HEAT CAPACITY AND CALORIMETRY

In his Glasgow laboratory in the 1760s, noticed something the caloric theory could not comfortably explain. Give a pound of water and a pound of mercury the same quantity of heat, and the mercury ends up about thirty times hotter. The same heat, the same mass, wildly different temperatures. Black concluded that substances differ in their capacity for heat — that warmth poured into matter is partly hidden, soaked up at different rates by different materials.

The modern name for that capacity is the Specific heat capacity: the energy needed to raise one gram of a substance by one kelvin. Water's is famously, anomalously large; metals' are small.

The relation is simple and is the workhorse of the whole subject:

$Q = m\,c\,\Delta T$

In words: the heat absorbed equals the mass times the specific heat times the temperature change. Rearranged, $\Delta T = Q/(mc)$ — for a fixed heat input, a large specific heat means a small temperature change. Water's large $c$ is why the oceans warm and cool slowly enough to govern climate, and why a cast-iron skillet ($c$ thirty times smaller) screams up to frying heat in a minute.

Specific heat per gram is convenient for kitchens but misleading for physics, because a gram of lead and a gram of aluminium contain very different numbers of atoms. The deeper quantity is the Molar heat capacity: the heat to warm one mole — a fixed count of atoms — by one kelvin.

In 1819 the French chemists and measured the molar heat capacities of a dozen solid elements and found something startling: they were all about the same. Lead, copper, iron, silver — element after element clustered near 25 J/(mol·K). This is the Dulong–Petit law:

$C_{\text{molar}} \approx 3R \approx 24.9\ \text{J/(mol·K)}$

In words: a mole of almost any simple solid needs roughly three times the gas constant of heat per kelvin, regardless of which element it is — light per gram for heavy atoms, heavy per gram for light ones, but the same per atom. The law is a hint of something profound: heat is shared out equally among the atoms' modes of vibration, three each, a foreshadowing of equipartition. Petit did not live to see its importance — he died of tuberculosis at 29, four months after the paper that carries his name.

For a gas, "the" heat capacity splits in two, because how you heat it changes the answer. Heat a gas in a sealed rigid box and all the heat goes into temperature. Heat it under a free piston — at constant pressure — and the gas expands, pushing the piston and doing work on the world, so some of the heat leaks out as mechanical work instead of warming.

So the same gas has a smaller Heat capacity at constant volume and pressure (C_v, C_p) at constant volume ($C_v$) than at constant pressure ($C_p$) — it takes more heat to achieve a given temperature rise when the gas is also doing work. For an ideal gas the difference is exactly the gas constant:

$C_p - C_v = R$

In words: per mole, the constant-pressure heat capacity exceeds the constant-volume one by precisely $R$ — the extra heat that goes into expansion work rather than temperature. That clean result, Mayer's relation, is one of the first places the gas constant shows up as something physical you can feel: it is the price, in heat, of letting a gas push back.

How were all these numbers measured, before electric heaters and digital thermometers? By Calorimetry: dropping a hot object into cool water inside an insulated vessel and reading the final temperature. Energy conservation does the rest.

If no heat escapes the calorimeter, the heat lost by the hot body equals the heat gained by the water:

$m_s\, c_s\, (T_s - T_{\text{eq}}) = m_w\, c_w\, (T_{\text{eq}} - T_w)$

Everything in that equation is measurable except the specific heat of the solid, $c_s$ — so you solve for it. This is exactly how Black, and a century of careful experimenters after him, built the table of specific heats in the first scene. The insulated cup is humble, but it is the instrument that turned heat from a quality into a quantity.

Black did not stop at specific heat. Watching ice melt, he found that heat poured into ice-water at its melting point produced no rise in temperature at all — the thermometer sat at 0 °C for as long as ice remained. The heat was going somewhere, but not into warmth. He called it latent heat — heat hidden in the act of changing state.

This breaks the tidy $Q = mc\,\Delta T$ picture, because here $\Delta T = 0$ while $Q$ is enormous. The energy is spent dismantling the bonds that hold a solid together, not speeding up its molecules. It is the same idea that lets a calorimeter work in reverse — and it is important enough, and strange enough, to be the subject of its own topic: phase changes and latent heat.

Return to the anomaly we started with. Water's specific heat is not just large; it is one of the largest of any common liquid, several times that of most organic liquids and metals. Why?

The answer is hydrogen bonding. Each water molecule links to its neighbours through a network of weak hydrogen bonds, and warming the liquid means not only speeding up the molecules but also flexing and breaking some of that network. That extra reservoir for energy — bonds to stretch as well as motion to quicken — is what makes water so reluctant to change temperature. It is not a universal feature of liquids; it is a quirk of the molecule.

We can now say how much energy it takes to warm a thing, why different substances bank heat so differently, and why a gas has two heat capacities split by $R$. But Black's latent heat hangs unfinished: what exactly happens to the energy while ice melts and water boils, when the thermometer refuses to move?

That is the physics of phase changes and latent heat — the energy that hides in melting and boiling, and the diagrams that map where each phase lives. It builds directly on the calorimetry here, and on the recognition, from heat and the caloric theory, that heat is energy in transit.