= Heat bath relaxation with finite capacity
{title2=$\tau=C_1C_2/[G(C_1+C_2)]$}
The equilibrium calculation determines where the two <temperatures> end, without determining how long they take. Stipulate a constant <thermal conductance> $G>0$ and the linear <heat transfer> law underlying <Newton's law of cooling>. With $Q$ positive from body 1 to bath 2, the <ordinary differential equations> are
$$
\dot Q=G(T_1-T_2),\qquad C_1\dot T_1=-G(T_1-T_2),\qquad C_2\dot T_2=G(T_1-T_2).
$$
The dot denotes a <derivative> with respect to time. This model assumes internal equilibration is fast compared with exchange between the bodies. Adding the two <energy> rates gives zero; subtracting the <temperature> rates gives
$$
\frac{d}{dt}(T_1-T_2)=-G\left(\frac1{C_1}+\frac1{C_2}\right)(T_1-T_2),\qquad
\tau=\frac{C_1C_2}{G(C_1+C_2)}.
$$
The <temperature> difference has <exponential decay> with <time constant> $\tau$. For the <finite heat bath> example and the stipulated $G=2\,\mathrm{W\,K^{-1}}=2\,\mathrm{J\,s^{-1}\,K^{-1}}$, $\tau=45\,\mathrm s$, and solving the <ordinary differential equations> gives
$$
T_1(t)=296\,\mathrm K+54\,\mathrm K\,e^{-t/(45\,\mathrm s)},\qquad
T_2(t)=296\,\mathrm K-6\,\mathrm K\,e^{-t/(45\,\mathrm s)},
$$
$$
Q(t)=5400\,\mathrm J\left(1-e^{-t/(45\,\mathrm s)}\right).
$$
At one <time constant>, $T_1\simeq315.865\,\mathrm K$, $T_2\simeq293.793\,\mathrm K$ and $Q\simeq3413.451\,\mathrm J$. Equal <temperatures> are approached as a <limit> at infinite time. The finite bath causes the <temperature> difference to shrink faster than the fixed-bath model, because both bodies change <temperature>.
\Image[/finite-heat-bath-newton-cooling.png]
{title=Finite bath and fixed bath <Newton's law of cooling> histories}
{description=Original analytical model predictions with $C_1=100\,\mathrm{J\,K^{-1}}$, $C_2=900\,\mathrm{J\,K^{-1}}$ and $G=2\,\mathrm{W\,K^{-1}}$; time is in seconds. Solid lines show the two finite bodies approaching $296\,\mathrm K$ with <time constant> $45\,\mathrm s$. Dashed lines show body 1 cooling toward a bath held at $290\,\mathrm K$, with <time constant> $50\,\mathrm s$. The two panels use different vertical scales so that the finite bath's $6\,\mathrm K$ warming remains visible.}
For an ideal <heat bath> with the same $G$, set $T_2=T_b=290\,\mathrm K$ rather than evolving its <temperature>. <Newton's law of cooling> then yields
$$
T_1(t)=290\,\mathrm K+60\,\mathrm K\,e^{-t/(50\,\mathrm s)},\qquad
\tau_\infty=C_1/G=50\,\mathrm s.
$$
Changing $G$ changes these <time constants>, while the final <temperature> is fixed by the capacities and initial conditions. The dynamical model independently checks the <Second law of thermodynamics>:
$$
\dot S_{\rm tot}=\frac{-\dot Q}{T_1}+\frac{\dot Q}{T_2}
=\frac{G(T_1-T_2)^2}{T_1T_2}\ge0.
$$
Its <integral> over the relaxation reproduces the endpoint <entropy> change. The useful separation is between capacity, which governs how much the bath <temperature> shifts, and <thermal conductance>, which governs the exchange rate. Neither alone establishes a reliable <heat bath>.
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