Properties of Substances and Mixtures

A gecko can run up a glass window and hang from the ceiling by one toe, with no glue and no suction. Its toes are covered in millions of tiny hairs that get extremely close to the surface, close enough for weak attractions between molecules to add up to a strong grip. Those same weak attractions decide whether a substance is a gas or a liquid at room temperature, why oil and water do not mix, and how a chromatography strip separates inks. This unit is about them. Intermolecular forces 分子间作用力 (IMFs) are attractions between molecules – much weaker than bonds, but they set melting/boiling points. The main types are listed below; their relative strengths depend on the particles and conditions:

• London dispersion forces 伦敦色散力: present in all molecules; they arise from Coulombic attraction between temporary, fluctuating dipoles, and are stronger for larger, more polarizable electron clouds - often the strongest net force between large molecules.
• Dipole–dipole 偶极-偶极: between polar molecules. Their strength depends on the size of the dipoles and their relative orientation - a $\delta+$ end lining up with a neighbour's $\delta-$ end attracts, so these act in addition to dispersion and make polar molecules stickier than nonpolar ones of similar size.
• Hydrogen bonding 氢键: a strong dipole force when H is bonded to N, O, or F.
• Ion–dipole 离子-偶极: between an ion and a polar molecule (the ion pulls on the oppositely-charged end of the dipole). Their strength depends on ion charge, ion size and the molecular dipole. They help stabilize dissolved ions, although not every ionic solid dissolves readily in water - each $\text{Na}^+$ is surrounded by the $\delta-$ oxygen ends of water molecules.

The whole ladder is understood qualitatively by looking at the sign of the partial charges: a larger or better-aligned partial charge, or a full ionic charge, gives a stronger attraction. Stronger IMFs mean higher boiling points and lower vapor pressure. This is why water ($18\ \text{g/mol}$, hydrogen-bonded) boils at $100\,{}^{\circ}\text{C}$ while methane ($16\ \text{g/mol}$, dispersion only) boils at $-162\,{}^{\circ}\text{C}$.

Dispersion forces grow with the number of electrons, because a larger electron cloud is more polarizable 易极化: it is easier to distort into a temporary dipole. The halogens show this clearly. $\text{F}_2$ boils at −188 °C, $\text{Cl}_2$ at −34 °C and $\text{Br}_2$ at 59 °C, and $\text{I}_2$ is a solid at room temperature. Shape matters too, because a larger contact area gives more attraction. Pentane, a long chain, boils at 36 °C; its isomer neopentane, with the same formula $\text{C}_5\text{H}_{12}$ but a compact ball shape, boils at 9.5 °C. Pi bonds also make electron clouds more polarizable.

A polar molecule can also induce a dipole in a nonpolar neighbour (a dipole–induced dipole force). This weak attraction is why a little oxygen dissolves in water, enough for fish to breathe.

Worked example (rank boiling points). Rank $\text{CH}_3\text{CH}_2\text{OH}$ (ethanol, 46 g/mol), $\text{CH}_3\text{OCH}_3$ (dimethyl ether, 46 g/mol) and $\text{CH}_3\text{CH}_2\text{CH}_3$ (propane, 44 g/mol). All three have similar numbers of electrons, so their dispersion forces are similar. Propane is nonpolar: dispersion only, lowest boiling point (−42 °C). Dimethyl ether is polar but has no H on its O: dipole–dipole added (−24 °C). Ethanol has an O–H, so it hydrogen-bonds: highest (78 °C). When molecules are similar in size, compare the extra forces.

These forces also work inside very large molecules. The two strands of DNA are held together by hydrogen bonds between bases, and a protein folds into its working shape through hydrogen bonds, dipole forces and dispersion forces between different parts of the same long chain.

A solid's properties reflect the particles and forces holding it: ionic, covalent-network (a 3D network like diamond, very hard and high-melting; graphite is a layered exception — high-melting but soft, because its 2D layers slide over one another), metallic, and molecular solids (held by weak IMFs, soft, low-melting). Matching a solid's properties to its structure is a common exam task.

