19-9 The Adiabatic Expansion of an Ideal Gas#

Prompts

In an adiabatic process, $Q = 0$. What happens to the temperature when an ideal gas expands adiabatically? Why?
Write $pV^\gamma = \text{constant}$ for an adiabatic process. What is $\gamma$? What is $\gamma$ for a monatomic gas?
Derive or state the relation $TV^{\gamma-1} = \text{constant}$. How does $T$ change when $V$ increases adiabatically?
Why does the relation $pV^\gamma = \text{constant}$ NOT apply to a free expansion? What happens to $T$ in a free expansion?

Lecture Notes#

Overview#

An adiabatic process has $Q = 0$—no heat exchange with the surroundings.
Achieved by rapid expansion/compression or a well-insulated container.
$pV^\gamma = \text{constant}$ and $TV^{\gamma-1} = \text{constant}$, where $\gamma = C_P/C_V$.
Adiabatic expansion → gas does work → internal energy decreases → temperature drops.

Adiabatic relations#

Pressure–volume: $$ p_i V_i^\gamma = p_f V_f^\gamma \quad \text{(adiabatic)} $$ (eq-adiabatic-pV)

Temperature–volume: $$ T_i V_i^{\gamma-1} = T_f V_f^{\gamma-1} \quad \text{(adiabatic)} $$ (eq-adiabatic-TV)

$\gamma = C_P/C_V$ (ratio of molar specific heats):

Monatomic: $\gamma = 5/3 \approx 1.67$
Diatomic: $\gamma = 7/5 = 1.4$

Physical picture#

Expansion ($V_f > V_i$): Gas does work on surroundings; $W > 0$. With $Q = 0$, first law gives $\Delta E_{\text{int}} = -W < 0$ → internal energy decreases → temperature drops.
Compression ($V_f < V_i$): Work done on gas; $W < 0$ → $\Delta E_{\text{int}} > 0$ → temperature rises.

On a $p$–$V$ diagram, an adiabat is steeper than an isotherm: $p \propto 1/V^\gamma$ vs $p \propto 1/V$.

Poll: Weather balloon—identify the process

A weather balloon is filled with helium and released. The balloon material is an excellent insulator. As it rises, what kind of process does the gas undergo?

(A) Isobaric
(B) Isochoric
(C) Isothermal
(D) Adiabatic

Poll: Balloon rising—what happens to $T$?

For the weather balloon above, the pressure decreases as it rises. What happens to the temperature of the helium gas?

(A) $T$ increases
(B) $T$ decreases
(C) $T$ remains the same

Free expansion (exception)#

In a free expansion (gas expands into a vacuum), $Q = 0$ and $W = 0$ (no external pressure to work against). So $\Delta E_{\text{int}} = 0$ → $T$ stays constant.

$p_i V_i = p_f V_f$ (ideal gas law; $T$ and $n$ unchanged).
$pV^\gamma = \text{constant}$ does not apply—free expansion is not a quasi-static process.

Poll: Adiabatic expansion

An ideal gas expands adiabatically. The final temperature $T_f$ is ___ the initial temperature $T_i$.

(A) Greater than
(B) Less than
(C) Equal to

Summary#

Adiabatic: $Q = 0$; $pV^\gamma = \text{constant}$; $TV^{\gamma-1} = \text{constant}$.
$\gamma = C_P/C_V$; monatomic $\gamma = 5/3$, diatomic $\gamma = 7/5$.
Expansion → cooling; compression → heating.
Free expansion: $Q = W = 0$ → $T$ constant; $pV^\gamma$ does not apply.