Thermal energy transfer and specific heat capacity

Solids, liquids and gases

Before any of the bookkeeping, the picture of matter itself. Three states, and everything separating them comes down to three questions. How far apart do the particles sit, how tidily are they arranged, and how do they move?

In a solid the particles sit close, touching their neighbours, held in a regular repeating arrangement by strong forces between them. They are not still. Each one vibrates about a fixed position and cannot swap places with the particles around it. Fixed positions in a fixed pattern are why a solid keeps its own shape and its own volume.

A liquid keeps the spacing and throws away the order. Its particles still touch, which is why melting barely changes the density of a substance, but past the nearest few neighbours the arrangement is disordered, and each particle carries enough energy to slide past the others while staying in contact. Contact fixes the volume. The lost order lets a liquid pour and take the shape of its container.

A gas throws away both. At ordinary pressures its particles sit roughly ten times as far apart as a liquid's, which is about a thousand times the volume for the same number of them, and in no arrangement whatsoever. They travel in rapid straight lines, in random directions, and feel almost no force from one another between collisions. Nothing holds them together, so a gas has neither a fixed shape nor a fixed volume and expands to fill whatever it is put in.

Those three paragraphs are the simple kinetic model, and it earns the word kinetic because motion carries the argument. Supply energy and the particles move faster and range further. Give them enough to break out of the lattice and the solid melts; give them enough to escape their neighbours' attractions altogether and the liquid boils. The rest of this lesson is that thread pulled tight, first into internal energy and then into the two formulas that quantify a heating.

Internal energy

Every object is a crowd of particles in random motion, and its internal energy is the sum of their randomly distributed kinetic and potential energies. Kinetic in the jiggling, potential in the arrangement and separation.

One consequence is worth keeping to hand. An ideal gas has no intermolecular forces and so no potential-energy term, which leaves its internal energy as the total kinetic energy of its molecules and nothing else. The word randomly is doing spec-level work too. A thrown ball's ordered motion counts as kinetic energy without counting as internal energy.

Exactly two ways exist to change internal energy. Heating the system is one. Doing work on it is the other, and compressing a gas warms it with no flame in sight. Written down, that bookkeeping becomes the first law of thermodynamics, ΔU = q + w. The rise in internal energy equals the heat supplied to the system plus the work done on it, with energy leaving counted negative. A gas doing work as it expands takes w negative; a gas being squeezed takes w positive. CIE examines the equation with exactly this sign convention, so fix the direction of each term before you substitute.

Temperature, meanwhile, tracks only the kinetic account, and temperature is what decides where heat flows. Energy passes from the hotter body to the colder one until the two reach the same temperature and the net transfer stops. Two objects left in contact always drift towards that state, thermal equilibrium. Heat is energy in transfer, measured in joules. Temperature measures the average jiggle, in kelvin. A sparkler runs far hotter than a warm bath, while the bath, with vastly more molecules, holds far more internal energy.

Changes of state

Supply energy steadily and the temperature climbs, until a change of state begins. Then it stops climbing entirely. Through melting or boiling the incoming energy goes into potential energy, prising particles apart against their attractions, while the kinetic energy, and so the temperature, stays constant. Only when the rearrangement finishes does the plateau end. Explain a flat section in exactly those terms, potential energy rising and kinetic energy unchanged, and the marks are bankable.

Calculating a temperature rise

The specific heat capacity c of a material is the energy needed to raise 1 kg of it by 1 K.

Water's enormous c, about 4200 J kg⁻¹ K⁻¹, is what lets the sea moderate coastal climates and what makes a kettle take as long as it does. One practical method deserves knowing. In continuous-flow heating, liquid streams steadily past an electrical heater, and three things get measured. The heater power P, the mass m carried past each second (a mass flow rate, in kg s⁻¹), and the temperature rise Δθ from inlet to outlet. Each second the heater supplies P, the liquid takes mcΔθ of it, and the rest leaks to the surroundings as a loss h. One run therefore has two unknowns in it, c and h, so one run can never settle c.

The second run is what does. Change the flow rate and adjust the power until the inlet and outlet temperatures read exactly as before. The apparatus now sits at the same temperatures as in the first run, so the loss h is the same number, and subtracting the two power equations deletes it,

with m₁ and m₂ the masses carried past per second in the two runs, flow rates in kg s⁻¹ rather than plain masses, so that both sides of the equation are powers, in watts. Divide the power difference by Δθ and by the difference in those two masses and c drops out, loss and all. Both powers are the point of the method. The flow readings on their own carry no energy in them at all.

Latent heat and changes of state

The specific latent heat l is the energy to change the state of 1 kg with no temperature change.

Latent heat of fusion for melting, of vaporisation for boiling, and the second is far larger, since boiling must separate the particles almost completely. A multi-stage problem, warm the ice, melt it, warm the water, is nothing more than Q = mcΔθ and Q = ml summed stage by stage, and keeping the stages separate is the technique.

Both latent heats are measured electrically, and both measurements are built the same way. An immersion heater supplies a known power, read as P = VI from a voltmeter across the heater and an ammeter in series with it, so the energy delivered in a timed run is VIt. A balance then weighs the substance that changed state. What separates a good version of the experiment from a poor one is entirely what it does about the energy the room is quietly adding or taking away, and in both cases the answer is a second run rather than more lagging.

For fusion, the heater sits in crushed melting ice held in a funnel, and the meltwater it produces is collected in a beaker on the balance for a measured time. Warm air melts the ice as well, so an identical funnel of ice stands beside the first with its heater switched off, draining into its own beaker over the same time. That control run measures what the room does on its own, m₀, and the heater is then answerable only for the difference, so VIt = (m − m₀)l_f. The ice must be melting to begin with, since any energy spent warming it from a freezer temperature belongs to c and not to l.

For vaporisation, the liquid stands on a top-pan balance with the heater immersed in it, and readings begin only once it is boiling steadily, so that the apparatus has stopped warming up and every extra joule leaves as vapour or as loss. Record the mass boiled away in a measured time, then repeat at a different heater power while the liquid boils just as steadily. The apparatus sits at the same temperature in both runs, so the loss to the room is the same number in each, and subtracting the two equations deletes it exactly as the continuous-flow method did.

That leaves l_v = (P₁ - P₂)t/(m₁ - m₂)$l_{v} = (P_{1} - P_{2})t/(m_{1} - m_{2})$l_v = (P_1 - P_2)t/(m_1 - m_2), with m₁ and m₂ the masses boiled away in the same time t at the two heater powers, and the heat loss gone from the answer without ever having been measured.