Boyle's law ($pV=\mathrm{constant}$)

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:Here $p$ is measured relative to a vacuum, not relative to the surrounding atmospheric pressure. Halving the volume doubles the absolute pressure; doubling the volume halves it. A real gas approximately follows Boyle's law when it is sufficiently dilute and far from a phase transition.^[ref]

For an original calculated example, suppose a sealed gas sample has $p_1=100\mathrm{kPa}$ and $V_1=100\mathrm{c}{\mathrm{m}}^3$. After isothermal compression to $V_2=50\mathrm{c}{\mathrm{m}}^3$,Both states have $C=10000\mathrm{kPa}\mathrm{c}{\mathrm{m}}^3=10\mathrm{J}$, since $1\mathrm{kPa}\mathrm{c}{\mathrm{m}}^3=10^{-3}\mathrm{J}$. The volume ratio is dimensionless, so using the same volume 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.

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 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.

With gauge pressure $p_g$ referenced to atmospheric pressure $p_{\mathrm{atm}}$, absolute pressure is $p=p_g+p_{\mathrm{atm}}$. Thus Boyle's law becomesEven a constant atmospheric pressure cannot be dropped from these sums. For the calculated $100\mathrm{kPa}$ to $200\mathrm{kPa}$ example, take $p_{\mathrm{atm}}=100\mathrm{kPa}$. The initial and final gauge pressures are $0$ and $100\mathrm{kPa}$; their products with the respective volume are $0$ and $5000\mathrm{kPa}\mathrm{c}{\mathrm{m}}^3$, which are unequal. Zero gauge pressure means balance with the surroundings, not absence of gas or absence of wall momentum transfer.

For a frictionless piston of surface area $A$ with its outer face exposed to atmospheric pressure, the additional inward force required for mechanical equilibrium is $F=(p-p_{\mathrm{atm}})A=p_{g}A$. This is why a mechanical load can reveal a pressure difference even though the equation of state requires absolute pressure. With piston friction, the load alone no longer determines $p_{g}A$; a direct pressure sensor avoids that particular inference.

A linearized plot of Boyle's law uses $x=1/V$ as its horizontal coordinate. Then absolute pressure satisfies $p=Cx$: the slope is $C=nRT$, and the ideal intercept is zero. In the figure, $(V,p)=(100,100)$ and $(50,200)$ in $\mathrm{c}{\mathrm{m}}^3$ and $\mathrm{kPa}$ become $(x,p)=(0.010,100)$ and $(0.020,200)$ in $\mathrm{c}{\mathrm{m}}^{-3}$ and $\mathrm{kPa}$. Both give slope $10000\mathrm{kPa}\mathrm{c}{\mathrm{m}}^3$. The Vernier experiment includes this reciprocal-volume graph as its extension.^[ref]

At constant atmospheric pressure, gauge pressure instead obeys $p_g=Cx-p_{\mathrm{atm}}$, with the same slope and a negative intercept. A nonzero fitted intercept in an intended absolute pressure plot can indicate sensor bias or an incorrect volume correction, but is not by itself a unique diagnosis. The origin is a mathematical extrapolation to infinite volume and zero absolute pressure, rather than an additional measured state.

Vernier's Boyle's Law: Pressure-Volume Relationship in Gases, Experiment 6 of Chemistry with Vernier, specifies a graduated $20\mathrm{mL}$ gas syringe, a pressure sensor and LabQuest recording. Preparation sets the disconnected syringe to $10.0\mathrm{mL}$, then attaches it to the sensor valve, trapping air. At several plunger positions, the displayed pressure is allowed to stabilize before the paired volume is entered. The analysis tests whether $pV$ remains approximately constant. It adds $0.8\mathrm{mL}$ to each syringe reading for the sensor's internal volume; this correction belongs to that apparatus. The worksheet supplies a blank results table, not measured values.^[ref]

The manufacturer's Gas Pressure Sensor User Manual identifies a membrane sensor referenced to a vacuum, hence an absolute pressure readout. Its calibration section uses the actual local atmospheric pressure for an ambient reference check, rather than a weather report corrected to sea level. Checking this baseline before sealing separates an offset error from compression physics.^[ref]

Additional controls follow from the model. Keep the same sealed amount of substance, connections and ambient temperature; persistent pressure drift at held volume after thermal equilibrium suggests a leak or another uncontrolled change. Small steps and waiting allow compression heat to leave the gas; avoiding hand warming reduces another heat input. Compare settled readings when approaching a position from either direction. Plunger friction affects the applied force and position reproducibility, so record actual pressure and actual plunger position rather than predicting pressure from the load. These are proposed controls, not reported experimental outcomes.

Nearly constant $pV$ across equilibrated states supports the fixed-temperature ideal gas approximation over the tested range. Stabilized pressure alone does not demonstrate uniform or constant temperature; the isothermal gas syringe experiment adds a temperature readout to distinguish a transient from its cooled endpoint. Finite sensor accuracy and uncertain total volume limit the test, and agreement does not establish ideal behavior at every pressure. Repeated return readings help distinguish reproducibility from a one-direction trend.

In a gas syringe, the graduated barrel reading $V_{\mathrm{read}}$ can omit a fixed connecting or sensor volume $V_d$ occupied by the same sealed gas. For a common settled temperature, Boyle's law then requiresThis second relation makes a graph of $1/p$ against $V_{\mathrm{read}}$ a straight line; its intercept divided by its slope estimates $V_d$ under these assumptions. Unequal temperatures between the barrel and the connecting space require a different correction, analogous to the gas thermometer dead-space correction.

For an original hypothetical example, let $V_d=1\mathrm{c}{\mathrm{m}}^3$, $V_{\mathrm{read},1}=20\mathrm{c}{\mathrm{m}}^3$, $p_1=100\mathrm{kPa}$, and $V_{\mathrm{read},2}=10\mathrm{c}{\mathrm{m}}^3$. The total volume falls from $21$ to $11\mathrm{c}{\mathrm{m}}^3$, so $p_2=100(21/11)\mathrm{kPa}\approx 190.91\mathrm{kPa}$, rather than $200\mathrm{kPa}$. The omitted volume makes the total compression ratio smaller than the barrel-reading ratio. These stipulated values are independent of the manufacturer's apparatus correction and are not measurements.