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What is the difference between dead volume and dwell volume?

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The dead volume

The dead volume is the volume of an HPLC system between the point of injection to the point of detection, excluding the column.
Thus it includes the injection volume, the volume of the injector, the volume of the connecting tubing before and after the column, the volume of the end-fittings and frits, and the volume of the detector flow cell.
The dead volume can be measured by replacing the column with a zero dead-volume connector.
By injecting a very small sample amount, the time can be measured between the moment of injection and the maximum peak height. This time multiplied by the flow rate gives you a very good estimation of the system dead volume.

The dwell volume

The dwell volume is responsible for the time delay for a gradient.
By definition it is the volume of a gradient HPLC system between the mixing chamber and the column inlet.
This volume does of course also exist under isocratic elution, but in that case, it has no impact on the separation.
It includes the volume of the gradient mixer, the connecting tubing to the pump, the pump head and check-valves, the tubing between the pump and the injector, the injector itself, and the tubing between the injector and the column inlet.
When initiating a gradient, the column is not subjected to the change in eluent composition until the gradient has passed the dwell volume.
During that time, the column is operated under isocratic elution. Attention has to be given when transferring a gradient method from one instrument to another.
If the dwell volume differs, the retention times will likely differ as well, despite an identical method and column.
The dwell volume can be measured by running a step gradient from 100 % methanol to 100 % methanol + 10 mg/L acetophenone.
The UV detector will detect an S-shaped trace.
The dwell volume is equal to the time between the injection and half height of the detector trace, multiplied by the flow rate.