Exercise set

Vacuum Gauge Pressure Measurement and Pumpdown Exercises

Solved vacuum gauge exercises for pressure units, mean free path, Knudsen number, gas correction, conductance, pumpdown and release evidence.

Branch: Engineering Physics
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

These exercises focus on vacuum gauges as local pressure sensors in conductance-limited systems. They cover pressure units, mean free path, Knudsen number, gas correction, gauge range, conductance pressure drop, pumpdown trend, leak-rate acceptance and release evidence.

Use the calculations as screening checks. Real vacuum measurements also need gauge type, gas species, calibration, gauge location, conductance path, pump state, temperature, outgassing, transient condition and acceptance criteria tied to the process volume.

How to use these exercises

Use the set as a vacuum measurement release review. Exercises 1 to 5 establish pressure units, pressure decades, mean free path and flow regime. Exercises 6 to 10 check gas correction, gauge range, conductance pressure drop, chamber pressure and effective pumping speed. Exercises 11 to 17 add pumpdown time, pressure-rise leak rate, acceptance margin, gauge offset, gauge agreement and guard bands. Exercise 18 combines the measurement-boundary evidence into a release decision.

Before calculating, name the gauge type, gas species, calibration basis, gauge location, chamber volume, conductance path, valve state, pump state and acceptance boundary. A vacuum reading is local: a pump-port gauge, a chamber gauge and a gauge behind a restriction may not support the same release decision. The engineering comment below each exercise identifies the boundary condition that must be controlled before accepting the pressure value.

Release Evidence Notes

Vacuum measurement evidence should state gauge type, calibration basis, gas correction, location, chamber volume, pump state, conductance restrictions, isolation-valve state, process gas, pressure range and whether the reading is steady-state or transient. A pressure number is local unless the system boundary proves otherwise.

The evidence package should separate gauge validity, system validity and acceptance validity. Gauge validity covers calibration, range, gas basis, offset and uncertainty. System validity covers location, conductance, pump configuration, valve state, outgassing and leak behavior. Acceptance validity covers whether the corrected chamber condition meets the process limit with guard band and documented configuration.

Pumpdown evidence should include the curve, not only the final pressure. A smooth curve, pressure plateau, pressure-rise test or gauge disagreement can distinguish normal outgassing, virtual leaks, real leaks, conductance limits and wrong valve state.

Engineering Boundary Notes

The models below use idealized gas, conductance and pumpdown screens. They do not replace molecular-flow conductance calculations, species-specific calibration, outgassing models, leak detection procedure, gauge cross-calibration or acceptance testing on the actual chamber. Treat pass results as configuration screens, not as proof that every process condition is acceptable.

The main boundary is species and range. Pirani, capacitance, cold-cathode and ion gauges respond differently across gases and pressure ranges. The second boundary is conductance: in molecular flow, a small line or valve can make pump-side pressure look acceptable while chamber pressure remains too high.

Common Release Mistakes

reading pressure near the pump and assuming it equals chamber pressure;
using an ion gauge gas correction for the wrong gas;
comparing Pirani and capacitance gauges without range and gas-basis review;
declaring pumpdown complete from a smooth curve while a leak or virtual leak remains;
ignoring conductance limits through long small-diameter lines;
omitting valve state and gauge location from release evidence.

Another common mistake is treating gauge agreement as proof without checking whether both gauges are valid in the same range and gas. Two gauges can agree and still be wrong for the process gas or location; they can also disagree because one is outside its useful range.

Do not use final pressure alone for contamination-sensitive, plasma, high-voltage or coating processes. Acceptance should include corrected pressure, pumpdown behavior, leak-rate or pressure-rise evidence, gauge placement and uncertainty guard band.

Scenario Map

The exercises move from unit conversion and gas regime to gauge correction, conductance drops, pumpdown, leak rate, placement error and release acceptance.

Exercise 1: Torr to Pascal Conversion

A vacuum gauge reads $7.5\times10^{-3}\ \text{Torr}$ . Use $1\ \text{Torr}=133.322\ \text{Pa}$ . Convert to pascals.

Solution

p=7.5\times10^{-3}(133.322)=1.00\ \text{Pa}

Engineering Comment

Pressure-unit conversion is simple, but release records should state the unit explicitly because vacuum ranges span many decades.

Plausibility Check

One pascal is about $7.5$ millitorr, matching the conversion.

Exercise 2: Pascal to Millibar

A capacitance gauge reads $0.42\ \text{Pa}$ . Convert to millibar using $1\ \text{mbar}=100\ \text{Pa}$ .

Solution

p=\dfrac{0.42}{100}=0.0042\ \text{mbar}

Engineering Comment

Gauge displays often mix Torr, Pa and mbar. Unit mistakes can create false pass/fail decisions.

Plausibility Check

Pascals are one hundredth of a millibar, so a sub-pascal reading is a few thousandths of a mbar.

