A flow can be rotationally cold while molecules inside it remain vibrationally hot. This paper shows why that mismatch can matter when scientists use “temperature” to interpret low-temperature reaction rates. Chemical reactions at low temperatures play a central role in understanding astronomical phenomena in naturally occurring low-temperature environments, such as the interstellar medium and planetary atmospheres.

In dense interstellar clouds, collisions are often governed by long-range interactions, and rate coefficients can display non-Arrhenius behaviour as a function of temperature. Capture theory can sometimes predict this behaviour, but not all reaction systems are well-described.

Because laboratory data are scarce, discrepancies often arise between observations and astrochemical models. Techniques for creating cold environments on Earth, together with careful laboratory study, have started to address this gap. One of the most prolific techniques in low-temperature reaction kinetics is the CRESU method, and it has measured rate coefficients for over two hundred ion–molecule and radical–neutral reactions.

CRESU expands a free jet through a carefully designed Laval nozzle with an axisymmetric convergent-divergent internal profile. At the nozzle exit, the supersonic jet contains a warm, sparse boundary layer and a cold, dense isentropic core. That core is uniform in temperature and density for tens of centimetres, so it is called a uniform supersonic flow, or USF.

The abundance of collisions leads to local thermal equilibrium, where motional degrees of freedom are thermalised. In theory, the particles can then be described by a Maxwell–Boltzmann distribution with a single temperature defining where reactions occur. The temperature measurements were taken in HILTRAC using Pitot tube measurements, infrared direct frequency comb spectroscopy, and LIF excitation spectroscopy, and they were compared with computational fluid dynamics simulations.

The study explored a large parameter space, varying total nozzle densities, Pitot temperatures, OCS concentration, buffer gas, and the region of the uniform supersonic flow probed. Under all conditions, a vibrational hot band appeared, corresponding to population in the low-lying doubly degenerate OCS bending mode.

Across the full range of conditions, the vibrational temperature was higher than the rotational temperature, which is clear evidence that the local thermal equilibrium assumption was not valid for OCS in CRESU experiments. In panel b, the temperature map spans roughly fifty to two hundred sixty kelvin, with the strongest variation concentrated near the flow edge and nozzle entrance.

The companion density map ranges from about one to six times ten to the power of sixteen molecules per cubic centimeter, while the core region is visibly more uniform than the surrounding boundary. The rotational temperatures for the main band and hot band were about fifty point six and forty-five point six kelvin, respectively, showing consistent rotational cooling despite different initial vibrational states.

The equivalent vibrational temperature was two hundred nineteen kelvin, and the population ratio of the zero one one zero to zero zero zero zero states was zero point zero six five five to one. So the excess reagent was rotationally cold but still vibrationally much hotter than thermal equilibrium would suggest.

A boundary measurement increased the rotational temperature to about seventy-six point two kelvin, while the vibrational temperature remained approximately the same at about two hundred twenty-three kelvin. The high vibrational temperature was therefore not an artefact of the measurement technique, but a genuine feature of O-C-S even within the isentropic core.

The hot band also persisted across different Pitot temperatures and USF conditions, with analyses across concentrations, densities, buffer gases, nozzles, and probed USF regions. Figure two shows how the authors extract rotational and vibrational populations from OCS absorption.

In panel a, blue peaks represent the combination band from ground-state OCS, while red peaks identify a vibrational hot band originating from the excited bend. Panel b plots the column-integrated populations across lower rotational state J, and panel c recasts those populations against lower-state energy, where Boltzmann fits provide rotational temperatures and their uncertainties.

For an argon flow using Nozzle 9, the average flow velocity was five hundred twelve metres per second. That gave molecules in the uniform supersonic flow approximately forty microseconds to four hundred microseconds to travel before the laser probed them.

With more travel time, the vibrational temperature and the zero one one zero to zero zero zero zero population ratio decreased from about two hundred thirty-one kelvin to two hundred seventeen kelvin. Meanwhile, the rotational temperature increased from about forty-three point one kelvin to fifty-one point nine kelvin.

