Read heating curves as kinetic and potential energy changes, classify phase changes, and calculate density from mass and volume

Learning goal: Interpret a temperature-versus-time graph of a substance being heated or cooled; explain the flat and sloped parts with the kinetic and potential energy of particles; classify each phase change as endothermic or exothermic; explain why most substances expand when heated and why solids are usually denser than liquids and liquids than gases; and calculate mass, volume or density from the other two, including from the slope of a mass-versus-volume graph.

Particles, energy and state. Middle school introduced solids, liquids and gases as arrangements of particles: tightly packed and vibrating in place, close but free to slide, far apart and moving freely. This course connects that picture to energy. Particles are in constant motion, and their kinetic energy is what temperature measures: the higher the temperature, the greater the average kinetic energy of the particles. Particles also attract one another, and the potential energy of the sample depends on how far apart they are held against those attractions. A phase change is a change in the potential energy of the particles, not their kinetic energy, and that single idea explains the shape of a heating curve.

Reading a heating curve. Suppose ice at −20 °C is heated steadily at ordinary atmospheric pressure (an ideal curve for a pure sample; to record the last part, the steam must be kept in the container rather than escaping). The temperature rises through the ice, stops at 0 °C and stays there while the ice melts, rises again through the liquid, stops at 100 °C while the water boils, and rises again once all of it is vapor. On the sloped parts, energy added to the sample increases the kinetic energy of the particles, so the temperature rises. On the flat parts (the plateaus), the temperature does not change even though energy is still being added, so the kinetic energy is not changing. The added energy is instead going into potential energy: it is being used to pull particles apart against their attractions, changing their positions from a rigid arrangement to a mobile one (melting) or from close together to far apart (boiling). The plateaus are at the melting point and the boiling point, and they are horizontal only while both phases are present. The moment the last crystal melts, the curve slopes upward again. Cooling reverses the picture: the temperature falls, holds at the condensation point while gas becomes liquid, falls, holds at the freezing point while liquid becomes solid, and falls again. A substance melts when the kinetic energy of its particles becomes great enough to overcome the attractive forces that hold them in place; it freezes or condenses when cooling leaves the particles without enough kinetic energy to resist those attractions.

Endothermic and exothermic phase changes. A change that absorbs energy from the surroundings is endothermic; one that releases energy to the surroundings is exothermic. Melting, boiling (and evaporation) and sublimation (solid straight to gas, as dry ice does) require energy input, so they are endothermic; that is why sweat evaporating from skin cools the skin, since the skin supplies the energy. Freezing, condensation and deposition (gas straight to solid, as frost forms) release energy, so they are exothermic; that is why fruit growers spray water on crops before a frost, since the water releases energy to the surroundings as it freezes. Note the direction: the sample's energy goes up in an endothermic change and down in an exothermic one, while the surroundings do the opposite.

Thermal expansion and density. Heat almost any solid, liquid or gas and it takes up a little more room: its particles move faster and, on average, sit farther apart. The mass is unchanged, so the density falls. Generally solids are denser than liquids, and liquids far denser than gases, because of the spacing of the particles. Water is the well-known exception between its solid and liquid: ice floats because water's particles lock into an open arrangement when they freeze, so ice is less dense than liquid water. The general rule holds for most substances; a claim about a particular substance must be checked against its measured densities.

Calculating density. Density is mass divided by volume, d = m / V, usually in g/cm³ (the same number as g/mL) or kg/m³. Given any two of mass, volume and density, the third can be found: m = d × V and V = m / d. On a graph of mass (vertical axis) against volume (horizontal axis) for several samples of one substance, the points lie on a straight line through the origin, and the slope of that line is the density, because slope is rise over run and rise over run here is mass over volume. Two substances graphed on the same axes give two lines, and the steeper line is the denser substance. A worked model: three pieces of the same metal measure 10.0 cm³ and 27.0 g, 20.0 cm³ and 54.0 g, 30.0 cm³ and 81.0 g. Each ratio is 27.0 ÷ 10.0 = 2.70, 54.0 ÷ 20.0 = 2.70 and 81.0 ÷ 30.0 = 2.70 g/cm³, so the points lie on one line of slope 2.70 g/cm³ and the density of the metal is 2.70 g/cm³. A fourth piece of this metal with mass 135 g would have volume 135 ÷ 2.70 = 50.0 cm³. The common error is to read the slope from the axes the wrong way round; volume over mass gives 0.370 cm³/g, which is not the density.

