# Why does capillary in gas thermometer have a temperature and pressure gradient?

I have been learning thermodynamics using Heat and Thermodynamics by Zemansky, in which he mentions

1. The gas in the capillary connecting the bulb with the manometer has a temperature gradient, that is, it is not at a uniform temperature. Sec (1.9)

I am not sure why there should be a temperature gradient. I think the gas in the bulb and the Dead space doesn't have the same temperature (1st point in sec 1.9). So there might be a temperature gradient for transitioning from one temperature to another. But this sounds more like conduction. I am not sure of this. I am entirely new to thermodynamics.

Can you explain why the temperature gradient exist in capillary?

Diagram of the Gas Thermometer:

Edit: And if possible can you also explain the fourth point

1. A pressure gradient exists in the capillary when the diameter of the capillary is comparable to the mean free path of the gas particles.

Why should it be near mean free path to happen?(Sorry for adding it after edit.)

Thanks!

• Where is the statement in Zemansky Commented Jul 11 at 12:47
• @BobD section 1.9 gas thermometer Commented Jul 11 at 12:51
• Since the capillary does. not appear to be in a constant temperature environment, wouldn't you expect there to be temperature gradients? Commented Jul 11 at 12:58
• @BobD Do you mean the endpoints not being at the same temperature? Commented Jul 11 at 13:02
• Yes, but I must admit I am not familiar with gas thermometers and have a hard time following Zemansky's description Commented Jul 11 at 13:05

When the mean free path is much smaller than an orifice or tube radius/diameter, the gas molecules mostly collide between themselves. In this continuum regime, a pressure difference can be reliably coupled to a flow (e.g., Hagen–Poiseuille flow) that eliminates that pressure difference. (Although the pressure head needed to maintain a certain flow rate increases very rapidly as the radius $$R$$ decreases, as evidenced by the $$R^{-4}$$ term). As the mean free path grows to match or exceed the orifice/tube/container size, the gas molecules interact much more with the surrounding walls, and the details of flow start to depend strongly on the thermodynamics and kinetics of attachment, detachment, solubility, and permeability with regard to the wall material, with the pressure difference becoming less important. Thus, Zemansky's intended meaning here seems to be that one can't conclude that a pressure head will be reliably eliminated by corresponding flow if one is in the Knudsen regime.