with one another and with the walls of the container, but collisions are perfectly elastic; they do not change the average kinetic energy of the molecules.
5. The average kinetic energy of the molecules of any gas depends on only the temperature, and at a given temperature, all
gaseous molecules have exactly the same average kinetic energy.
A Molecular Description
• Postulates 1 and 4 state that molecules are in constant motion and collide frequently with the walls of their container and are an
explanation for pressure
1. Anything that increases the frequency with which the molecules strike the walls or increases the momentum of the gas molecules increases the pressure.
2. Anything that decreases that frequency or the momentum of the molecules decreases the pressure.
• Postulates 2 and 3 state that all gaseous particles behave
identically, regardless of the chemical nature of their component molecules — this is the essence of the ideal gas law.
• Postulate 2 explains how to compress a gas — simply decrease the
distance between the gas molecules.
A Molecular Description
• Postulate 5 provides a molecular explanation for the temperature of a gas and states that at a given temperature, all gases have the same average kinetic energy.
• The average kinetic energy of the molecules of a gas is KE = ½ m
2where
2is the average of the squares of the speeds of the particles and m is the mass of the object.
• The square root of
2is the root mean square (rms) speed (
rms)• All gases have the same average kinetic energy at a given temperature but they do not all possess the same root mean square (rms) speed.
• Root mean square speed increases with temperature.
• At any given time, what fraction of the molecules in a particular sample have a given speed; some of the molecules will be moving more slowly than average and some will be moving faster than average.
• Graphs of the number of gas molecules versus speed give curves that show the distributions of speeds of molecules at a given temperature.
• Increasing the temperature has two effects:
1. Peak of the curve moves to the right because the most probable speed increases
2. The curve becomes broader because of the increased spread of the speeds
• Increased temperature increases the value of the most probable speed but decreases the relative number of molecules that have that speed.
• Curves are referred to as Boltzmann distributions.
Boltzmann Distribution
• Pressure versus volume
– At constant temperature, the kinetic energy of the molecules of a gas and the root mean square speed remain unchanged.
– If a given gas sample is allowed to occupy a larger volume, the speed of the molecules doesn’t change, but the density of the gas decreases and the average distance between the molecules increases: they
collide with one another and with the walls of the container less often, leading to a decrease in pressure.
– Increasing the pressure forces the molecules closer together and
increases the density, until the collective impact of the collisions of the molecules with the walls of the container balances the applied
pressure.
The Relationships between Pressure, Volume,
and Temperature
• Volume versus temperature
– Raising the temperature of a gas increases the average kinetic energy and the root mean square speed (and the average speed) of the gas molecules.
– As the temperature increases, the molecules collide with the walls of the container more frequently and with greater force, thereby increasing the pressure unless the volume increases to reduce the pressure
– An increase in temperature must be offset by an increase in volume for the net impact (pressure) of the gas molecules on the container walls to remain unchanged.
•
Pressure of gas mixtures
– If gaseous molecules do not interact, then the presence of one gas in a gas mixture will have no effect on the pressure exerted by another, and Dalton’s law of partial pressures holds.
The Relationships between Pressure, Volume,
and Temperature
• Diffusion is the gradual mixing of gases due to the motion of their component particles even in the absence of mechanical agitation such as stirring.
– Result is a gas mixture with uniform composition
• The rate of diffusion of a gaseous substance is inversely
proportional to the square root of its molar mass (rate 1/ M) and is referred to as Graham’s law.
• The ratio of the diffusion rates of two gases is the square root of the inverse ratio of their molar masses.
– If r is the diffusion rate and M is the molar mass, then
r
1/r
2= M
2/M
1• If M
1 M
2, then gas #1 will diffuse more rapidly than gas #2.
Diffusion
• Effusion is the escape of a gas through a small (usually microscopic) opening into an evacuated space.
• Rates of effusion of gases are inversely proportional to the square root of their molar masses.
• Heavy molecules effuse through a porous material more slowly than light molecules.
Effusion
• Graham’s law states that the ratio of the rates of diffusion or effusion of two gases is the square root of the inverse ratio of their molar
masses.
– Relationship is based on the postulate that all gases at the same temperature have the same average kinetic energy
• The expression for the average kinetic energy of two gases with different molar masses is
KE = ½M
1
2rms1= ½M
2
2rms2.
Multiplying both sides by 2 and rearranging gives
2rms2= M
1.
2rms1M
2
Taking the square root of both sides gives
rms2/
rms1= M
1/M
2.
• Thus the rate at which a molecule diffuses or effuses is directly related to the speed at which it moves.
