Rate constants are converted by substituting the ideal-gas relationship between partial pressure and molar concentration. For a second-order gas-phase rate law, (-dp_A/dt = k_p p_A^2), and the concentration form, (-dC_A/dt = k_C C_A^2), the conversion at constant temperature is (k_C = k_pRT). The gas constant (R) and temperature (T) must use units consistent with the pressure and concentration units.

The conversion factor depends on reaction order: for an (n)-th-order rate law, (k_C = k_p(RT)^{n-1}) under constant-temperature ideal-gas conditions. For second-order kinetics, multiply the pressure-basis rate constant by (RT).

Why the Conversion Is Necessary

Pressure and concentration describe the same gas differently

For an ideal gas, the partial pressure of species (A) is related to its molar concentration by

[ p_A = C_A RT ]

Therefore,

[ C_A = \frac{p_A}{RT} ]

For a gas mixture, the partial pressure is

[ p_A = y_A P ]

where (y_A) is the mole fraction of (A) and (P) is the total pressure. Thus,

[ C_A = \frac{y_A P}{RT} ]

This relationship allows pressure-sensor measurements to be converted into the concentration data required by concentration-based rate laws.

Temperature must be absolute

The temperature in (p_A=C_ART) must be expressed in kelvin. Using Celsius directly produces an incorrect conversion factor because the ideal-gas relationship depends on absolute temperature.

Converting a Second-Order Rate Constant

Begin with the pressure-based rate law

Suppose pilot-plant data are represented by

[ -\frac{dp_A}{dt}=k_p p_A^2 ]

Here, (k_p) is the rate constant on a partial-pressure basis. For example, if pressure is measured in atm and time in hours, its units are

[ \frac{1}{\text{atm}\cdot\text{h}} ]

Substitute partial pressure in terms of concentration

At constant temperature,

[ p_A=C_ART ]

Differentiating with respect to time gives

[ \frac{dp_A}{dt}=RT\frac{dC_A}{dt} ]

Substituting both relationships into the pressure-based rate law gives

[ -RT\frac{dC_A}{dt}

k_p(RTC_A)^2 ]

Dividing by (RT),

[ -\frac{dC_A}{dt}

k_pRT,C_A^2 ]

Comparing this with

[ -\frac{dC_A}{dt}=k_CC_A^2 ]

produces the conversion:

[ \boxed{k_C=k_pRT} ]

Check the units

Using (R=0.082057\ \text{L atm}/(\text{mol K})), the product (RT) has units of

[ \frac{\text{L atm}}{\text{mol}} ]

Therefore,

[ \left(\frac{1}{\text{atm}\cdot\text{h}}\right) \left(\frac{\text{L atm}}{\text{mol}}\right)

\frac{\text{L}}{\text{mol}\cdot\text{h}} ]

which is the expected unit for a second-order concentration-based rate constant.

Applying the Conversion in a Pilot Plant

Calculate the relevant partial pressure

Pressure sensors may measure total pressure rather than the partial pressure of the reacting species. In that case, (p_A) must first be determined from the reaction stoichiometry, feed composition, and measured total pressure.

For a known gas composition,

[ p_A=y_AP ]

In constant-volume systems with changing total mole numbers, the relationship between (p_A) and total pressure may instead be obtained from the stoichiometric pressure balance. For example, for (2B(g)\rightarrow B_2(g)),

[ p_B=2P-P_0 ]

where (P_0) is the initial total pressure under the same temperature and volume conditions.

Convert pressure data to concentration

Once (p_A) is known, calculate

[ C_A=\frac{p_A}{RT} ]

Use the operating temperature for the experiment and ensure that the pressure units in (p_A) match those used in (R).

Determine the pressure-basis constant

For second-order kinetics, the integrated pressure-based expression is

[ \frac{1}{p_A}-\frac{1}{p_{A0}}=k_pt ]

Thus, a plot of (1/p_A) against time gives a slope equal to (k_p), assuming constant temperature and the applicable reactor assumptions.

The concentration-based form is

[ \frac{1}{C_A}-\frac{1}{C_{A0}}=k_Ct ]

The two slopes are related by

[ k_C=k_pRT ]

Example at 400 K

Assume a pressure-basis constant is reported as

[ k_p=0.020\ \frac{1}{\text{atm}\cdot\text{h}} ]

At (400\ \text{K}), using

[ R=0.082057\ \frac{\text{L atm}}{\text{mol K}} ]

gives

[ RT=(0.082057)(400) =32.823\ \frac{\text{L atm}}{\text{mol}} ]

Therefore,

[ k_C=(0.020)(32.823) =0.656\ \frac{\text{L}}{\text{mol}\cdot\text{h}} ]

The numerical value changes because the basis has changed from pressure squared to concentration squared.