Solid	Particles	Held by	Melting point	Conducts electricity?	Example
ionic	cations and anions	ionic attraction	high	only when melted or dissolved	NaCl
covalent network	atoms	covalent bonds throughout	very high	no (graphite: yes)	diamond, $\text{SiO}_2$
molecular	molecules	intermolecular forces	low	no	ice, sugar, $\text{I}_2$
metallic	cations in an electron sea	metallic bonding	usually high	yes, as a solid	copper

Vaporizing a molecular substance separates its molecules completely, so its boiling point is a direct measure of the forces between them. Strong forces mean a high boiling point and a low vapor pressure 蒸气压 (few molecules escape from the liquid).

Metallic solids conduct electricity and heat and are malleable 可锻的 and ductile 可延展的, all because their free valence electrons 自由价电子 move easily and let the metal ions slide past one another without breaking the bonding. Solids are also either crystalline 晶体 (particles in a regular, repeating 3-D arrangement) or amorphous 非晶体 (no long-range order, like glass).

The three states differ in how tightly particles are held. Rising temperature raises average kinetic energy; when it overcomes the attractions, the substance melts or boils. Gases are mostly empty space, so they are compressible and fill their container.

In a solid and in a liquid the particles are in contact, so the two phases have similar volumes per mole: a mole of liquid water (18 mL) and a mole of ice (about 20 mL) are close. A mole of steam at 100 °C and 1 atm fills about 30 L, over 1500 times more. In a gas the spacing and the rate of collisions depend on the temperature, pressure and volume.

An ideal gas obeys

linking pressure, volume, moles, and absolute temperature. Use it to find any one quantity from the others, or (holding some constant) to predict how a gas responds to a change.

Worked example. How many moles of gas fill a $2.0\ \text{L}$ container at $300\ \text{K}$ and $1.5\ \text{atm}$? Using $R=0.0821\ \text{L atm/(mol K)}$,

Always use kelvin for $T$ and match the units of $R$ to your pressure and volume.

In a mixture of gases, each gas pushes on the walls as if it were alone. Its partial pressure is set by its share of the moles, its mole fraction 摩尔分数 $X$:

Worked example (gas collected over water). Hydrogen collected over water at $25\,^{\circ}\text{C}$ has a total pressure of $0.980$ atm, but the gas is a mixture of $\text{H}_2$ and water vapor. The vapor pressure of water at $25\,^{\circ}\text{C}$ is $0.031$ atm, so $P_{\text{H}_2}=0.980-0.031=0.949$ atm. Use $0.949$ atm, not $0.980$, in $PV=nRT$ to find the moles of $\text{H}_2$.

Worked example (partial pressures). A tank holds 2.0 mol $\text{O}_2$ and 6.0 mol He at a total pressure of 8.0 atm, a mixture used by deep-sea divers. $X_{\text{O}_2}=2.0/8.0=0.25$, so $P_{\text{O}_2}=0.25\times8.0=2.0$ atm and $P_{\text{He}}=6.0$ atm. The two add up to 8.0 atm.

Graphs show the simple relationships hidden in $PV=nRT$. At fixed $n$ and $T$, $P$ against $V$ is a curve ($P\propto1/V$: halve the volume, double the pressure). At fixed $n$ and $V$, $P$ against kelvin temperature is a straight line through the origin.

Kinetic molecular theory 分子运动论 explains gas behavior: particles are tiny, in constant random motion, with negligible volume and no attractions, and collisions are elastic. Temperature is proportional to average kinetic energy, so at a given temperature lighter molecules move faster (Graham's law of effusion).

Worked example (who moves faster?). At the same temperature, $\text{H}_2$ (2 g/mol) and $\text{O}_2$ (32 g/mol) have the same average kinetic energy, $\tfrac12mv^2$. Since the oxygen molecule is 16 times heavier, its $v^2$ must be 16 times smaller, so its average speed is $\sqrt{16}=4$ times smaller. Hydrogen molecules move about four times faster. This is why a helium balloon goes down faster than an air-filled one: the light atoms escape through tiny holes more often.

Real gases deviate from ideal behavior at high pressure and low temperature, where molecules are close enough that their real volume and their attractions matter. Attractions lower the pressure below ideal; molecular volume raises it.

