Equations for respirometry without gas scrubbing
Equations used for computing exchange rates are derived in part from Depocas and Hart (1957; J.  Appl.  Physiol.  10:388-392), Hill (1972, J.  Appl.  Physiol.  33:261-263) and Withers (1977, J.  Appl.  Physiol.  42:120-123); others I derived myself.  If you want to see a (relatively) simple example of deriving an equation for VO2, look at this page -- but note that the example shown assumes scrubbing of both water and CO2 from excurrent gas.

The following symbols are used: FRadj =adjusted flow rate, V = exchange rate for the gas in question (oxygen, CO2, or water vapor), STP = factor for converting to standard conditions of temperature and pressure, Fi = input fractional concentration, Fe = excurrent fractional concentration, RQ = respiratory quotient.

 Note: if you have already done the STP correction to the a fixed flow rate or flow rate data in a separate channel -- for example, if you measured flow with a mass flow controller with STP-corrected output, or if you used the STP converter in the OUTPUT menu and saved the results-- make sure the temperature and pressure are set to 0 °C and 760 torr, respectively.

The calculations for VO2 and VCO2 depend on the position of the flowmeter, although the effect is relatively small for VCO2 .   Flow rate (FR) through the metabolism chamber is best measured upstream from the chamber in most cases -- this provides greater control (or understanding) of incurrent gas concentrations.

To account for the presence of water vapor in the analyzed excurrent gas stream, two adjustments need to be made.   First, the animal is assumed to have added water vapor, thereby increasing excurrent flowrate.   To account for this, the program subtracts the fraction of excurrent flow comprised of water vapor (FeH2O) and uses the adjusted flow rate in calculations of VO2 and VCO2:

    FeH2O = water vapor pressure (kilopascals) / ambient pressure (kilopascals)

Second, water vapor dilutes both excurrent oxygen concentration (FeO2 ) and excurrent CO2 concentration (FeCO2), so these values have to be adjusted upward, as follows:

       'real' FeO2 = measured FeCO2 / (1 - FeH2O)
       'real' FeCO2 = measured FeCO2 / (1 - FeH2O)

      Evaporative water loss (mH2O):

The addition of CO2 and extraction of O2 by the animal affects flow rate, which must be accounted for (athough the effect on mH2O is generally quite minor). The adjusted flowrate (FRadj) = measured flow rate * (1 - (FiO2-FeO2)) * (1 + (FeCO2-FiCO2)

Since excurrent gas water content is measured as kilopascals water vapor pressure and one liter of gas contains 7.926 milligrams of water vapor per kilopascal vapor pressure:

     mH2O in mg/min = FRadj in liters/min * 7.926

      Oxygen consumption (VO2):

• adjusted flowrate (FRadj) = measured flow rate * (1 - FeH2O)

If the flowmeter is upstream from the chamber (Mode 1):

VO2 = STP * FRadj * ((FiO2 - FeO2) - FeO2* (FeCO2 - FiCO2)) / (1 - FeO2)


If the flowmeter is dowmstream (Mode 2):

VO2 = STP * FRadj * ((FiO2 - FeO2) - FiO2 * (FeCO2 - FiCO2))/ (1 - FiO2)

      Carbon dioxide production (VCO2):

adjusted flowrate (FRadj) = measured flow rate * (1 - FeH2O)

If the flowmeter is upstream from the chamber (Mode 1):

VCO2 = STP * FRadj * ((FeCO2 - FiCO2) - FeCO2* (FiO2 - FeO2)) / (1 - FeCO2)


If the flowmeter is dowmstream (Mode 2):

VCO2 = STP * FRadj * ((FeCO2 - FiCO2) + FiCO2* (FiO2 - FeO2)) / (1 + FiCO2)

CAUTION:   you may need to synchronize the vapor pressure, oxygen and CO2 channels in time.   This is especially important in serial configuration because the different analyzers are 'looking' at a given portion of the gas stream in succession, not simultaneously (the lag correction option in the EDIT menu can help fix this problem).   But note that if your subject's metabolism is changing rapidly, exact synchronization may not be possible even if you split your sample gas stream and read vapor pressure, O2, and CO2 in parallel, because the response times of different gas analyzers are usually unequal.   For example, the response time of a typical CO2 analyzer or a fuel cell O2 analyzer is seconds, while some O2 analyzers (like those from Applied Electrochemistry) respond in milliseconds.


The algorithms used to compute vapor density are derived from Properties of Air, by Tracy, Welch, and Porter (1980; University of Wisconsin; you can find a pdf on the Web via Google Scholar).  In turn, these are based on the Smithsonian Meteorological Tables.   For those interested, the formulae used are as follows:

• Vapor pressure (pw) at temperatures over liquid water (Smithsonian Tables, 1984, after Goff and Gratch, 1946):

Log10 pw = -7.90298 (373.16 T-1)
                    + 5.02808 Log10(373.16 / T)
                    - 1.3816 10-7 (1011.344 (1-T / 373.16) -1)
                    + 8.1328 10^-3 (10-3.49149 (373.16 / T-1) -1)
                    + Log10(1013.246)
with T in °K and pw in hPa

• Vapor pressure (pi) at temperatures below 0 °C (over ice; Smithsonian Tables, 1984):

Log10 pi =   -9.09718 (273.16/T - 1)
                    - 3.56654 Log10(273.16/ T)
                    + 0.876793 (1 - T/ 273.16)
                    + Log10(6.1071)
with T in °K and pi in hPa

The Goff-Gratch equation (for air over liquid water) covers a temperature range of -50 °C to about 100 °C, but is mostly theoretical for very low temperatures.   Accuracy is probably ±0.5% or better at temperatures between -20 and 70 °C.

• CAUTION: regardless of the accuracy of the equations, technically it is very difficult to avoid condensation or freezing of water vapor when working at subzero temperatures.


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