Exercise 3: Pressure Decade Ratio

Pumpdown reduces pressure from $1.0\ \text{Pa}$ to $1.0\times10^{-3}\ \text{Pa}$ . How many pressure decades is this?

Solution

N=\log_{10}\left(\dfrac{1.0}{1.0\times10^{-3}}\right)=3

Engineering Comment

Vacuum acceptance often depends on logarithmic pressure progress, not linear difference.

Plausibility Check

From $10^0$ to $10^{-3}$ is three decades.

Exercise 4: Mean Free Path Screen

Mean free path is $6.8\ \text{mm}$ at $1\ \text{Pa}$ for a given gas. Estimate mean free path at $0.02\ \text{Pa}$ assuming inverse pressure scaling.

Solution

\lambda_2=\lambda_1\dfrac{p_1}{p_2}=6.8\left(\dfrac{1}{0.02}\right)=340\ \text{mm}

Engineering Comment

As pressure falls, molecular effects dominate and gauge placement can become more sensitive to conductance paths.

Plausibility Check

Reducing pressure by a factor of 50 increases mean free path by a factor of 50.

Exercise 5: Knudsen Number

A channel has characteristic length $L=25\ \text{mm}$ and mean free path $\lambda=340\ \text{mm}$ . Compute Knudsen number.

Solution

Kn=\dfrac{\lambda}{L}=\dfrac{340}{25}=13.6

Engineering Comment

A large Knudsen number indicates molecular-flow behavior, where continuum intuition and pressure-drop assumptions can fail.

Plausibility Check

The mean free path is much larger than the channel, so $Kn$ should be much greater than one.

Exercise 6: Gauge Gas Correction

An ion gauge calibrated for nitrogen reads $2.0\times10^{-5}\ \text{Pa}$ . For the process gas, the correction factor is 1.6. Estimate corrected pressure.

Solution

p_\mathrm{corr}=1.6(2.0\times10^{-5})=3.2\times10^{-5}\ \text{Pa}

Engineering Comment

Ion gauges are species-sensitive. The gas basis must be recorded whenever the reading supports release.

Plausibility Check

A correction factor above one should raise the reported pressure.

Exercise 7: Gauge Range Check

A Pirani gauge is specified down to $1.0\times10^{-1}\ \text{Pa}$ . The process acceptance limit is $2.0\times10^{-3}\ \text{Pa}$ . Is the Pirani gauge sufficient?

Solution

The acceptance pressure is

\dfrac{1.0\times10^{-1}}{2.0\times10^{-3}}=50

times below the gauge lower useful range. The gauge is not sufficient.

Engineering Comment

The right gauge type depends on pressure range and gas. A stable displayed number outside the useful range is not release evidence.

Plausibility Check

The required pressure is two orders of magnitude below the Pirani lower range.

Exercise 8: Pressure Drop across Conductance

Throughput is $Q=3.0\times10^{-4}\ \text{Pa m}^3/\text{s}$ and conductance is $C=0.020\ \text{m}^3/\text{s}$ . Estimate pressure difference.

Solution

\Delta p=\dfrac{Q}{C}=\dfrac{3.0\times10^{-4}}{0.020}=0.015\ \text{Pa}

Engineering Comment

Even when a gauge is accurate locally, conductance restrictions can make chamber pressure different from pump-side pressure.

Plausibility Check

Small throughput divided by modest conductance gives hundredths of a pascal.

Exercise 9: Chamber Pressure from Pump-Side Gauge

A pump-side gauge reads $0.020\ \text{Pa}$ and estimated conductance drop from chamber to gauge is $0.015\ \text{Pa}$ . Estimate chamber pressure.

Solution

p_\mathrm{chamber}=0.020+0.015=0.035\ \text{Pa}

Engineering Comment

Gauge placement can change the release decision when the acceptance limit applies to the chamber, not the pump port.

Plausibility Check

The chamber pressure is higher than the pump-side reading, as expected for gas flowing toward the pump.

Exercise 10: Effective Pumping Speed

A pump has speed $0.080\ \text{m}^3/\text{s}$ and connecting conductance is $0.040\ \text{m}^3/\text{s}$ . Estimate effective pumping speed at the chamber.

Solution

\dfrac{1}{S_\mathrm{eff}}=\dfrac{1}{S}+\dfrac{1}{C} =\dfrac{1}{0.080}+\dfrac{1}{0.040}=37.5

S_\mathrm{eff}=0.0267\ \text{m}^3/\text{s}

Engineering Comment

Conductance can dominate pumpdown. A larger pump does little if the restriction remains small.

Plausibility Check

The effective speed is below both pump speed and conductance.

Exercise 11: Ideal Pumpdown Time Constant

Chamber volume is $0.60\ \text{m}^3$ and effective pumping speed is $0.030\ \text{m}^3/\text{s}$ . Estimate ideal time constant.

Solution

\tau=\dfrac{V}{S}=\dfrac{0.60}{0.030}=20\ \text{s}

Engineering Comment

Real pumpdown slows because of outgassing, conductance changes and gauge range transitions.