Rates of Diffusion or Effusion
• Gaseous molecules have a speed of hundreds of meters per second (hundreds of miles per hour).
• The effect of molar mass on these speeds is dramatic.
• Molecules with lower masses have a wider distribution of speeds.
• Postulates of the kinetic molecular theory lead to the following equation, which directly relates molar mass, temperature, and rms speed:
rms = 3RT/M
rms has units of m/s, the units of molar mass M are kg/mol, temperature T is expressed in K, and the ideal gas constant R has the value 8.3144 J/(K•mol).
• The average distance traveled by a molecule between collisions is the mean free path; the denser the gas, the shorter the mean free path.
• As density decreases, the mean free path becomes longer because collisions occur less frequently.
Rates of Diffusion or Effusion
Chemistry: Principles, Patterns, and Applications, 1e
10.8 The Behavior of
Real Gases
10.8 The Behavior of Real Gases
• Postulates of the kinetic molecular theory of gases ignore both the volume occupied by the molecules of a gas and all interactions between molecules, whether attractive or repulsive.
• In reality, all gases have nonzero molecular volumes and
the molecules of real gases interact with one another in
ways that depend on the structure of the molecules and
differ for each gaseous substance.
Pressure, Volume, and Temperature Relationships in Real Gases
• For an ideal gas, a plot of PV/nRT versus P gives a horizontal line with an intercept of 1 on the PV/nRT axis.
• Real gases show significant deviations from the behavior expected for an ideal gas, particularly at high pressures, but at low pressures (less than 1 atm) and at higher temperatures, real gases
approximate ideal gas behavior.
Pressure, Volume, and Temperature Relationships in Real Gases
• Real gases behave differently from ideal gases at high pressures and low temperatures because the basic assumptions behind the ideal gas law — that gas molecules have negative volume and that intermolecular interactions are negligible — are no longer valid.
• Molecules of an ideal gas are assumed to have zero volume; volume available to them for motion is the same as the volume of the
container.
Pressure, Volume, and Temperature Relationships in Real Gases
• Molecules of a real gas have small but measurable volumes.
– At low pressures, gaseous molecules are far apart.
– As pressure increases, intermolecular distances become smaller and smaller and the volume occupied by the molecules
becomes significant compared with the volume of the container.
– Total volume occupied by gas is greater than the volume
predicted by the ideal gas law, and at very high pressures, the experimentally measured value of PV/nRT is greater than the value predicted by the ideal gas law.
Pressure, Volume, and Temperature Relationships in Real Gases
• Molecules are attracted to one another by a combination of forces that are important for gases at low temperatures and high pressures where intermolecular distances are shorter.
– As the intermolecular distances decrease, the pressure exerted by the gas on the container wall decreases and the observed pressure is less than expected.
– At low temperatures, the ratio of PV/nRT is lower than predicted for an ideal gas.
– At high pressures, the effect of nonzero molecular volume predominates.
– At high temperatures, molecules have sufficient kinetic energy to overcome intermolecular attractive forces and the effects of nonzero molecular volume predominates.
The van der Waals Equation
• Van der Waals modified the ideal gas law to describe the
behavior of real gases by including the effects of molecular size and intermolecular forces.
• Van der Waals equation
(P + an2/V2) (V – nb) = nRT
– a and b are empirical constants that differ for each gas
– Pressure term, P + (an2/V2), corrects for intermolecular attractive forces that tend to reduce the pressure from that predicted by the ideal gas law – n2/V2 represents the concentration of the gas (n/V) squared because
it takes two particles to engage in the pairwise intermolecular interactions – Volume term, V – nb, corrects for the volume occupied by the gaseous
molecules
Liquefaction of Gases
• Liquefaction of gases is the condensation of gases into a liquid form, which is neither anticipated nor explained by the kinetic molecular theory of gases.
• Both the theory and the ideal gas law predict that gases compressed to very high pressures and cooled to very low temperatures should still behave like gases.
• However, as gases are compressed and cooled, they condense to form liquids.
Liquefaction of Gases
• Liquefaction — extreme deviation from ideal gas behavior
– Occurs when the molecules of a gas are cooled to the point
where they no longer possess sufficient kinetic energy to overcome the intermolecular attractive forces
– Precise combination of temperature and pressure needed to
liquefy a gas depends on its molar mass and structure, with heavier and more complex molecules liquefying at higher temperatures
– Substances with large van der Waals a coefficients are easy to liquefy because large a coefficients indicate strong intermolecular attractive interactions
– Small molecules that contain light elements have small a coefficients, indicating weak intermolecular attractive interactions, and are difficult to liquefy
