Thermal reservoir

A thermal reservoir (or heat bath) is an idealized part of the surroundings that can exchange energy with a thermodynamic system as heat while maintaining an (effectively) constant temperature $T_R$. Operationally, it is a macroscopic body with such a large heat capacity (and fast internal relaxation) that exchanging a finite amount of heat produces a negligible change in its temperature, so it stays in (approximate) internal equilibrium at all times.

Physically, a thermal reservoir models a very large environment (e.g., a large water bath, a laboratory “thermostat,” or an effectively infinite environment) that fixes temperature via the zeroth law : when the system is placed in contact with the reservoir through a diathermal wall , it can relax toward thermal equilibrium with $T = T_R$ (provided other constraints don’t prevent equilibration).

A reservoir is typically assumed to exchange only heat with the system (no matter transfer, and no controlled work transfer). If the coupling allows energy transfer as heat, the transferred amount is described by the inexact differential of heat $\delta Q$.

Key relations (constant-$T_R$ idealization).
Let $Q$ denote the net heat absorbed by the system from the reservoir over some interaction. Under the reservoir idealization, the reservoir’s entropy change is

because the reservoir remains at temperature $T_R$ while its energy decreases by $Q$.

Combined with the second law , this yields a standard “system + reservoir” entropy balance:

with equality in the reversible limit (see Clausius inequality and reversible process ). This is one of the main practical reasons thermal reservoirs are useful: they convert the second law into a concrete inequality involving $Q$ and $T_R$.

In statistical mechanics language, a thermal reservoir is what underlies the canonical description: the reservoir fixes the system’s temperature, equivalently fixing the inverse temperature $\beta = 1/(k_B T_R)$ with Boltzmann constant $k_B$ (see canonical ensemble conventions ).

Contrast: a work reservoir idealizes controlled exchange of work rather than heat.