How Reaction Order Changes the Conversion

General (n)-th-order relationship

For a pressure-based rate law

[ -\frac{dp_A}{dt}=k_p p_A^n ]

substitute (p_A=C_ART):

[ -RT\frac{dC_A}{dt}

k_p(RTC_A)^n ]

After division by (RT),

[ -\frac{dC_A}{dt}

k_p(RT)^{n-1}C_A^n ]

Therefore,

[ \boxed{k_C=k_p(RT)^{n-1}} ]

First-order kinetics

For (n=1),

[ k_C=k_p ]

The numerical rate constant is unchanged when temperature, time basis, and ideal-gas assumptions are consistent, although its units are expressed differently.

Second-order kinetics

For (n=2),

[ k_C=k_pRT ]

This is the conversion most commonly needed when comparing pressure-based pilot-plant results with concentration-based kinetic models.

Zero-order kinetics

For (n=0),

[ k_C=\frac{k_p}{RT} ]

The units change accordingly because the rate law contains no concentration or pressure term.

Understanding the Trade-offs

The ideal-gas assumption may be inadequate

The conversion (p_A=C_ART) is exact only for an ideal gas. At high pressures or near condensation conditions, a compressibility factor may be needed:

[ p_A=ZC_ART ]

If (Z) varies significantly with operating conditions, a single constant conversion factor may not accurately represent the experiment.

Variable temperature breaks the simple constant factor

The expression (k_C=k_pRT) assumes that temperature is constant during the rate measurement. If the pilot plant is nonisothermal, (RT) changes with time and pressure data must be converted point by point.

Temperature also affects the intrinsic kinetic constant through the Arrhenius relationship, so basis conversion and temperature correction are separate issues.

Total pressure is not automatically reactant pressure

A pressure sensor may report total system pressure, while the rate law requires the reactant partial pressure. When the reaction changes the total number of gas moles, (P) and (p_A) do not have the same time dependence.

Using total pressure directly in place of (p_A) can produce an incorrect reaction order and rate constant.

Units can create hidden errors

A rate constant reported in (\text{atm}^{-1}\text{h}^{-1}) cannot be converted with an (R) expressed in (\text{kPa L}/(\text{mol K})) unless the pressure units are first reconciled.

Time units must also remain consistent. Converting hours to seconds requires the corresponding conversion of the rate constant.

Rate definitions must be consistent

Some sources define the disappearance rate as (-dC_A/dt), while others include a stoichiometric coefficient in the reaction-rate definition. The pressure-to-concentration conversion should be applied to equivalent rate definitions before comparing constants.

Making the Right Choice for Your Goal

Use the same kinetic basis throughout a calculation, then convert only when comparing results across experiments, models, or reactor designs.

If your primary focus is analyzing pressure-sensor data: Determine the reactant partial pressure from total pressure, composition, and stoichiometry before fitting the rate law.
If your primary focus is comparing with concentration-based models: Calculate (C_A=p_A/(RT)) and use (k_C=k_p(RT)^{n-1}), with units checked explicitly.
If your primary focus is second-order kinetics: Multiply the pressure-basis constant by (RT) to obtain the concentration-basis constant.
If your primary focus is nonideal or nonisothermal operation: Apply the appropriate compressibility and temperature corrections rather than using one constant (RT) factor.

With the reaction order, gas-law assumptions, temperature, and units handled consistently, pressure-based and concentration-based rate constants describe the same underlying reaction kinetics.

Summary Table:

Key topic	Main takeaway
Ideal-gas relationship	(p_A=C_ART) and (C_A=p_A/(RT))
Second-order conversion	(k_C=k_pRT)
General (n)-th-order conversion	(k_C=k_p(RT)^{n-1})
Partial pressure	Use (p_A=y_AP), not total pressure alone
Temperature	Use absolute temperature in kelvin
Unit consistency	Match pressure, volume, amount, and time units in (R) and (k)
Nonideal operation	Include the compressibility factor (Z) when required
Variable temperature	Convert data point by point because (RT) changes

Need reliable pilot-plant data for kinetic modeling or process scale-up? Contact LABPARK to discuss Educational and Vocational Unit Operations Pilot Plants for chemical engineering, bioprocess and biotech, and environmental and water treatment applications. LABPARK helps universities, research institutes, and enterprises obtain practical systems for pressure and concentration measurements, reaction studies, process training, and technology validation. Our team can help match the pilot plant configuration to your reaction model, operating conditions, teaching goals, and research requirements.