= Boyle's law
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{title2=$pV=\mathrm{constant}$}
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= Boyle law
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<Boyle's law> states that a fixed <amount of substance> of an <ideal gas> at constant absolute <thermodynamic temperature> has constant product of <absolute pressure> and total <volume (thermodynamics)>:
$$
pV=C,\qquad p_1V_1=p_2V_2,\qquad p_2=p_1\frac{V_1}{V_2}.
$$
Here $p$ is measured relative to a <vacuum>, not relative to the surrounding <atmospheric pressure>. Halving the <volume (thermodynamics)> doubles the <absolute pressure>; doubling the <volume (thermodynamics)> halves it. A <real gas> approximately follows <Boyle's law> when it is sufficiently dilute and far from a <phase transition>.<University Physics Volume 2 by Samuel J. Ling and others/2.1 Molecular Model of an Ideal Gas>[§2.1, The Gas Laws, Figure 2.4, and equations 2.1–2.2: fixed temperature and particle number, absolute pressure, and the dilute-gas limit]{ref}
For an original calculated example, suppose a sealed <gas> sample has $p_1=100\,\mathrm{kPa}$ and $V_1=100\,\mathrm{cm^3}$. After <isothermal compression> to $V_2=50\,\mathrm{cm^3}$,
$$
p_2=(100\,\mathrm{kPa})\frac{100\,\mathrm{cm^3}}{50\,\mathrm{cm^3}}
=200\,\mathrm{kPa}.
$$
Both states have $C=10\,000\,\mathrm{kPa\,cm^3}=10\,\mathrm J$, since $1\,\mathrm{kPa\,cm^3}=10^{-3}\,\mathrm J$. The <volume (thermodynamics)> ratio is dimensionless, so using the same <volume (thermodynamics)> units in numerator and denominator leaves the <pressure> in kilopascals. Although $pV$ has units of <energy>, its value is not the compression <work>: <isothermal ideal-gas work> involves an <integral> along the path.
\Image[/boyles-law-pressure-volume-linearization.png]
{title=Boyle's law: a pressure-volume curve becomes a straight line against reciprocal volume}
The blue curves show calculated <ideal gas> states, not measurements. The left panel is a branch of a <hyperbola>; the right panel plots exactly the same <absolute pressures> against $1/V$. The dashed line uses <gauge pressure> with a stipulated constant <atmospheric pressure> of $100\,\mathrm{kPa}$. Its displaced intercept explains why <absolute pressure in Boyle's law> matters.
A short derivation starts from the <ideal gas law>, $pV=nRT$. Fixing the <amount of substance> $n$ and absolute <thermodynamic temperature> $T$ fixes $C=nRT$, where $R$ is the <molar gas constant>. The microscopic picture in <ideal gas pressure from wall collisions> is equally appealing: at fixed <temperature>, the typical particle <kinetic energy> stays fixed, while compression increases the <number density> and wall <momentum> transfer per unit <surface area> and time. A larger <absolute pressure> follows without requiring hotter particles.
The constraints are essential. A leak changes the <amount of substance>. Rapid compression can raise the <temperature> before <heat transfer> restores <thermal equilibrium>, as in the <isothermal gas syringe experiment>. Molecular size and interactions in a dense <real gas> invalidate the <ideal gas> approximation; the <Van der Waals equation> describes one such correction. Equal endpoint <temperatures> allow an endpoint test of <Boyle's law>, but do not establish that every intervening state lies on an <isothermal process>.
For fixed <mass> $m$, <mass density> satisfies $\rho=m/V=mp/C$, so it is proportional to <absolute pressure>. In a dilute bubbly flow, <Boyle's law> allows bubble <volume (thermodynamics)> and its contribution to <buoyancy> to increase as <hydrostatic pressure> decreases, provided <gas> <mass> is retained, dissolution is negligible, and the <temperature> stays approximately fixed. Its usefulness is that a simple ratio predicts a substantial size change without tracking individual <molecules>.
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