In panel b, the red DFCS curve and blue CFD curve track closely across the three probe positions, with the largest separation appearing at the first point and near-overlap at the last. Panel c shows why a single Pitot value is only a reference: the blue measurements span well beyond the shaded DFCS sampling window, including pronounced peaks and troughs along the flow.

Regardless of the relaxation mechanism or whether argon or nitrogen was the collision partner, the measured vibrational temperatures suggest that OCS would require longer than the available flow length to fully thermalise. Near the end of the flow, the vibrational temperature was still substantially higher than the rotational temperature.

The underlying mechanism is competition between vibrational relaxation timescales and the residence time available in the expansion and uniform flow, and it is not specific to pulsed operation. The non-equilibrium was established directly for a pulsed uniform supersonic flow, while direct measurements in continuous flows would also be valuable because similar vibrational non-equilibrium may arise.

The temperature derived from impact pressure measurements using the Pitot tube failed to reflect the true energy possessed by molecules in a range of cases. In argon, the seeded Pitot temperature deviated from the buffer-only temperature, with the point of curvature beginning at about zero point two three percent density of the uniform supersonic flow.

That decrease in temperature was non-physical and likely resulted from the Rayleigh–Pitot equation no longer being applicable. In argon, increasing O-C-S concentration raised the rotational temperature from about fifty point six to sixty-seven point five kelvin, while vibrational temperature slightly decreased from about two hundred nineteen to two hundred fifteen kelvin.

In nitrogen, the seeded Pitot temperature showed no deviation from the buffer-only value, but direct frequency comb spectroscopy still found rotational temperature increasing from about seventy-one point three to ninety-three point four kelvin and vibrational temperature increasing from about two hundred thirteen to two hundred twenty-nine kelvin.

In panel a, the blue Pitot points stay near the dashed reference through the lower-concentration region, then bend downward toward the right-hand dashed marker. In panel b, the points remain clustered around the reference line across the plotted concentration range, with only small point-to-point variation.

Uniform supersonic flows are commonly assumed to be in local thermal equilibrium because the many collisions at the Laval nozzle throat should equilibrate all degrees of freedom. However, a vibrational hot band was present in OCS under all conditions probed, even when its concentration was an order of magnitude below the one percent rule of thumb often used in the community.

N2O also showed a vibrational hot band at a flow temperature of about two hundred sixty-seven kelvin, even at a concentration comprising zero point one percent of the flow. Because black-body thermalisation typically occurs on timescales of seconds, far longer than uniform supersonic flow timescales, the excitation is unlikely to come from surrounding room-temperature equipment or the environment.

When low-lying vibrational modes are present, a single equilibrium temperature may not be well represented by the conventional rotational or translational temperature measurement. The experiments established this behaviour directly for OCS and N2O and suggest that analogous non-equilibrium may arise more generally for seeded reagents with low-lying vibrational modes and slow vibrational relaxation.

For OCS, the expected population of the vibrationally excited state was essentially zero at a thirty-two kelvin Pitot temperature, but the measured population was around six to seven percent. For N2O, the expected population at about one hundred twelve point seven kelvin was zero point one percent, but the measured population was four percent.

Population in vibrationally excited states may affect reaction kinetics by placing reagents at a different starting point on the potential energy surface and changing their ability to overcome barriers along the reaction coordinate. The study cautions against relying on Pitot tubes and the Rayleigh–Pitot equation as the sole method of temperature measurement.

Changes in rotational temperature are not always well represented by Pitot temperature, and deviations can occur below one percent of the uniform supersonic flow, depending on the molecule and its vibrational-mode energies. These findings do not imply that existing low-temperature rate coefficients are broadly invalid; instead, the reported temperature may not uniquely define the reacting ensemble’s thermodynamic state.

The distinction matters when vibrational excitation influences reactivity, and it highlights the need for direct measurements of internal-state distributions in low-temperature kinetics experiments. The findings motivate more careful assessment of the temperature reported alongside rate coefficients in astrochemical databases and of when non-equilibrium effects should be explicitly considered.

CRESU flows cannot always be described by one thermodynamic temperature: vibrational excitation persists in OCS and N2O, while Pitot temperatures can miss the reacting molecules’ energetic state. Rate databases therefore need closer attention to internal-state distributions.