Air Squeezed Without Heat

solved AdiabaticProcess

Q. Air at 1 atm $P_{0} = 1$ atm fills 8 L $V_{0} = 8$ L of a diesel engine's cylinder. It is compressed so fast that no heat escapes, until its pressure is 32 atm $P_{1} = 32$ atm. Taking γ for air as 1.4 $γ = 1.4$ , what volume does it then fill, in litres?

Inputs

starting pressure

P_{0}

1 atm

starting volume

V_{0}

8 L

final pressure

P_{1}

32 atm

gamma

γ

1.4

Outputs

final volume

V_{1}

How it's solved

Solved with AdiabaticProcess

Given

$P_{0} = 1$ atm
$V_{0} = 8$ L
$P_{1} = 32$ atm
$γ = 1.4$

Find $V_{1}$ .

Working

$P_{0} = 1$ atm $= 101325$ Pa
$V_{0} = 8$ L $= 0.008$ m³
$P_{1} = 32$ atm $= 3.2424 \times 10^{6}$ Pa
pv to the gamma is kept(compressed) $P_{0} \cdot {V_{0}}^{γ} = P_{1} \cdot {V_{1}}^{γ}$
$101325 \cdot {0.008}^{1.4} = 3.2424 \times 10^{6} \cdot {V_{1}}^{1.4}$
$V_{1} = 0.673$ L
$V_{1} = 6.72950096 \times 10^{−4}$ m³ $= 0.672950096$ L
$V_{1} \approx 0.673$ Lto 3 significant figures

Conclusion

With no heat let out the gas warms as it is squeezed, so its pressure climbs faster than the volume falls, and a 32-fold rise in pressure needs only a 12-fold fall in volume.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :air_squeezed_without_heat_to_thirty_two_atmospheres do4  kind :numerical5  describes "Air at %{starting_pressure} fills %{starting_volume} of a diesel engine's cylinder. It is compressed so " \6            "fast that no heat escapes, until its pressure is %{final_pressure}. Taking γ for air as %{gamma}, " \7            "%{final_volume}, in litres?"8  given starting_pressure: "1 atm"9  given starting_volume: "8 L"10  given final_pressure: "32 atm"11  given gamma: "1.4"12  asks final_volume: "what volume does it then fill"13end

solution.rb

1require_relative "../questions/air_squeezed_without_heat_to_thirty_two_atmospheres"2require_relative "../science/thermodynamics/adiabatic_process"34solve :air_squeezed_without_heat_to_thirty_two_atmospheres, using: AdiabaticProcess do5  objects :air6  air.goes(:compressed)78  take starting_pressure: air.initial_pressure, starting_volume: air.initial_volume9  take final_pressure: air.final_pressure, gamma: air.ratio_of_specific_heats1011  final_volume = air.final_volume12  answer final_volume:, in: "L", figures: 31314  conclusion "With no heat let out the gas warms as it is squeezed, so its pressure climbs faster than the " \15             "volume falls, and a 32-fold rise in pressure needs only a 12-fold fall in volume."16end

playground.rb

1require_relative "../solutions/air_squeezed_without_heat_to_thirty_two_atmospheres"23play_with :air_squeezed_without_heat_to_thirty_two_atmospheres do4  title "Air Squeezed Without Heat"5end

The laws it drew on

quantity	written	units	within	from
pressure	$P$	Pa	0 … ∞	AdiabaticProcess
volume	$V$	m³	0 … ∞	AdiabaticProcess
ratio of specific heats	$γ$	—	—	AdiabaticProcess
starting pressure	$P_{1}$	Pa	0 … ∞	AdiabaticProcess
final pressure	$P_{2}$	Pa	0 … ∞	AdiabaticProcess
starting volume	$V_{1}$	m³	0 … ∞	AdiabaticProcess
final volume	$V_{2}$	m³	0 … ∞	AdiabaticProcess
pressure of air at start of compressed	$P_{0}$	Pa	0 … ∞
volume of air at start of compressed	$V_{0}$	m³	0 … ∞
ratio of specific heats of air	$γ$	—	—
pressure of air at end of compressed	$P_{1}$	Pa	0 … ∞
volume of air at end of compressed	$V_{1}$	m³	0 … ∞

P_{1} \equiv P_{i} (start of motion)

starting pressure

P_{2} \equiv P_{i} (end of motion)

final pressure

V_{1} \equiv V_{i} (start of motion)

starting volume

V_{2} \equiv V_{i} (end of motion)

final volume

P_{0} \cdot {V_{0}}^{γ} = P_{1} \cdot {V_{1}}^{γ}

pv to the gamma is kept(compressed)