Chapter 3: Phase equilibrium (C1160445)

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1 Gas phasePhase

Phase is a region of material, where all physical and chemical properties of the material are essentially homogeneous/uniform. Although it is commonly thought of as the states of matter, solid, liquid or gas; a compound or mixture may have several solid or liquid phases.

“For example, water, ice and steam are different phases,” Mandy remarked, “and water ice can be found in the hexagonal ice Ih, the cubic ice Ic, the rhombohedral ice II, and so forth.”

Frequently asked questions
What is phase?
Region of material, where all physical and chemical properties of the material are homogeneous.

What's homogeneous?
Where it's uniform, all the same, analogous.

What are the states of matter?
Solid, liquid, gas.

Is phase the same thing as states of matter?
No, because several phases, can have the same state of matter.

Whereas chemical reactions transform a set of chemical compounds into another, physical reactions do not create new compounds. An example of a physical reaction is phase transition, which is the change from one phase to another. These include:

  • Melting, which is conversion from solid to liquid
  • Sublimation, which is conversion from solid to gas
  • Freezing, which is conversion from liquid to solid
  • Boiling, evaporation, or condensation, which is conversion from liquid to gas
  • Deposition, which is conversion from gas to solid
  • Condensation, which is conversion of gas to liquid

Frequently asked questions
What are chemical reactions?
Where chemical compounds transform into another.

What are physical reactions?
Where new compounds are not created.

What is a phase transition?
Where one phase changes into another phase.

Is a phase transition a chemical or physical reaction?
Physical reaction. That's because they're the same compound.

Intermolecular forces

Phase transition is dependent on the ability to strengthen or weaken intermolecular forces. Intermolecular forces (aka secondary forces) are forces between molecules. Intermolecular forces are electrostatic, but far weaker than intermolecular forces. They are caused by dipoles. Dipole is the separation of (equal but) positive and negative charge across a molecule. Dipole moment is a vector pointing from positive to negative charge, and has the magnitude [mathjax]p=q.d[/mathjax], where [mathjax]d[/mathjax] is the distance between the charges, and [mathjax]q[/mathjax] is the charge of either [mathjax]-q[/mathjax] or [mathjax]+q[/mathjax], but not both. The various intermolecular forces, also known as van der Waals forces, include:

  • London dispersion forces, which are caused by a temporary dipole, which induces dipoles in neighboring molecules. These forces are unique as they apply in nonpolar molecules. Increased molar mass, increases the numbers of electrons per molecule, and hence delta positive and negative charges, which can induce dipoles
  • Dipole-dipole forces, which are permanent dipoles due to alignment of polar molecules
    • Hydrogen bond, an extreme type of dipole-dipole force. It occurs when a hydrogen covalently bonded with an oxygen, nitrogen or fluorine, is in turn dipole-dipole bonded with another oxygen, nitrogen or fluorine. As hydrogen only has a single electron, and its electron spends time around the very electronegative oxygen, nitrogen or fluorine, its positive nucleus is exposed, and is more strongly positive. In hydrogen bonding, the oxygen, nitrogen or fluorine [of another molecule] attracts to the hydrogen. For example, water has a molar mass of [mathjax]18g/mol[/mathjax], which is analogous with methane which has a molar mass of [mathjax]16g/mol[/mathjax]. However, water vaporizes at [mathjax]100^{\circ} C[/mathjax], far higher than methane which vaporizes at [mathjax]-160^{\circ} C[/mathjax]. This is due the strong hydrogen bond capable on the two hydrogen’s attached to very electronegative oxygen
  • Ion-dipole forces, which are permanent dipoles between ions and polar molecules
Kinetic theory

Kinetic [molecular] theory is that gases are made up of a large number of particles, which are in constant, random motion.

