Formula sheet

Vacuum and Rarefied Gas Systems Formula Sheet

Vacuum system formulas for pressure units, pumping speed, conductance, throughput, pumpdown, gas load, leak rate, outgassing, mean free path, Knudsen number, and validation.

Branch: Engineering Physics
Content: Formula sheet
Updated: Jun 23, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects first-pass relationships used to size, review and diagnose vacuum systems. It focuses on pressure, gas load, effective pumping speed, conductance, pumpdown time, leak rate, outgassing, pressure-rise tests, mean free path and rarefied-flow regime checks.

Use these formulas with stated gas species, temperature, pressure range, chamber volume, pump type, valve state, conductance path, gauge location and surface condition. Vacuum calculations are weak when the measurement boundary is unclear.

Notation

Symbol	Meaning	Typical unit
$p$	absolute pressure	Pa
$V$	chamber or isolated volume	$\text{m}^3$
$T$	absolute temperature	K
$S_p$	pump rated speed	$\text{m}^3/\text{s}$
$S_{eff}$	effective pumping speed at chamber	$\text{m}^3/\text{s}$
$C$	conductance of duct, valve, baffle or restriction	$\text{m}^3/\text{s}$
$Q$	throughput, gas load or leak rate	$\text{Pa m}^3/\text{s}$
$q_A$	outgassing rate per area	$\text{Pa m}^3/(\text{s m}^2)$
$A$	outgassing surface area	$\text{m}^2$
$\lambda$	molecular mean free path	m
$d_m$	molecular diameter	m
$L$	characteristic length	m
$Kn$	Knudsen number	dimensionless
$k_B$	Boltzmann constant	$\text{J/K}$

Pressure Units and Absolute Pressure

Pressure must be absolute for vacuum physics:

p_{abs}=p_{atm}+p_{gauge}

Common conversions:

1\ \text{mbar}=100\ \text{Pa}

1\ \text{Torr}=133.322\ \text{Pa}

1\ \text{Pa}=7.5006\times10^{-3}\ \text{Torr}

Throughput conversion:

1\ \text{mbar L/s}=0.1\ \text{Pa m}^3/\text{s}

Use

Do not mix gauge pressure, absolute pressure and pressure below atmosphere. Vacuum gauges, leak detectors and pump curves usually refer to absolute pressure.

Ideal Gas Inventory

Moles in a volume:

\displaystyle n=\frac{pV}{RT}

Molecules in a volume:

\displaystyle N=\frac{pV}{k_BT}

Pressure change from molecule or mole change:

\displaystyle \Delta p=\frac{\Delta nRT}{V}

Use

This inventory check is useful for interpreting venting, trapped volume and pressure-rise tests. It does not describe adsorption, desorption, condensation or reactive gas removal.

Throughput and Pumping Speed

Throughput:

Q=pS

For a gas load $Q_g$ removed by effective pumping speed $S_{eff}$ , steady pressure is:

\displaystyle p_{\infty}=\frac{Q_g}{S_{eff}}

Required effective speed for a target pressure:

\displaystyle S_{eff}\ge\frac{Q_g}{p_{target}}

Use

Throughput uses pressure at the location where the speed is defined. A pump inlet speed is not the same as chamber effective speed when conductance restrictions exist.

Effective Pumping Speed and Conductance

Series conductance relation:

\displaystyle \frac{1}{S_{eff}}=\frac{1}{S_p}+\frac{1}{C}

Equivalent form:

\displaystyle S_{eff}=\frac{S_pC}{S_p+C}

If several conductance restrictions are in series:

\displaystyle \frac{1}{C_{eq}}=\sum_i\frac{1}{C_i}

Parallel paths:

C_{eq}=\sum_i C_i

Use

Conductance depends on gas species, temperature, pressure regime and geometry. In high vacuum, a short restrictive duct can dominate the whole system even when the pump nameplate speed looks large.

Pumpdown with Constant Gas Load

For a volume with constant effective speed and constant gas load:

\displaystyle \frac{dp}{dt}=-\frac{S_{eff}}{V}p+\frac{Q_g}{V}

Steady pressure:

\displaystyle p_{\infty}=\frac{Q_g}{S_{eff}}

Pressure versus time:

p(t)=p_{\infty}+\left(p_0-p_{\infty}\right)e^{-S_{eff}t/V}

Time to reach target pressure:

\displaystyle t=\frac{V}{S_{eff}}\ln\left(\frac{p_0-p_{\infty}}{p_t-p_{\infty}}\right)

This expression is meaningful only when $p_t>p_{\infty}$ .

