Can Negative Heat Capacity Exist?

Settled in 1968 by classical thermodynamics of polytropic processes (negative heat capacity for certain process paths of an ideal gas); formalized for self-gravitating systems by D. Lynden-Bell and R. Wood.

Negative heat capacity for special processes of an ordinary gas (e.g., certain polytropic expansions where the gas absorbs heat yet cools, or releases heat yet warms) is a standard, if surprising, result in classical thermodynamics predating Lange's book. The deeper, now-famous illustration — that self-gravitating systems such as stars and globular clusters generically have negative heat capacity — was formalized by Lynden-Bell and Wood in 1968, shortly after Lange's first edition, and has since become central to understanding stellar evolution and the 'gravothermal catastrophe.' Lange's Chapter 6 ('Thermodynamics') problem most likely uses the classical polytropic-process version available to a 1960s popular-science audience.

It shows that 'more energy in, hotter out' is a property of familiar materials, not a law of physics — some systems get colder precisely because you added energy.

Explanations that circulated

• Assuming heat capacity C = dQ/dT must always be positive simply because 'obviously' adding energy makes something hotter — a rule that holds for ordinary bulk substances like water or metal at constant volume or pressure, but is not a universal law of physics.
• Overlooking that for a gas undergoing certain polytropic (non-adiabatic, non-isothermal) expansions, or for self-gravitating systems like stars and star clusters (where the virial theorem ties kinetic/thermal energy to gravitational potential energy in a fixed ratio), removing total energy from the system can force it to contract and its temperature to RISE — giving an effective negative heat capacity that seems to contradict everyday experience but does not violate any conservation law.

References

• The gravo-thermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems, D. Lynden-Bell, R. Wood, Monthly Notices of the Royal Astronomical Society, 1968