The ideal gas law is [mathjax]PV=nRT[/mathjax], where [mathjax]P[/mathjax] is the pressure of the gas (in Pascals), [mathjax]V[/mathjax] is the volume of gas (in cubic meters), [mathjax]n[/mathjax] is the number of moles of gas, [mathjax]R[/mathjax] is the ideal gas constant [mathjax]8.314 J/K.mol[/mathjax] (this provides volume in [mathjax]m^3[/mathjax], so also keep in mind [mathjax]1m^3 = 1,000L[/mathjax]) if using Pascals [or [mathjax]0.08206 L.atm/K.mol[/mathjax] if using atm; this will provide volume in L], and T is the temperature (in Kelvins). Embedded within the ideal gas law is Charles’ law which is that gases expand when heated ([mathjax]V\propto T[/mathjax]), Boyle’s law which is that pressure and volume are inversely proportional ([mathjax]P\propto \dfrac{1}{V}[/mathjax]) (this thus means that ceteris paribus, the product of [mathjax]PV[/mathjax] is constant), and Avogadro’s law which is that ceteris paribus (for example, at STP), all gases with the same volume contain the same number of molecules ([mathjax]V\propto n[/mathjax]). Standard temperature and pressure (STP) is a temperature of [mathjax]0^{\circ}C[/mathjax] and pressure of [mathjax]1 atm[/mathjax]. Keep in mind [mathjax]0^{\circ}C=273.15K[/mathjax], and [mathjax]1[/mathjax]. At STP, [mathjax]1 mol[/mathjax] of substance has the volume [mathjax]V=\dfrac{nRT}{P}=\dfrac{1\times 0.08206\times 273.15}{1}=22.4L[/mathjax], known as the standard molar volume.

The ideal gas law makes four assumptions, including:

  • No volume, which is an assumption that the total volume of the individual gas molecules are negligible compared to the volume of the gas itself. This approximation is made because the distance separating the gas particles is large compared to the size of the gas particles
  • No attractive force, which is an assumption there are not even weak London dispersion forces. In reality, because gases are nonpolar, they do have London dispersion forces, but these are very weak and thus essentially negligible attractive forces
  • Perfectly elastic collisions, which is an assumption that gas particles are perfectly spherical in shape, and elastically collide with each other and the walls of the container. Remember from , that elastic means no mechanical energy is lost, such that kinetic energy before and after the collision are the same. In reality, gases rotate and vibrate, which can alter kinetic energy
  • Dependence of kinetic energy only on temperature

Dalton’s law is that the total pressure exerted by a mixture of gases (which are assumed not to react with each other), is equal to the sum of the partial pressures of individual gases. Partial pressure is the pressure a gas would have if it alone occupied the volume of the mixture. Essentially, it is the concentration of a gas, in a mixture of gases. This is possible because of the assumptions made by the ideal gas law. An alternative formula is [mathjax]p_{i}=P.y_{i}[/mathjax], where [mathjax]p_{i}[/mathjax] is the pressure of a particle gas, [mathjax]P[/mathjax] is the total pressure, and [mathjax]y[/mathjax] is the mole fraction of a particular gas. Mole fraction is the total amount of a constituent divided by the total amount of all constituents, or [mathjax]y_{i}=\dfrac{n_{i}}{n_{total}}[/mathjax].

The principle that kinetic energy depends only on temperature (for an ideal gas), is true for any fluid system. Remember that based on the kinetic molecular theory, this kinetic energy depends on random translational motion, not uniform translational motion of a fluid (also discussed ). Also, given the random translational speed of oxygen molecules is approximately 1,800kmph, the effects of uniform translational motion are negligible anyway. Average kinetic energy in the kinetic theory can be determined by [mathjax]KE=\dfrac{3}{2}RT[/mathjax], where [mathjax]R[/mathjax] is the ideal gas constant, and [mathjax]T[/mathjax] is temperature in Kelvin. Note therefore, that the only variable in determining [average] kinetic energy is temperature, meaning all molecules under the same temperature have the same [average] kinetic energy. However, given [mathjax]KE=\dfrac{1}{2}mv^2[/mathjax], and kinetic energy is constant for molecules under the same temperature, if temperature is held constant, mass and velocity [squared] are inversely proportional. Therefore, heavier molecules [on average] will travel more slowly. For example, nitrogen has a standard atomic weight of 14, and oxygen has a standard atomic weight of 16. Thus, as oxygen is heavier, oxygen [on average] travels more slowly. Graham’s law states that the relationship between the velocities of two gases is [mathjax]\dfrac{v_{1}}{v_{2}}=\sqrt{\dfrac{M_{2}}{M_{1}}}[/mathjax]. It can be derived by equating kinetic energies since they are the same under constant temperature, meaning [mathjax]\dfrac{1}{2}M_{1}v_{1}^2=\dfrac{1}{2}M_{2}v_{2}^2[/mathjax], or reducing, [mathjax]M_{1}v_{1}^2=M_{2}v_{2}^2[/mathjax]. Reshuffling, [mathjax]\dfrac{v_{1}^2}{v_{2}^2}=\dfrac{M_{2}}{M_{1}}[/mathjax], or [mathjax](\dfrac{v_{1}}{v_{2}})^2=\dfrac{M_{2}}{M_{1}}[/mathjax], meaning [mathjax]\dfrac{v_{1}}{v_{2}}=\sqrt{\dfrac{M_{2}}{M_{1}}}[/mathjax], quod erat demonstrandum.