Use

Real pumpdown curves often have multiple regions: roughing, high-vacuum valve opening, water-vapor removal, outgassing tail and possible virtual leaks. Use the exponential model for screening and baseline comparison, not as proof that every gas load is constant.

Pressure-Rise Gas Load

For an isolated volume:

\displaystyle Q_{rise}=V\frac{\Delta p}{\Delta t}

If pressure rise is nonlinear, use local slope:

\displaystyle Q_{rise}=V\frac{dp}{dt}

Total gas load can include:

Q_g=Q_{leak}+Q_{outgas}+Q_{permeation}+Q_{virtual}+Q_{process}

Use

Pressure-rise testing measures total gas load. It does not separate external leakage from outgassing, permeation, trapped volumes or desorption unless supported by additional evidence.

Outgassing Load

Surface outgassing load:

Q_{outgas}=q_AA

Multiple material regions:

Q_{outgas,total}=\sum_i q_{A,i}A_i

Base pressure from outgassing:

\displaystyle p_{base}\approx\frac{Q_{outgas}+Q_{leak}}{S_{eff}}

Use

Outgassing rates depend on material, cleaning, humidity exposure, bakeout, temperature, surface area, time under vacuum and contamination history. A clean metal chamber and a loaded chamber with polymers, cables or adhesives can have very different gas loads.

Leak-Rate Allocation

External leak contribution from pressure rise:

Q_{leak}\le Q_{rise}

Fraction of total gas load explained by measured external leak:

\displaystyle f_{leak}=\frac{Q_{leak}}{Q_{rise}}

Guarded acceptance:

Q_{meas}+U_Q\le Q_{limit}

Use

A helium leak detector checks external leak paths. It does not fully measure water vapor outgassing, trapped gas, virtual leaks or gas released from warm surfaces during operation.

Mean Free Path and Knudsen Number

Mean free path for a simple gas model:

\displaystyle \lambda=\frac{k_BT}{\sqrt{2}\pi d_m^2p}

Knudsen number:

\displaystyle Kn=\frac{\lambda}{L}

Common regime screen:

Regime	Approximate $Kn$ range	Engineering implication
continuum	$Kn<0.01$	ordinary continuum models often usable
slip	$0.01<Kn<0.1$	wall slip may matter
transitional	$0.1<Kn<10$	continuum intuition becomes weak
free molecular	$Kn>10$	wall interactions dominate gas transport

Use

Choose $L$ from the feature that controls the physics: sensor gap, aperture, duct diameter, microchannel, coating gap or distance from source surface. A chamber can be continuum in one feature and molecular in another.

Viscous-Flow Screening

Reynolds number:

\displaystyle Re=\frac{\rho vD}{\mu}

Continuity:

Q_v=vA

Mass flow:

\dot{m}=\rho Q_v

Use

These continuum expressions are useful during roughing, venting, purge and gas-feed operation. They may be inappropriate in high-vacuum molecular flow.

Gauge Location and Pressure Drop

If gas load flows through a conductance path:

\displaystyle \Delta p=\frac{Q}{C}

Approximate local pressure:

\displaystyle p_{local}=p_{pump}+ \frac{Q}{C}

Use

This simple relation explains why a pump-side gauge can report an acceptable pressure while the sensitive region remains worse. It assumes the conductance and gas load are defined for the relevant gas and flow regime.

Thermal and Process Effects

Gas load often changes with temperature:

Q_g(T)\ne Q_g(T_{room})

Thermal expansion or stress checks may be needed for chambers, windows and feedthroughs:

\sigma_{thermal}\approx E\alpha\Delta T

Heat load from radiation or conduction can change outgassing and local pressure:

\dot{Q}_{heat}=q''A

Use

Vacuum release should state temperature condition. A chamber accepted cold may fail during bakeout, plasma exposure, x-ray operation, electronics heating or thermal-vacuum cycling.