Deviations are also larger close to the point where the gas would condense. A gas with strong attractions, such as $\text{NH}_3$ or $\text{H}_2\text{O}$ (hydrogen bonding), deviates far more than He, whose tiny atoms barely attract each other. So of He, $\text{N}_2$ and $\text{NH}_3$ at the same high pressure, $\text{NH}_3$ will show a pressure furthest below the ideal value.

A solution 溶液 is a homogeneous mixture of a solute 溶质 dissolved in a solvent 溶剂. Concentration is usually molarity 摩尔浓度:

Dilution conserves moles: $M_1V_1=M_2V_2$.

Worked example. What volume of water must you add to $50\ \text{mL}$ of $6.0\ \text{M}$ HCl to make it $2.0\ \text{M}$? The moles of HCl are unchanged, so $M_1V_1=M_2V_2$ gives the final volume $V_2=\dfrac{M_1V_1}{M_2}=\dfrac{6.0\times50}{2.0}=150\ \text{mL}$. You therefore add $150-50=100\ \text{mL}$ of water.

Worked example (ions in solution). How many moles of ions are in 250 mL of 0.40 M $\text{CaCl}_2$? Moles of $\text{CaCl}_2$: $0.250\times0.40=0.10$ mol. Each formula unit gives one $\text{Ca}^{2+}$ and two $\text{Cl}^-$, so $[\text{Cl}^-]=0.80$ M and there are $0.10+0.20=0.30$ mol of ions in total. In a particle drawing, show twice as many $\text{Cl}^-$ as $\text{Ca}^{2+}$.

A particulate diagram shows the solute and solvent particles. For an ionic solute, show it fully dissociated into separate ions surrounded by solvent; count particles to reason about concentration and conductivity.

Because a mixture's components keep their properties, physical methods separate them: filtration (by particle size), distillation (by boiling point), and chromatography 色谱法 (by how strongly each component sticks to a stationary phase versus moving with a solvent).

Filtration cannot separate a solution, because the dissolved particles are as small as the solvent's. The other two methods both work through intermolecular forces:

• Distillation: the component with weaker intermolecular forces has the higher vapor pressure, so it boils off first. Ethanol (78 °C) distils away from water (100 °C).
• Chromatography: a component that is attracted more strongly to the stationary phase (the paper, or the solid in a column) moves slowly; one attracted more to the moving mobile phase (the solvent) travels further. On polar paper with a nonpolar solvent, the most polar dye moves the shortest distance and has the smallest $R_f$ (distance moved by the spot divided by distance moved by the solvent).

Solubility 溶解度 is how much solute dissolves. "Like dissolves like" is a useful guide: polar solutes often dissolve in polar solvents, and nonpolar solutes in nonpolar solvents. It is not a guarantee, especially for ionic solids. Both interaction energies and entropy affect dissolution.

Worked example (why do some pairs mix?). Ethanol mixes with water in any amount: both hydrogen-bond, and ethanol–water hydrogen bonds replace the ones broken. Hexane does not mix with water. Hexane can only offer dispersion forces, which cannot make up for the strong hydrogen bonds between water molecules that would have to break. NaCl dissolves in water because strong ion–dipole attractions replace the ionic attractions in the crystal, but it does not dissolve in hexane, which offers no such attraction.

Spectroscopy 光谱学 studies how matter absorbs or emits light. Different regions of the electromagnetic spectrum probe different changes: microwaves (rotation), infrared (bond vibrations), ultraviolet–visible (electron transitions). The light absorbed reveals structure.

Region	Photon energy	What it changes	Use
microwave	lowest	how fast molecules rotate	microwave ovens heat water this way
infrared	medium	how bonds vibrate	identifying bonds such as O–H and C=O
ultraviolet–visible	highest of the three	electron energy levels	colour, Beer–Lambert analysis

Light is carried by photons 光子, each with energy $E=h\nu=\dfrac{hc}{\lambda}$. Higher frequency (shorter wavelength) means higher energy. A molecule absorbs a photon only when its energy matches an allowed energy gap.

When an atom or molecule absorbs a photon, its energy rises by exactly the photon's energy. When it emits one, its energy falls by the same amount. That is why each element gives a line spectrum of particular colours, as in flame tests.