Effusion is the process where gas flows (as natural, from higher pressure to lower pressure) through a pinhole. Pinhole is an opening much smaller than the average distance between gas molecules. Diffusion of gases, is where a gas spreads itself around its container, [and if applicable] through other gases. An example of diffusion is odor of perfume spreading throughout a room. Because both the rate of effusion and diffusion depends directly on the average velocity of its particles, Graham’s law can be rewritten as [mathjax]\dfrac{rate_{effusion of gas 1}}{rate_{effusion of gas 2}}=\sqrt{\dfrac{M_{2}}{M_{1}}}[/mathjax], and [mathjax]\dfrac{rate_{diffusion of gas 1}}{rate_{diffusion of gas 2}}=\sqrt{\dfrac{M_{2}}{M_{1}}}[/mathjax]. Analogous to the relationship between mass and velocity, lighter particles diffuse and effuse more quickly, and vice versa.

Real gases are distinct to ideal gases, due to deviations in pressure and volume. Real gas molecules do exert attractive forces on each other, and therefore decreases the impact of molecules against the wall, than otherwise, thus decreasing pressure. Real gas thus deviates from ideal gas by having less pressure than expected. Also, real gas molecules have volume. Real gas thus deviates from ideal gas by having greater volume than expected. When molecules are closer to each other, London dispersion forces increase, thereby exacerbating the reduction of pressure even more. Additionally, as the distance between molecules decreases, the volume assumptions become weaker, thereby increasing volume. Increased molecule proximity is caused by high pressure, and low temperature. Therefore, more proximate molecules, and therefore higher pressure and lower temperature, exacerbates deviation from ideal behavior.

“The Epicurean paradox: Premise (1) God and evil cannot coexist. Premise (2) Evil exists. Conclusion (3) Thus, God cannot exist,” Em asked, “how do we respond to the criticism that in an ideal world, where God is all good and all powerful, there should be no evil? In reality, we observe evil, and at that, seemingly needless evil!”

“The fallacy lies in the 1st premise,” Mandy replied, “The atheist has to argue they not only emotionally dislike God and evil coexisting, but that it is logically impossible. C.S. Lewis says to those who believe God and suffering are incompatible, ‘have they never been to a dentist?’”

“Also, have they never heard of S&M ?” Mandy giggled.

“You naughty girl!!” Blaire replied, “Totally something we’d expect coming from your mouth, Mandy!”

“Because of our limitation of perspective, a God who providentially orders history through free decisions of humans is simply, incomprehensible,” Blaire adds, “akin to Chaos Theory explaining how a butterfly fluttering can cause a hurricane.”

“God has given humans free will, which must severely restrict Him,” Mandy replied, “there’s nothing to say this isn’t the best of all possible worlds.”

"Also, the Christian purpose of life is not happiness, but knowledge of God, which is happiness not only here on earth, but into eternity,” Mandy continued, “in the context of eternity, living 60 instead of 100 years may be analogous to saying ‘oh dang, just then, I blinked for one second rather than two’. It’s just as silly!”

“By looking at the evil in the world; and the bitterly cruel and shameful suffering Jesus endured on the cross so that we could overcome sin and death,” Mandy continued, “it becomes evident the true problem of evil is our evil, and not how God can justify Himself to us, but paradoxically, how we can justify ourselves to Him.”

    Formative learning activityMaps to RK3.A
    What is gas phase?

    2 Intramolecular forces

    In contrast, intramolecular forces (aka primary forces) are forces within a molecule. This can be memorized with the mnemonic, that interschool competitions are competitions between schools. The different intramolecular forces include ionic, covalent and metallic.

