Final answer:
A manometer measures the pressure difference between a gas and the atmosphere by using the formula P = hρg. The diameter of the tubes in the manometer does not affect the height measurement because the pressure equation involves height, which balances regardless of the tube's diameter.
Step-by-step explanation:
A manometer measures the difference in pressure between the gas supply and atmospheric pressure because it uses the principle that the pressure exerted by a column of fluid due to its weight is P = hρg (Pressure = Height x Density x Gravitational Force). In a closed manometer, when a gas is introduced into one side, it exerts pressure and causes a difference in the mercury column heights, directly measuring the gas pressure. An open-end manometer measures either the excess pressure over atmospheric pressure or the deficit under atmospheric pressure (depending on whether the gas pressure is greater or less than atmospheric pressure), by allowing the atmospheric pressure to act on one side of the U-shaped tube.
Regarding why the diameter of the manometer tubes does not affect the height difference of the liquid, the concept of force balance comes into play. Imagine the two sides of the U-tube have different diameters. While the pressures on both sides need to balance, a wider tube would require a larger volume of fluid to achieve the same height as a narrower tube. However, because only the height of the fluid columns is measured, and the equation P = hρg involves only height, not volume, the diameter of the tubes is irrelevant. Whether the diameter is wide or narrow, the fluid will adjust until the pressure due to the fluid's height, density, and gravity on both sides is the same, thus resulting in equal height levels on both sides of the manometer when open to the atmosphere.