= Adiabatic compression
<Adiabatic compression> decreases the <volume (thermodynamics)> of a sample without <heat transfer> across its boundary. For the fixed-composition <ideal gas> and endpoint assumptions in <adiabatic process>, compression <work> increases <internal energy>; positive <heat capacity at constant volume> then means a higher final <temperature>. The effect comes from supplied <work>, even though no <heat> enters. The amount of <temperature> increase depends on the <work> actually delivered to the <gas>, not on the <adiabatic process> condition alone.
The <reversible ideal-gas adiabat> supplies a particularly useful quantitative benchmark. Assume a fixed <amount of substance> $n$, a <quasistatic process>, negligible <friction>, only <pressure>-<volume (thermodynamics)> <work>, and constant <heat capacity ratio> $\gamma=C_p/C_V>1$. Here $C_p$ and $C_V$ are the whole sample's <heat capacity at constant pressure> and <heat capacity at constant volume>. With $R$ the <molar gas constant>, <Mayer's relation> gives $C_V=nR/(\gamma-1)$. The proof outline is $C_V\,dT=-p\,dV$ from the <first law of thermodynamics>, followed by $p=nRT/V$ from the <ideal gas law> and integration. The result is
$$
pV^\gamma=\text{constant},\qquad TV^{\gamma-1}=\text{constant}.
$$
These finite power laws belong to this reversible, constant-$\gamma$ <ideal gas> model. If <heat capacity ratio> varies, use the integral relation in <heat capacity ratio>; if the compression is irreversible, use the actual <work> balance, as in <irreversible adiabatic piston compression>. A short stroke by itself justifies neither power law. https://openstax.org/books/university-physics-volume-2/pages/3-6-adiabatic-processes-for-an-ideal-gas[Ling, Moebs and Sanny, University Physics Volume 2, §3.6: quasi-static ideal-gas adiabat]{ref}
Writing $r=V_1/V_2>1$ for the compression ratio, the reversible model predicts
$$
\frac{T_2}{T_1}=r^{\gamma-1},\qquad
\frac{p_2}{p_1}=r^\gamma,\qquad
W_{\rm in}=-W_{\rm out}=C_V(T_2-T_1)
=\frac{p_1V_1}{\gamma-1}\left(r^{\gamma-1}-1\right).
$$
All <temperatures> in these ratios are absolute. The larger <pressure> rise than in an <isothermal compression>, where $p_2/p_1=r$, explains the greater local stiffness described by <isentropic compressibility>.
For an original stipulated example, set $\gamma=7/5$, $p_1V_1=nRT_1=1\,\mathrm{kJ}$ and $r=2$. Then $T_2/T_1=2^{0.4}\simeq1.31951$ and $p_2/p_1=2^{1.4}\simeq2.63902$. The input <work> and <internal energy> increase are both $0.79877\,\mathrm{kJ}$, while $Q_{\rm in}=0$. No initial <temperature> or apparatus performance is implied by the chosen <energy> scale. Independent integration gives the signed outward <work>
$$
W_{\rm out}=\int_{V_1}^{V_2}p\,dV
=\frac{p_2V_2-p_1V_1}{1-\gamma},
$$
which is negative during compression and agrees with $-\Delta U$. This <integral> uses equilibrium gas <pressure> only because the path is reversible.
\Image[/adiabatic-processes-work-and-temperature.png]
{title=Reversible <adiabatic compression> and <adiabatic expansion>: <temperature> changes balance signed <work>}
{description=Original analytical <ideal gas> model with constant $\gamma=1.4$ and common initial state $p_1V_1=1\,\mathrm{kJ}$. The two runs go from $V_1$ separately to $V_1/2$ and $2V_1$; they are not a compression followed by its reversal. Shaded <pressure>-<volume (thermodynamics)> areas give <work> magnitudes, with signs supplied by the direction of motion. The dashed <isothermal process> lines are comparisons, not additional <adiabatic processes>. The right panel displays $\Delta U=-W_{\rm out}$ for each run. No curves are measured data.}
In the sensor experiment of <adiabatic process>, compare the <pressure>-<volume (thermodynamics)> <integral> with $C_V\Delta T$ as an <energy> check only when internal <pressure> is sufficiently uniform and piston dynamics permit that boundary-<work> estimate. Account for <friction> separately before comparing it with hand input <work>. PASCO's manual supplies a compression procedure and an illustrative logarithmic-slope result $1.39$, compared with the model value $1.40$ for a diatomic <gas>; that is its sample result, not new measurements here. https://cdn.pasco.com/product_document/Adiabatic-Gas-Law-Apparatus-TD-8565-Manual.pdf[PASCO scientific, guide 012-05110D, pp. 5–6: work comparison and sample fitted exponent]{ref}
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