    Solids can either be crystalline, which have a regular pattern of repeating units; or amorphous, which have no long-range regular pattern. Crystals can have repeating units of atoms, molecules (e.g. ice made of [mathjax]H_{2}O[/mathjax]) or ions (e.g. salt made of [mathjax]NaCl[/mathjax]). Crystals can be further divided into:

    • Ionic
    • Network covalent, which is the use of covalent bonds in a continuous network. There are no individual molecules. Examples include diamond, which is a continuous network of carbon atoms
    • Covalent
    • Metallic, which has a lattice of positively charged ions, with a sea of free electrons. The freedom provided to electrons, also provides metal atoms the ability to break and remake bonds, and hence provide malleability and ductility
    • Molecular

    Whereas crystals have a sharp melting point and a characteristic shape, amorphous solids don’t have a sharp melting point nor a characteristic shape. An example of an amorphous solid is glass. Polymers are compounds made of repeating structural units, created from the reaction of monomer molecules to form polymer chains, known as polymerization. Examples of polymers include DNA and protein. When melted quickly, an amorphous solid; and when melted slowly, a crystalline solid, will result. This is because there is insufficient time for quick melting to permit structural creation.

    Formative learning activityMaps to RK3.B
    What are intermolecular forces?

    3 Phase equilibrium

    Heat capacity is the amount of energy [or the former definition, heat] required to change the temperature of a substance [in a given phase], by [mathjax]1^{\circ}C[/mathjax]. Despite the name persists, it is outdated, and based on the incorrect caloric theory, which holds that heat was an invisible fluid called caloric, which would flow from hotter to colder bodies.

    “Why doesn’t everybody know God?” Emily asked.

    “Although Christians have classically thought of God has omnipresent, His manifestation definitely isn't everywhere,” Mandy replied, “Even in the Garden of Eden, God could be encountered only in certain locations (Genesis 3:9). Isaiah 45:15 says, ‘Truly you are a God who has been hiding himself.’ Following resurrection, Jesus veiled himself so His disciples failed to recognize Him until prompted (Luke 24:13-35).”

    “God hides Himself so (1) only those who want to know Him do, and others aren't coerced to know Him by the vividness of God's presence; (2) so that we can live life independent of Him if we want to, and can choose wrong if we like; and (3) so we have the opportunity to seek Him with all our hearts. Evidently, you can't seek something that isn't hidden to some degree. Proverbs 25:2 says 'It is the glory of God to conceal a matter; to search out a matter is the glory of kings'.”

    Since heat was a material substance, it could neither be created nor destroyed, affirming the conservation of heat. It is more accurate to replace the word “heat” with “internal energy” (discussed ). Energy increases temperature, because, [as it will be defined ,] increasing kinetic energy increases temperature. Specifically, for an ideal gas, the average kinetic energy of gas molecules can be determined by, [mathjax]KE=\dfrac{3}{2}RT[/mathjax]. If the gas is held at constant pressure, given the ideal gas law [mathjax]PV=nRT[/mathjax], specifically, Charles’ law, the volume of the gas must increase (expand) when heated. Given that work done is [mathjax]W=F\cdot d[/mathjax] (from ), where [mathjax]F[/mathjax] is the force against the container wall, and [mathjax]d[/mathjax] is the distance the walls move. Because work is the transfer of energy (from ), energy is transferred to the surrounding [walls], thereby slowing its increase in temperature. However, if gas is held at constant volume, no PV work can be done on the surroundings, and it is all energy is transferred to temperature increase. Therefore, gases at constant pressure have higher heat capacities, than at constant volume, because constant pressure can include heat energy that conducts PV work. This is true also for solid and liquids, because if constant pressure permits expansion [i.e. change in volume], the intramolecular forces are increased, and thus electrostatic potential energy is increased. Because some energy becomes potential energy, less energy is kinetic energy, thereby reducing temperature increase. Heat capacity is defined as [mathjax]C=\dfrac{C}{\Delta T}[/mathjax], where [mathjax]C[/mathjax] is heat capacity, [mathjax]Q[/mathjax] is amount of heat, and [mathjax]\Delta T[/mathjax] is change in temperature, measured in the units [mathjax]J/K[/mathjax], although it only works if the energy transferred is only heat.

    The mechanical equivalent of heat can be found by Joules’ apparatus, which converted the work of a falling weight, into heat increase of the water. Rather than using the formula [mathjax]C=\dfrac{C}{\Delta T}[/mathjax], Joule replaced [mathjax]Q[/mathjax] with work done [mathjax]WD=mgh[/mathjax]. The heat was contained so it could not escape into its environment, known as an adiabatic process.