Worked example (photon energy). Green light has a wavelength of 500 nm. Its frequency is

so one photon carries $E=h\nu=(6.626\times10^{-34})(6.00\times10^{14})=3.98\times10^{-19}$ J. A mole of such photons carries $3.98\times10^{-19}\times6.022\times10^{23}=2.40\times10^5$ J, about 240 kJ, similar to the energy of a weak chemical bond. That is why ultraviolet light, with even more energy per photon, can break bonds and damage skin.

The Beer–Lambert law 比尔-朗伯定律 relates how much light a solution absorbs to its concentration:

where $A$ is absorbance, $\varepsilon$ the molar absorptivity, $b$ the path length, and $c$ the concentration. Since $A$ is proportional to $c$, measuring absorbance is a fast way to find an unknown concentration.

Worked example. A dye has molar absorptivity $\varepsilon=2000\ \text{L/(mol cm)}$; in a $1.0\ \text{cm}$ cell a sample reads absorbance $A=0.40$. Its concentration is $c=\dfrac{A}{\varepsilon b}=\dfrac{0.40}{2000\times1.0}=2.0\times10^{-4}\ \text{M}$. Because $A\propto c$, a solution twice as concentrated would read $A=0.80$ – the basis of a calibration curve.

In the lab, you usually do not know $\varepsilon$. Instead, you measure the absorbance of several standard solutions of known concentration, all in the same cell and at the same wavelength, and plot $A$ against $c$. The unknown's absorbance is then read off the straight line. Choose the wavelength the substance absorbs most strongly: the measurement is then most sensitive. Errors to watch for include fingerprints or scratches on the cell (which make $A$ too high), and a cell not rinsed with the solution being measured.

Exam skill (lab errors). Many free-response questions ask how a mistake changes the result. Follow the error through $A=\varepsilon bc$. Drops of water left in the cuvette dilute the sample, so $A$ is too low and the calculated concentration is too low. Fingerprints or scratches on the cuvette absorb or scatter light, so $A$ is too high. Measuring at a wavelength the substance hardly absorbs makes the line shallow, so small reading errors become large concentration errors: always use the wavelength of maximum absorbance.

Exam skill (making a solution). To make 100.0 mL of 0.340 M NaCl: calculate the mass ($0.1000\times0.340\times58.44=1.99$ g), weigh it on an analytical balance, and dissolve it in less than 100 mL of distilled water in a 100.0 mL volumetric flask. Then add water up to the mark and mix. To dilute instead, measure the stock with a volumetric pipette or burette into the volumetric flask, then fill to the mark. A beaker or graduated cylinder is not precise enough for the final volume.

Common mistakes.

• Using °C in $PV=nRT$. Convert to kelvin first ($T/\text{K}=t/{}^{\circ}\text{C}+273$); at $0\,{}^{\circ}\text{C}$ the Celsius value would give a volume of zero.
• Saying a hydrogen bond is a bond between the H and the O inside one water molecule. That O–H is a covalent bond; the hydrogen bond is the attraction from that H to a lone pair on a neighbouring molecule.
• Saying any molecule containing H can hydrogen-bond. The H must be bonded to N, O or F; the H atoms in $\text{CH}_4$ cannot.
• Assuming dispersion forces are always the weakest. Between large, highly polarizable molecules, dispersion can outweigh the dipole–dipole forces of a smaller polar molecule.
• Using $V_2$ as the volume of water to add in a dilution. $M_1V_1=M_2V_2$ gives the final volume; subtract the starting volume to find the water added.
• Saying real gases deviate most at low pressure and high temperature. Deviations are largest at high pressure and low temperature, where the particles are close and slow enough for their volume and attractions to matter.
• Reading a higher absorbance as a more dilute solution. In $A=\varepsilon bc$, absorbance is proportional to concentration.

• Intermolecular forces (dispersion, dipole–dipole and hydrogen bonding) affect boiling points — they are much weaker than the bonds inside a molecule.
• Boiling breaks the forces between molecules, not the covalent bonds within them.
• Use $PV=nRT$ with temperature in kelvin and $R$ matched to your pressure/volume units.
• For solutions use molarity $M=\text{mol}/\text{L}$; dilution conserves moles, so $M_1V_1=M_2V_2$.
• "Like dissolves like" is a guide; check the particular solute and solvent.