Entropy maximization under thermal contact

{title2=$T_f=(C_1T_{1,0}+C_2T_{2,0})/(C_1+C_2)$}

Consider two rigid bodies forming an <isolated system>, with fixed composition and constant whole-body <heat capacity at constant volume> values $C_1=1\,\mathrm{kJ\,K^{-1}}$ and $C_2=3\,\mathrm{kJ\,K^{-1}}$. Their initial <temperatures> are $T_{1,0}=300\,\mathrm K$ and $T_{2,0}=400\,\mathrm K$. Assume each body remains internally uniform, their contact stores negligible <energy>, no <phase transition> occurs and no <work> is done or extracted. These are stipulated model values, not material measurements.

Because $dU_i=C_i\,dT_i$, write $U_i=a_i+C_iT_i$ with arbitrary fixed reference constants $a_i$. The conserved relative <internal energy> is
$$
E=U_1+U_2-a_1-a_2=C_1T_1+C_2T_2=1500\,\mathrm{kJ}.
$$
This is an <energy> reference convention, not a claim about actual <heat capacities> down to zero <temperature>. At <thermal equilibrium> the two <temperatures> equal $T_f$, so the <first law of thermodynamics> gives
$$
T_f=\frac{C_1T_{1,0}+C_2T_{2,0}}{C_1+C_2}=375\,\mathrm K,\qquad
\Delta U_1=75\,\mathrm{kJ},\qquad \Delta U_2=-75\,\mathrm{kJ}.
$$
The larger <heat capacity> pulls the common <temperature> closer to its initial value: body 1 warms by $75\,\mathrm K$, while body 2 cools by $25\,\mathrm K$.

\Image[/thermal-equilibrium-two-bodies.png]
{title=Two finite bodies approach <thermal equilibrium> at $375\,\mathrm K$}
{description=Ideal model with $C_1=1$ and $C_2=3\,\mathrm{kJ\,K^{-1}}$ and contact conductance $G=0.1\,\mathrm{kJ\,s^{-1}\,K^{-1}}$. The weighted <temperature> stays at $375\,\mathrm K$ and the relative <internal energy> stays at $1500\,\mathrm{kJ}$; the plotted curves are model predictions.}

To predict a timescale, stipulate a constant contact conductance $G=1/R_{\rm th}=0.1\,\mathrm{kJ\,s^{-1}\,K^{-1}}$, the reciprocal of the contact's <thermal resistance>. The rate of <heat transfer> into body 1 is $G(T_2-T_1)$, giving
$$
C_1\dot T_1=G(T_2-T_1),\qquad C_2\dot T_2=-G(T_2-T_1).
$$
Adding proves conservation of $E$. Subtracting gives <exponential decay> of the <temperature> difference, with <time constant>
$$
\tau=\frac{C_1C_2}{G(C_1+C_2)}=7.5\,\mathrm s,\qquad
T_2-T_1=100\,\mathrm K\,e^{-t/\tau},
$$
and therefore
$$
T_1(t)=375\,\mathrm K-75\,\mathrm K\,e^{-t/\tau},\qquad
T_2(t)=375\,\mathrm K+25\,\mathrm K\,e^{-t/\tau}.
$$
Exact equality is approached as $t\to\infty$ in this ideal model. Changing $G$ changes the rate but not $T_f$: equilibrium and the speed of reaching it are separate predictions. Real bodies can have internal <temperature gradients>, changing <heat capacities> and contact losses, so the plotted $7.5\,\mathrm s$ is an assumed model timescale.


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 Incoming links: Thermal equilibration of two finite bodies