Concentration at Time Constant Multiples
Full Results Table
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How to Use This Calculator
Buildup mode models concentration rising toward steady state while a source is active. Decay mode models concentration falling back toward background after the source stops, which is useful for purge time planning.
In buildup mode, enter the generation rate as concentration units times m³ per hour (see the tooltip for ppm, µg/m³, and mg/m³). In decay mode, generation rate is ignored and the initial concentration represents the starting peak. Room volume and ventilation airflow together determine the time constant.
If you have a specific guideline or alarm threshold, enter it to see exactly when the space reaches that concentration. This works in both buildup and decay modes.
The curve shows concentration over your selected time range. The time constant table shows concentration at 1, 2, 3, 4, and 5 time constants, the standard reference points for understanding exponential approach to steady state.
The Universal Model Behind Indoor Pollutant Concentration
Nearly every indoor air quality calculation on this site, including CO2, CO, VOCs, formaldehyde, and PM2.5, relies on the same underlying mathematical model. Understanding that model directly, rather than through each pollutant-specific tool, gives you the flexibility to analyze any contaminant, including ones without a dedicated calculator, and to understand why the specific numbers in the other tools behave the way they do.
The First-Order Mixing Model
The model assumes a well-mixed space: the pollutant concentration is uniform throughout the room rather than concentrated near the source. Under this assumption, the rate of change of concentration equals the generation rate divided by room volume, minus the removal rate, which is proportional to how far current concentration is above background, times the air change rate. This differential equation has a closed-form exponential solution: concentration at time t equals steady-state concentration plus the difference between initial and steady-state concentration, decaying exponentially with a time constant tau equal to room volume divided by ventilation flow rate. The steady state itself is the background plus the generation rate divided by the airflow. This is the same mathematics used in pharmacokinetics for drug concentration in blood, RC circuit charging in electronics, and countless other first-order systems. In this calculator the generation rate is entered as concentration units times m³ per hour, so a source releasing 3.6 litres of pure vapour per hour is 3,600 ppm·m³/hr. In the default example that source, in a 60 m³ room with 20 L/s of ventilation (72 m³/hr), settles at 50 ppm and crosses 25 ppm after about 35 minutes.
Reading the Time Constant
Tau tells you the characteristic timescale of the system. A room with a 30-minute time constant responds to changes in about half an hour, and a room with a 4-hour time constant takes most of a workday to settle into a new equilibrium after a change in source or ventilation. This has direct practical implications for scheduling: pre-occupancy ventilation flush times, how long after a renovation before re-testing makes sense, or how quickly a CO2 sensor should be expected to respond after occupancy changes. Because tau is room volume divided by airflow, it is simply the reciprocal of the air change rate, and the ACH calculator converts between airflow, room volume, and air changes per hour. The CO2 level calculator and carbon monoxide calculator both use this exact math with pollutant-specific defaults built in.
Buildup vs. Decay: Same Math, Different Direction
Buildup and decay are mathematically identical, since both are exponential approaches to a steady-state value, just starting from different initial conditions. Buildup typically starts near background and approaches a higher steady state while a source is active. Decay starts at a higher peak and approaches background once the source stops. The same time constant governs both directions equally: a space that builds up slowly also purges slowly, and vice versa. That symmetry is useful when comparing a purge with a target concentration. Health Canada publishes residential indoor air quality guidelines for individual pollutants such as CO2, CO, formaldehyde, and nitrogen dioxide, and entering one of those values as the target shows how long the space takes to reach it or to clear back below it. The guidelines are averaged over specific periods, so a brief excursion above a value is not the same as an exceedance.
When the Model Breaks Down
The well-mixed assumption is the model's main limitation. In practice, spaces are never perfectly mixed, and concentration gradients exist near sources, in stagnant corners, and in poorly ventilated zones. The ventilation effectiveness calculator and dilution air calculator both apply correction factors (Ez and K, respectively) to account for this imperfect mixing. Multizone programs such as NIST's CONTAM extend the same well-mixed idea to several connected zones, with airflows between them, and add filtration, deposition, and chemical transformation. The model also assumes constant generation rate and constant ventilation flow, which is a reasonable approximation for many practical scenarios but breaks down for highly variable sources like intermittent cooking or rapidly cycling exhaust fans. For those situations, the steady-state and time-constant values from this calculator are still useful as bounding estimates even if the exact transient behavior is more complex.
Practical Applications Beyond the Named Calculators
This general-purpose tool is useful for scenarios that don't fit neatly into a single named contaminant calculator: modeling a specific solvent not covered elsewhere, projecting how quickly a commercial kitchen clears grease smoke after closing, estimating how long CO2 from a gathering takes to clear, or verifying purge time for a laboratory fume hood incident. It also suits enclosed spaces with a known ventilation rate, and the garage ventilation calculator covers the specific case of vehicle exhaust in a garage. Any scenario with a roughly constant source (or a stopped source and known peak concentration), a defined room volume, and a known ventilation rate can be modeled here using the same physics that underlies every pollutant-specific calculator on this site. Sources driven by pressure differences, such as radon entering through the soil, are a poor fit, because the entry rate depends on building pressure rather than on ventilation alone.
Frequently Asked Questions
Indoor concentration follows a first-order exponential approach to steady state: C(t) = C-steady-state + (C-initial - C-steady-state) × e^(-t/τ), where τ is the room time constant (volume divided by airflow). This applies to any pollutant with a roughly constant generation and ventilation rate in a well-mixed space, and the same model underlies the CO2, CO, and VOC calculators on this site. See the CO2 level calculator for a specific application.
The time constant τ equals room volume divided by ventilation airflow. It represents how fast a space responds to changes. After 1τ, concentration is 63% of the way to steady state. After 3τ, it's 95%. After 5τ, it's effectively complete at 99.3%. A small poorly-ventilated room has a long τ and responds slowly; a large well-ventilated space has a short τ and reaches equilibrium quickly. This governs both buildup and decay behavior identically.
Practical steady state is reached after 3-5 time constants (95-99%). A 50 m³ room with 20 L/s ventilation has a τ of about 42 minutes, reaching steady state in 2-3.5 hours. Larger spaces or lower ventilation rates produce longer time constants. A short-duration source may never reach the steady-state concentration a continuous source would produce, since it stops before the exponential curve gets there. Use the dilution air calculator for ventilation sizing to hit a specific steady-state target.
Solve the buildup equation for t: t = -τ × ln[(target - steady-state) / (initial - steady-state)]. This only works if the target lies between the initial and steady-state values. If the target exceeds steady-state, the space will never reach it under constant conditions, since the curve asymptotically approaches but never exceeds steady state. This calculator computes this automatically when you enter a target concentration.
Related Air Quality Tools
Ventilation sizing for any contaminant
CO2-specific buildup and ventilation
CO buildup and time-to-threshold
Ez corrections for imperfect mixing
VOC-specific dilution ventilation
Sub-slab depressurization sizing