Investigation: record a cooling curve and find a freezing point. Materials: a test tube about one-third full of a low-melting solid suitable for school use such as lauric acid or stearic acid, a beaker of hot water from a kettle or warm-water bath (no open flame needed), a thermometer or temperature probe, a ring stand and clamp, a timer, goggles. Steps: stand the tube in hot water until the solid has melted completely and the reading is at least 10 °C above its melting point; lift the tube into a dry empty beaker so it cools in air; record the temperature every 30 seconds for 15 minutes, stirring gently with a separate wire stirrer (not the thermometer) only while liquid is present; stop stirring as soon as the sample becomes stiff. Record: a table of time and temperature and a graph of temperature against time. A result that supports the claim: the curve falls, then flattens for several readings while solid forms, then falls again; the plateau temperature is the freezing point, and it equals the melting point measured on reheating. Safety: goggles; hot water scalds; do not heat the solid directly over a flame; keep the thermometer in the tube while the sample solidifies rather than pulling it out. What the model cannot show: the graph shows that energy is leaving the sample during the plateau (the surroundings keep taking it) but does not measure how much; quantifying the energy of a temperature change or a phase change is left for chemistry.

Worked example 1 — read a heating curve. The diagram's graph for a small sample of water heated steadily shows a plateau at 0 °C from minute 0.5 to minute 4.5 and a plateau at 100 °C from minute 9.5 to minute 36.5. From minute 0 to 0.5 the sample is solid ice and its temperature rises, so the particles' kinetic energy is increasing. From minute 0.5 to 4.5 ice and liquid water are both present; the temperature does not change, so the kinetic energy does not change, and the energy added is increasing the potential energy of the particles as they leave their fixed positions. From minute 4.5 to 9.5 the sample is entirely liquid and warming. From minute 9.5 to 36.5 liquid and vapor are both present at 100 °C. At minute 8 the sample is liquid at about 70 °C, and at minute 20 it is a mixture of boiling liquid and vapor, both at 100 °C. The lengths are in realistic proportion: the boiling plateau lasts almost seven times as long as the melting plateau, because separating the particles completely into a gas takes far more energy than loosening them into a liquid, and ice and steam warm about twice as fast as liquid water because liquid water needs more energy per degree. This applies at ordinary atmospheric pressure; at lower pressure water boils at a lower temperature.

Worked example 2 — classify phase changes. Dew forming on grass overnight is condensation, gas to liquid: exothermic, because the water releases energy to the surroundings as its particles move closer together. A puddle drying is evaporation, liquid to gas: endothermic, because the water absorbs energy from the ground and air to separate its particles. Frost forming directly from water vapor on a cold windshield is deposition: exothermic. Dry ice shrinking in a warm room is sublimation: endothermic. The pattern is that changes in which particles pull farther apart from their neighbors (solid to liquid, liquid to gas, solid to gas) absorb energy, and changes in which they become more tightly held (the reverse three) release it. This is about how strongly particles are held, not about density: ice is less dense than liquid water, yet freezing still releases energy.

Worked example 3 — density three ways. A block has mass 54.0 g and volume 20.0 cm³, so its density is 54.0 ÷ 20.0 = 2.70 g/cm³. A liquid has density 1.00 g/cm³; the volume of 250 g of it is 250 ÷ 1.00 = 250 cm³. A stone of density 2.50 g/cm³ displaces 12.0 cm³ of water in a graduated cylinder, so its mass is 2.50 × 12.0 = 30.0 g. Units must be carried in every line; a density in g/cm³ multiplied by a volume in cm³ gives grams, and the cm³ cancel.

Worked example 4 — compare two lines on a mass-versus-volume graph. Substance A has points (10 cm³, 8 g) and (30 cm³, 24 g); substance B has points (10 cm³, 20 g) and (20 cm³, 40 g). Slope of A: (24 − 8) ÷ (30 − 10) = 16 ÷ 20 = 0.80 g/cm³. Slope of B: (40 − 20) ÷ (20 − 10) = 20 ÷ 10 = 2.0 g/cm³. B is denser, and its line is steeper. A would float on water (0.80 is less than 1.00 g/cm³) and B would sink, provided neither dissolves in or reacts with water.

Representation task: annotate a heating curve. Draw temperature (vertical) against time (horizontal) for a substance heated from solid at −40 °C to gas, with a melting point of −10 °C and a boiling point of 60 °C (a hypothetical substance). Label each of the five segments with the phase or phases present, write "KE rising" on the sloped segments and "PE rising, KE constant" on the plateaus, and mark each plateau as endothermic. Below the graph, sketch a particle diagram for one point on each segment.

Independent performance: find and use a density from data. Using a balance and a graduated cylinder (water displacement), measure the mass and volume of at least four different-sized pieces of one material (glass marbles, steel nuts or aluminum pieces work well). Plot mass against volume, draw the best straight line through the origin, and find its slope. Success criteria: the table has four or more rows with units in the headings; the graph's axes are labeled with units and the slope is calculated from two points on the line, not from a single data point; the slope is reported as the density with units; a prediction is made for the mass of a fifth piece from its measured volume and then checked by weighing, with the percentage difference stated; and one paragraph explains why the line should pass through the origin and what it would mean if the points did not lie on a single line.