Plausibility Check

A chamber volume twenty times the pumping speed gives a twenty-second ideal time constant.

Exercise 12: Exponential Pumpdown

Starting pressure is $100\ \text{Pa}$ , target pressure is $1.0\ \text{Pa}$ and ideal time constant is 20 s. Estimate ideal pumpdown time using $p=p_0e^{-t/\tau}$ .

Solution

t=\tau\ln\left(\dfrac{p_0}{p}\right)=20\ln(100)=92.1\ \text{s}

Engineering Comment

If the measured curve is much slower, investigate leaks, outgassing, restrictions or incorrect valve state.

Plausibility Check

Two pressure decades require about $4.6$ time constants.

Exercise 13: Leak Rate from Pressure Rise

An isolated $0.15\ \text{m}^3$ chamber rises from $0.020\ \text{Pa}$ to $0.080\ \text{Pa}$ in 600 s. Estimate leak or outgassing throughput.

Solution

Q=V\dfrac{\Delta p}{\Delta t} =0.15\dfrac{0.080-0.020}{600} =1.5\times10^{-5}\ \text{Pa m}^3/\text{s}

Engineering Comment

A pressure-rise test measures total gas load from leaks, permeation and outgassing unless the procedure separates them.

Plausibility Check

Small volume and slow pressure rise give a small throughput.

Exercise 14: Leak Acceptance Margin

Measured leak rate is $1.5\times10^{-5}\ \text{Pa m}^3/\text{s}$ and acceptance limit is $2.0\times10^{-5}\ \text{Pa m}^3/\text{s}$ . Find margin.

Solution

\mathrm{margin}=\dfrac{2.0-1.5}{1.5}=0.333

The margin is 33.3 percent relative to measured leak rate.

Engineering Comment

The procedure should state temperature, isolation time and whether virtual leaks are acceptable.

Plausibility Check

The measured value is three-quarters of the limit, so one-third relative margin is correct.

Exercise 15: Gauge Offset Error

A capacitance gauge has zero offset $0.006\ \text{Pa}$ . It reads $0.045\ \text{Pa}$ . Estimate corrected pressure.

Solution

p_\mathrm{corr}=0.045-0.006=0.039\ \text{Pa}

Engineering Comment

Offset matters most near the low end of the gauge range. Zero checks should be part of release evidence.

Plausibility Check

The correction is small but more than 10 percent of the reading.

Exercise 16: Gauge Agreement

Gauge A reads $0.052\ \text{Pa}$ and Gauge B reads $0.058\ \text{Pa}$ at the same port. Estimate percent difference relative to their average.

Solution

\bar{p}=\dfrac{0.052+0.058}{2}=0.055\ \text{Pa}

\epsilon=\dfrac{0.058-0.052}{0.055}(100)=10.9\ \text{percent}

Engineering Comment

Cross-checks are useful only if the gauges have overlapping valid ranges and the same gas basis.

Plausibility Check

A difference of 0.006 on about 0.055 is roughly 11 percent.

Exercise 17: Pumpdown Acceptance Guard Band

Acceptance limit is $5.0\times10^{-3}\ \text{Pa}$ and expanded pressure uncertainty is $0.8\times10^{-3}\ \text{Pa}$ . What maximum indicated pressure should be accepted?

Solution

p_\mathrm{ind,max}=5.0\times10^{-3}-0.8\times10^{-3} =4.2\times10^{-3}\ \text{Pa}

Engineering Comment

Guard bands are important when vacuum acceptance protects contamination, plasma process quality or high-voltage insulation.

Plausibility Check

The acceptance indication is below the limit by the uncertainty allowance.

Exercise 18: Vacuum Gauge Release Gate

A pumpdown release package has 16 required evidence items. Fourteen are complete, but gas correction and gauge-location drawing are missing. Should release pass?

Solution

\mathrm{completion}=\dfrac{14}{16}=0.875

The package is 87.5 percent complete, but release should not pass because the missing items affect the validity of the pressure reading.

Engineering Comment

Vacuum release depends on where and what the gauge measured, not only on whether the displayed pressure was low.

Plausibility Check

Two missing boundary items are enough to invalidate the acceptance record.

Validation Package Checklist

Before accepting a vacuum gauge or pumpdown measurement, collect:

gauge type, range, serial number, calibration and gas correction;
gauge location, chamber volume, valve state and pump configuration;
pressure units, corrected reading, uncertainty and guard band;
conductance boundary and any pump-side versus chamber-side pressure estimate;
pumpdown curve, pressure-rise or leak-rate evidence and acceptance limit;
gauge offset, cross-check status and useful-range confirmation;
outgassing, virtual-leak and real-leak disposition where relevant;
release authority, process boundary and retained configuration record;
release decision tied to the exact chamber configuration and process gas.

A complete validation package should make the pumpdown decision reproducible. Another engineer should be able to see what the gauge measured, where it measured, which gas basis applied, how uncertainty was guarded and why the chamber was accepted, retested, baked, leak-checked or held.

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