Video Transcript
In this lesson, we will learn how
to describe the process of pressure measurement using the height of a liquid column
in a U-shaped tube. Before we learn about this process,
letβs first refresh our memory on how to find the pressure in a liquid.
We need to recall that when weβre
looking for the pressure inside a fluid at different heights, we can use the formula
π, the pressure of the fluid, is equal to π, the density of the fluid, times π,
the acceleration due to gravity, times β, the height of the fluid above the point we
are considering. Going deeper into our liquid is the
same thing as saying that the height of the liquid above us is increasing, which
means that the pressure of the liquid is also increasing.
We can connect this to the fact
that the pressure at the bottom of a swimming pool is greater than the pressure near
the top of a swimming pool. The height of the water above us
when weβre at the bottom of the swimming pool is much greater than the height of the
water above us when weβre near the top. Letβs apply what we just recalled
about the pressure in a liquid to a manometer.
A manometer is a U-shaped tube
filled with a liquid that is used to measure the pressure of a gas. How does a manometer help us
measure the pressure of a gas? Letβs look at a manometer when the
left tube and the right tube are both open to the atmosphere. In both tubes of the manometer, the
atmosphere will apply a pressure to the liquid. In the left tube, weβll call that
pressure π subscript πΏ. In the right tube, weβll call that
pressure π subscript π
.
We can use the equation for the
pressure of a liquid π equals ππβ that we discussed earlier to compare the
pressures in both of the tubes. We can see that the liquid column
in both of the tubes is at the same height, which tells us that the β is the same
for both the left and the right tubes, where we measure the β in both tubes from the
same arbitrarily chosen vertical position. We also know that we are using a
manometer on Earth. Therefore, acceleration due to
gravity in both the left and the right tube is going to be the same.
And our manometer has been filled
with same liquid in both the left and the right tube. Therefore, the density of the fluid
is also gonna be the same in the left and right tube. Since all three of our variables
happen to be the same in both the left and the right tube, this would imply that the
pressure is the same in both the left and the right tube. This should make sense as both of
our ends are open to the atmosphere.
In general, when we have a
manometer, assuming that the liquid is the same throughout, when the height of the
liquid in the left tube is equal to the height of the liquid in the right tube, then
the gas pressure that the left tube is attached to is equal to the gas pressure that
the right tube is attached to.
What happens if they are both not
open to the atmosphere? Letβs look at an example where we
attach our left tube to a gas chamber of unknown gas pressure. And our right tube will be open to
the atmosphere such that the height of the left column is lower than the height of
the right column with the difference between the two heights being Ξβ. If the liquid on the right rises to
a higher height than it does on the left tube, then this would imply that the liquid
feels a greater push from the pressure in the left tube than it does from the
pressure in the right tube.
Or another way to say that, the
pressure of the gas attached to the left tube is greater than the pressure of the
gas attached to the right tube. But by how much is the pressure
greater? To determine that, we have to come
back to our equation for the pressure of a liquid is equal to ππβ. The difference in the pressure of
our gases will be equal to the density of the liquid times the acceleration due to
gravity times the difference in height that the liquid goes in each tube.
In general, when we have a
manometer that is filled with a liquid that is the same throughout, and the height
of the liquid in the left tube is less than the height of the liquid in the right
tube, then we can say that the pressure of the gas attached to the left tube will be
greater than the pressure of the gas attached to the right tube.
In our final example, weβll attach
our left tube to a different unknown gas pressure such that the height of the liquid
in the left tube is gonna be greater than the height of the liquid in the right tube
with the difference in the height of the liquids in each tube being Ξβ. Because the height of the liquid in
the left tube is greater than the height of the liquid in the right tube, we can say
that the pressure in the right tube is gonna be greater than the pressure in the
left tube. This is because the push from the
pressure in the right tube must be greater than the push from the pressure in the
left tube.