Worked Example 1: Conductance-Limited Pumping Speed

A chamber uses a pump rated for nitrogen at:

S_p=0.20\ \text{m}^3/\text{s}

The connecting valve, screen and duct have estimated conductance:

C=0.030\ \text{m}^3/\text{s}

Effective pumping speed:

\displaystyle S_{eff}=\frac{S_pC}{S_p+C}=\frac{(0.20)(0.030)}{0.20+0.030}=0.0261\ \text{m}^3/\text{s}

Fraction of pump nameplate speed available at the chamber:

\displaystyle \frac{S_{eff}}{S_p}=\frac{0.0261}{0.20}=0.131

Engineering comment: only about 13 percent of the pump nameplate speed is available at the chamber. The conductance path, not the pump, controls performance.

Worked Example 2: Pumpdown Time with Base-Pressure Limit

Use:

V=0.12\ \text{m}^3

S_{eff}=0.0143\ \text{m}^3/\text{s}

Q_g=2.0\times10^{-4}\ \text{Pa m}^3/\text{s}

Base pressure:

\displaystyle p_{\infty}=\frac{Q_g}{S_{eff}}=\frac{2.0\times10^{-4}}{0.0143}=0.0140\ \text{Pa}

Time to pump from:

p_0=100\ \text{Pa}

to:

p_t=1.0\ \text{Pa}

is:

\displaystyle t=\frac{0.12}{0.0143}\ln\left(\frac{100-0.0140}{1.0-0.0140}\right)=38.8\ \text{s}

Engineering comment: this is an idealized high-speed calculation. Real pumpdown from atmosphere includes roughing limits, valve sequencing, water vapor, conductance changes, gauge range changes and outgassing tails. Compare model and measured curve by region.

Worked Example 3: Pressure-Rise Leak and Gas Load

An isolated chamber has:

V=0.080\ \text{m}^3

Pressure rises by:

\Delta p=6.0\ \text{Pa}

over:

\Delta t=300\ \text{s}

Gas load:

\displaystyle Q_{rise}=V\frac{\Delta p}{\Delta t}=0.080\frac{6.0}{300}=1.60\times10^{-3}\ \text{Pa m}^3/\text{s}

In mbar L/s:

\displaystyle Q_{rise}=\frac{1.60\times10^{-3}}{0.1}=1.60\times10^{-2}\ \text{mbar L/s}

Engineering comment: this is total gas load. If helium leak testing finds only $1.0\times10^{-4}\ \text{mbar L/s}$ of external leakage, then most of the pressure rise is likely outgassing, trapped gas or permeation, not a single open leak.

Worked Example 4: Outgassing-Limited Base Pressure

A small test fixture has:

A=4.0\ \text{m}^2

Average outgassing estimate:

q_A=1.0\times10^{-6}\ \text{Pa m}^3/(\text{s m}^2)

External leak allowance:

Q_{leak}=2.0\times10^{-5}\ \text{Pa m}^3/\text{s}

Effective pumping speed:

S_{eff}=0.020\ \text{m}^3/\text{s}

Outgassing load:

Q_{outgas}=q_AA=(1.0\times10^{-6})(4.0)=4.0\times10^{-6}\ \text{Pa m}^3/\text{s}

Total gas load:

Q_g=4.0\times10^{-6}+2.0\times10^{-5}=2.4\times10^{-5}\ \text{Pa m}^3/\text{s}

Base pressure estimate:

\displaystyle p_{base}=\frac{2.4\times10^{-5}}{0.020}=1.2\times10^{-3}\ \text{Pa}

Engineering comment: the leak allowance dominates this example. If the pressure requirement were $1.0\times10^{-3}\ \text{Pa}$ , the design would need more speed, lower leak allowance, lower outgassing or a guarded acceptance rule.

Worked Example 5: Knudsen Regime Near a Vacuum Sensor Gap

Air is approximated with:

d_m=0.37\ \text{nm}=0.37\times10^{-9}\ \text{m}

At:

T=293\ \text{K}

and:

p=5.0\ \text{Pa}

mean free path is:

\displaystyle \lambda=\frac{k_BT}{\sqrt{2}\pi d_m^2p}

\displaystyle \lambda=\frac{(1.381\times10^{-23})(293)}{\sqrt{2}\pi(0.37\times10^{-9})^2(5.0)}=1.33\times10^{-3}\ \text{m}

For a sensor gap:

L=1.0\ \text{mm}=1.0\times10^{-3}\ \text{m}