= Adiabatic expansion
<Adiabatic expansion> increases the accessible <volume (thermodynamics)> without <heat transfer>. In a loaded piston expansion of the fixed-composition <ideal gas> considered in <adiabatic process>, positive outward <work> comes from <internal energy>, so the final <temperature> decreases. This provides a cooling mechanism without removing <heat> during the stroke. In the <Carnot cycle>, reversible <adiabatic expansion> connects two different <temperatures> between its <isothermal processes>.<Statistical Physics by David Tong/4 Classical Thermodynamics>[§4.3.1: adiabatic expansion between the isothermal legs]{ref}
For the same constant-$\gamma$ <reversible ideal-gas adiabat>, put $b=V_2/V_1>1$. The scaling and <first law of thermodynamics> give
$$
\frac{T_2}{T_1}=b^{1-\gamma},\qquad
\frac{p_2}{p_1}=b^{-\gamma},\qquad
W_{\rm out}=C_V(T_1-T_2)
=\frac{p_1V_1}{\gamma-1}\left(1-b^{1-\gamma}\right)>0.
$$
With the figure's stipulated $\gamma=1.4$, $p_1V_1=1\,\mathrm{kJ}$ and $b=2$, the final ratios are $T_2/T_1\simeq0.75786$ and $p_2/p_1\simeq0.37893$. Outward <work> is $0.60535\,\mathrm{kJ}$ and $\Delta U=-0.60535\,\mathrm{kJ}$. Reversing this particular path from its cooled endpoint would require the same <work> input; the figure's separate compression starts at the original state, so its input is different. Endpoint <entropy> supplies another independent check:
$$
\Delta S=C_V\ln\frac{T_2}{T_1}+nR\ln\frac{V_2}{V_1}=0.
$$
The two <logarithm> terms cancel using $C_V(\gamma-1)=nR$, confirming the reversible path is an <isentropic process>. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node40.html[MIT Unified Engineering, §5.4: ideal-gas endpoint entropy and constant-entropy scaling]{ref}
PASCO's expansion demonstration first compresses trapped air and waits for <thermal equilibrium> before the rapid expansion. Its manual warns that <friction>-warmed cylinder regions contact the <gas> during expansion, compromising quantitative comparison. https://cdn.pasco.com/product_document/Adiabatic-Gas-Law-Apparatus-TD-8565-Manual.pdf[PASCO scientific, guide 012-05110D, p. 3: expansion preparation and frictional wall-heating limitation]{ref} The hold separates cooling caused by expansion from simply undoing the earlier hot compression. As proposed controls, observe the <temperature> dip and subsequent warming, compare rapid and slow strokes from matched initial states, and assess wall <heat transfer> and probe lag before fitting a reversible model.
Cooling is not a universal consequence of <adiabatic expansion>: <free expansion> of an <ideal gas> has no outward <work> and returns to the same equilibrium <temperature>. Its <entropy> increases, so the reversible power laws do not describe it. Reuse <free expansion>, <Joule's free-expansion experiment> and <temperature sensitivity of a free-expansion experiment> for that apparatus, its <energy> budget and the real-gas qualifications. The useful distinction is the load: expansion cools this <ideal gas> when it must spend <internal energy> doing <work>.
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