But how much bigger? Once again, we apply the pressure
in a liquid is equal to ππβ. And just as we saw last time, the
difference in the pressure of the gases is equal to the density of the liquid times
the acceleration due to gravity times the change in height of the liquid in each of
the tubes. In general, when we have a
manometer that is filled with a liquid that is the same throughout, and the height
of the liquid in the left tube is greater than the height of the liquid in the right
tube, then we can say that the pressure of the gas attached to the left tube is less
than the pressure of the gas attached to the right tube.
In a manometer, the preferable
liquid is mercury. This is because mercury is very
dense and we donβt need to use tall columns to measure significant pressure
differences. From a practical point of view,
mercury also has a low evaporation rate, which means that itβs easy to work with
without having to lose any of the material. If we donβt have access to mercury
or weβre working with gases that have low pressure differences, we can use either
oil or water as our liquid in the manometer. Manometers are widely used because
theyβre very simple.
Now that weβve learned how a
manometer can be used to find differences in pressure and compare our pressures to
each other, letβs go over two examples.
The diagram shows a liquid column
manometer connected at one end to a gas reservoir and at the opposite end to the
atmosphere. Which of the following correctly
relates the pressure of the gas and the pressure of the atmosphere, π gas and π
atmosphere? (A) π gas is equal to π
atmosphere. (B) π gas is greater than π
atmosphere. (C) π gas is less than π
atmosphere.
We need to remember that a
manometer is a U-shaped tube filled with a liquid that is used to measure the
pressure of a gas. For this problem, one end of the
tube is opened up to the atmosphere, and the other end is connected to a gas
reservoir. The atmosphere applies a pressure
π atmosphere onto the liquid in one side of the tube. On the other side of the tube, the
gas from the gas reservoir also applies a pressure onto the liquid. We can compare the pressure of the
atmosphere to the pressure of the gas by looking at the height that the liquid goes
to on either end of the tube.
We need to recall that the pressure
of a fluid π is equal to the density of the fluid π times the acceleration due to
gravity π times the height of the fluid β. If we wanna compare the two
pressures, we need to analyze the height that the liquid goes to on either end of
the tube. Because the liquid rises to the
same height on either end of the tube, this implies that both ends of the tube apply
the same amount of pressure. Or put another way, the pressure
that the gas applies on the liquid is gonna be equal to the pressure that the
atmosphere applies on the liquid. Therefore, we can say answer choice
(A) is correct. The pressure of the gas from the
gas reservoir is equal to the pressure from the atmosphere.
Next, weβll try an example where
our liquid is not going to the same height in either end of the tube.
The diagram shows a liquid column
manometer connected at one end to a gas reservoir and at the opposite end to the
atmosphere. Which of the following correctly
relates the pressure of the gas and the pressure of the atmosphere, π gas and π
atmosphere? (A) π gas equals π
atmosphere. (B) π gas is less than π
atmosphere. (C) π gas is greater than π
atmosphere.
We need to recall that a manometer
is a U-shaped tube filled with a liquid that is used to measure the pressure of a
gas. In a manometer, the pressure on one
side, in this case, the pressure of the atmosphere, pushes down on the liquid on one
end of the tube. And the pressure on the other end
of the tube, in this case, the pressure from the gas reservoir, pushes down on the
liquid on the other end. The relative pressure of the gases
will determine what height the liquid will go to on either end of the tube.
We can see in our diagram that we
have a difference in height between the liquid on one end of the tube to the liquid
on the other end of the tube. In this case, the height of the
liquid thatβs in the tube attached to the atmosphere is lower than the height of the
liquid in the tube thatβs attached to the gas reservoir. This implies that the pressure the
atmosphere puts on the liquid provides a greater push than the pressure the gas
reservoir puts on the liquid. Or another way to put it would be
that the pressure of the atmosphere is greater than the pressure of the gas in the
gas reservoir. Therefore, we can say that when we
compare the pressure of the gas in the gas reservoir to the pressure of the
atmosphere that the pressure of the gas is going to be less than the pressure of the
atmosphere or answer choice (B).
Key Points
We saw that a liquid manometer
works on the principle of different pressures being exerted on the liquid in each
side of the tube. The ratio of the column heights in
the U-shaped tube is equal to the ratio of the pressures on top of the two
columns.