= Temperature sensitivity of a free-expansion experiment
The <gas>-only condition $\Delta U=0$ requires negligible <heat> exchange with the apparatus. If the insulated <isolated system> instead includes the walls, sensors and water, the <first law of thermodynamics> fixes their combined <energy>. Once everything returns to a common final <temperature>, a constant apparatus <heat capacity> $C_a$ gives
$$
U_g(T_2,V_2)-U_g(T_1,V_1)+C_a(T_2-T_1)=0.
$$
Define $A=U_g(T_1,V_2)-U_g(T_1,V_1)$, the gas's <internal energy> increase for the same expansion evaluated at the initial <temperature>. For a small <temperature> change, linearizing at the final <volume (thermodynamics)> gives
$$
\Delta T\simeq-\frac{A}{C_{V,g}+C_a}.
$$
Here $C_{V,g}$ is the whole gas sample's <heat capacity at constant volume> at the reference state. Compared with an isolated gas sample's $-A/C_{V,g}$, the common final signal is reduced by $C_{V,g}/(C_{V,g}+C_a)$. This <energy> accounting explains why a massive bath can conceal a real-gas effect. Fitzpatrick gives the corresponding apparatus correction for the <Van der Waals equation>. <Thermodynamics and Statistical Mechanics by Richard Fitzpatrick/Van der Waals Gas>{ref}
For a toy calculation, stipulate $A=5\,\mathrm J$, $C_{V,g}=10\,\mathrm{J\,K^{-1}}$ and $C_a=990\,\mathrm{J\,K^{-1}}$, all constant in this example. The predicted cooling magnitude changes from $0.5\,\mathrm K$ for the gas alone to $0.005\,\mathrm K$ for the combined apparatus. An illustrative effective detection threshold of $0.01\,\mathrm K$ would miss the latter. These are chosen numbers, not Joule's measurements. In this approximation a null observation only bounds $|A|\lesssim(C_{V,g}+C_a)\epsilon_T$, where $\epsilon_T$ is the detectable <temperature> change after allowing for drift and other experimental uncertainties; a display's smallest digit is not automatically this threshold.
The historical limitation was recognized explicitly: Joule and Thomson later estimated that the common apparatus in Joule's experiment would cool by only about $0.003\,\mathrm K$, explaining why the earlier external thermal effect was undetectable. This was a later calculation about the apparatus, not a measured cooling trace. https://upload.wikimedia.org/wikipedia/commons/4/47/The_free_expansion_of_gases-_memoirs_by_Gay-Lussac%2C_Joule%2C_and_Joule_and_Thomson%3B_%28IA_cu31924012329300%29.pdf#page=75[Joule and Thomson, On the Thermal Effects of Elastic Fluids, Part II, reprint p. 63: estimated temperature change of the copper-and-water apparatus]{ref}
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