    Specific heat capacity is a variant of heat capacity, which is the heat capacity per unit mass of substance [in a given phase], [mathjax]Q=m.c.\Delta T[/mathjax], where [mathjax]m[/mathjax] is mass of substance, and [mathjax]c[/mathjax] is specific heat of substance, measured in the units [mathjax]J/kg/K[/mathjax]. Other variants include molar heat capacity, which is heat capacity per mole of substance, measured in the units [mathjax]J/mol/K[/mathjax]; and volumetric heat capacity, which is heat capacity per volume of substance, measured in the units [mathjax]J/L/K[/mathjax]. The specific heat of water is [mathjax]4.1855 J/g/K[/mathjax], but can be defined as [mathjax]1 cal/g/^{\circ}C[/mathjax]. The specific heat of ice and steam is approximately [mathjax]0.5 cal/g/^{\circ}C[/mathjax], therefore noting that the specific heat are specific for each phase. The specific heat also advises, that water requires more energy to cause a change in temperature, than steam and ice.

    Calorimeter can be used to measure the heat of chemical reactions. The common types of calorimeters include:

    • Constant-pressure calorimeter, which measures enthalpy [or heat] of solution (discussed ) at constant pressure. An example is the coffee-cup calorimeter, which consists of two nested Styrofoam cups. The temperature change before and after the reaction is measured. Specific heat capacity can be determined by [mathjax]Q=m.c.\Delta T[/mathjax], and at constant pressure, the heat of reaction [mathjax]Q=\Delta H[/mathjax] (discussed )
    • Constant-volume calorimeter, which measures heat of solution at constant volume. It measures the heat of combustion, of a particular reaction, in a rigid container, and measure the temperature change in water surrounding the container. The heat of the reaction [mathjax]Q=C.\Delta T[/mathjax]

    [Stepwise] heating curves plot temperature [on the y-axis] against time [on the x-axis], assuming that over time, heat is added. The typical segments include:

    • Line with positive slope, ending at the melting point of the substance. Along this line, the substance is solid. As heat is added, it increases the kinetic energy of the molecules, thereby increasing its temperature. Specific heat capacity can be determined by [mathjax]Q=m.c.\Delta T[/mathjax] along this line, where  is the specific heat of the solid
    • Line that is horizontal, along the melting point. Heat does not increase kinetic energy, so temperature remains constant. This confirms that heat and temperature are not analogous (discussed ). The additional energy goes into breaking the intermolecular bonds, so that a liquid can be formed. Since temperature does not change along this line, [mathjax]Q=m.c.\Delta T[/mathjax] cannot be used to find specific heat capacity. Rather, the enthalpy of fusion (aka specific latent heat) can be used, which is often provided in kJ/kg, where [mathjax]Q=mL[/mathjax], where [mathjax]m[/mathjax] is the mass of substance, and [mathjax]L[/mathjax] is the specific latent heat for a particular substance. The term “latent” means “lie hidden”, and was used because of the incorrect caloric theory of heat (discussed )
    • Line with positive slope, extending from the melting point to the boiling point. Along this line, the substance is liquid. As heat is added, energy contributes into increasing kinetic energy, so the temperature of the liquid increases. As , specific heat capacity can be determined by [mathjax]Q=m.c.\Delta T[/mathjax] along this line, where [mathjax]c[/mathjax] is the specific heat of the liquid. Note therefore, that the slope of this segment is different to the slope of the first segment
    • Line that is horizontal, along the boiling point. This line represents heat of vaporization, as the intermolecular bonds of the liquid are broken to form gas
    • Line with positive slope, extending from the boiling point. Along this line, the substance is gas. S heat is added, energy contributes into increasing kinetic energy, so the temperature of the gas increases

    Where there are horizontal lines, note that as the intermolecular bonds are weakened, there is work done, so there is increase in entropy ([mathjax]\Delta S[/mathjax] , discussed ). Also, enthalpy ([mathjax]H[/mathjax]) is positive, because it is an endothermic reaction, and heat is absorbed in order to break the bonds. Since Gibbs free energy [mathjax]\Delta G=H-T\Delta S[/mathjax] (all of which will be discussed ), Gibbs free energy can either be positive or negative, meaning the reaction can be either spontaneous ([mathjax]\Delta G[/mathjax] is negative) or non-spontaneous ([mathjax]\Delta G[/mathjax] is positive). By the Gibbs free energy formula, it is also evident that temperature will play a deterministic role, as expected in phase changes.

    Where substances directly transition from solid to gas, there is heat of sublimation. Where substances transition from one solid to another solid, there is heat of transition.

