= Calorimeter heat-loss correction
{title2=$Q_{\rm loss,out}=G\int(T-T_a)\,dt$}
A calorimeter heat-loss correction estimates the signed <heat> exchanged with the room during the measurement. Under <Newton's law of cooling>, a constant room <temperature> $T_a$ and a single representative assembly <temperature> $T$ give the model
$$
C_{\rm eff}\frac{dT}{dt}=P(t)-G(T-T_a),\qquad
Q_{\rm loss,out}=G\int_{t_0}^{t_1}(T(t)-T_a)\,dt.
$$
Here $P(t)$ is deposited input <power>, and $G$ is the apparatus loss coefficient in $\mathrm{W\,K^{-1}}$. After input ceases, the <temperature> excess shows <exponential decay> with <time constant> $\tau=C_{\rm eff}/G$:
$$
T(t)-T_a=(T(t_1)-T_a)e^{-(t-t_1)/\tau}.
$$
A cooling trace determines the ratio $G/C_{\rm eff}$, not the two factors separately. Combining the cooling trace with a known electrical <energy> input supplies the missing information. For constant input $P_0$, starting at $T_a$, the heating solution is $T(t)-T_a=(P_0/G)(1-e^{-t/\tau})$; these solutions generate the calibration figure.
For <mixing calorimetry>, infer cooling behavior only after the sample, <water> and vessel approach a common <temperature>. Applying a single-body cooling model while the sample remains much hotter than the <water> can misestimate the earlier loss. The Toronto handout's area correction assumes the same loss mechanism represents both the mixing interval and subsequent cooling.<Heat Capacity (University of Toronto laboratory handout)>[p. 3, Figure 1 and the area-ratio cooling correction]{ref} A backward extrapolation to the transfer time is therefore a model-dependent estimate, not the ultimate equilibrium with the room. Extend the model or report a limitation if sensor delay, substantial <temperature gradients>, changing room <temperature>, or escaping material invalidate that assumption. In particular, the relevant $C_{\rm eff}$ after adding a sample includes its <heat capacity>; an empty-receiver cooling rate need not apply to the loaded receiver.
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