    Phase diagrams plot pressure [on y-axis] against temperature [on x-axis]. The diagram generally has three areas: (1) solid; (2) liquid; and (3) gas. Solids can be found up and to the left, as they exist at higher pressure and low temperature [respectively]. Liquids are to the right of solids, as when temperature increases, solids melt. Gases are found bottom and to the right, as they exist at low pressure and high temperatures [respectively]. The lines marking out the phase boundaries, mark conditions at which multiple phases can coexist at equilibrium. The point at which all three lines meet, is uniquely where all three phases coexist, and is known as the triple point. For example, in water, at 1 atm (normal atmospheric levels), it can exist as a solid (ice), liquid (water) or gas (steam), depending on the temperature. Therefore, a horizontal line must pass through all three phases. Therefore, the triple point of water must be below 1 atm. The point at which the phase boundary of liquid and gas ends, is the critical point. Above the critical point, liquids and gases are indistinguishable, known as supercritical fluid.

    © 2014 MR. SHUM

    Colligative properties are properties [of solution] that depend on number, and not kind of chemical species present. The four important colligative properties are:

    • Vapor pressure, which is decreased by addition of a non-volatile solute to the solution
    • Boiling point, which is increased by addition of a non-volatile solute to the solution. A liquid boils [into a gas] when its vapor pressure equals the surrounding atmospheric pressure. Since a non-volatile solute will lower the vapor pressure (just ) of the liquid, it will take longer for the vapor pressure to reach the surrounding atmospheric pressure, thereby increasing boiling point. The elevation of the boiling point is [mathjax]\Delta T=K.b.i[/mathjax], where [mathjax]K[/mathjax] is the boiling point elevation constant, the ebullioscopic constant, of the [volatile] solvent, [mathjax]b[/mathjax] is the molality of the solution, and [mathjax]i[/mathjax] is the van ‘t Hoff factor. The van ‘t Hoff factor is the ratio of the concentration of particles when dissolved, and the concentration before it is dissolved, thereby providing information about how many pieces each solute molecule will break into when dissolved. For example, the theoretical van ‘t Hoff factor of NaCl is 2 pieces ([mathjax]Na^+[/mathjax] and [mathjax]Cl^-[/mathjax]), and for glucose [mathjax]C_{6}H_{12}O_{6}[/mathjax] is 1 piece only. It is theoretical, because, for example, the actual van ‘t Hoff factor of NaCl is 1.9, because some of the broken ions are still engaged in ionic bonding
    • Freezing point, which is decreased by addition of a non-volatile solute to the solution. A liquid freezes [into a solid] by forming crystalline structures. A non-volatile solute is therefore an impurity, which ruins the crystalline structure. Therefore, the freezing point has to be even lower, before capability to achieve freeze. The depression of the freezing point is calculated analogously with boiling point elevation [mathjax]\Delta T=K.b.i[/mathjax], except the constant [mathjax]K[/mathjax] is the freezing point depression constant, the cryoscopic constant.
    • Osmotic pressure, which is the pressure created by solutes in solution. Whereas hydrostatic pressure essentially pushes outward, osmotic pressure essentially pulls inward. The problem with this definition is that pressure is a scalar [and not a vector], and so shouldn’t have direction. What actually occurs is where there is a semipermeable membrane, such as a blood vessel, osmosis would like to distribute the concentration of solute. However, the semipermeable membrane prevents the solute from moving to the area of less solute. Rather, the solvent will move through the semipermeable membrane, to where there is more solute. Because there is more solvent on one side, there is increased pressure there. The solute molecules have therefore essentially created [an osmotic] pressure, which can use the ideal gas law, [mathjax]PV=nRT[/mathjax] (discussed ), except to replace [mathjax]\Pi[/mathjax] [instead of [mathjax]P[/mathjax]] for osmotic pressure, to obtain, reshuffling, [mathjax]\Pi=\dfrac{nRT}{V}[/mathjax]. Since [mathjax]c=\dfrac{n}{V}[/mathjax], this can be reduced further to [mathjax]\Pi=cRT[/mathjax]. However, since osmotic pressure is a colligative property, the van ’t Hoff factor ([mathjax]i[/mathjax]) is required, therefore, [mathjax]\Pi=cRTi[/mathjax]

    Formative learning activityMaps to RK3.C
    What is phase equilibria?




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    Chapter 3: Phase equilibrium - General chemistry - Pre-med science - MR. SHUM'S CLASSROOM