In this explainer, we will learn how to describe the process of pressure measurement using the height of a liquid column in a U-shaped tube.

A liquid column manometer is a U-shaped tube filled with a liquid that is used to measure the pressure difference of gases on either side of it. A manometer open to the atmosphere filled with water is shown below.

The water level is equal on both sides of the tube. Due to this, the pressure on the left side, , is equal to the pressure on the right side, (i.e., when the height of the water on both the left and right sides is equal, so is the pressure): when

If one side of the tube is attached to a pressure different from the atmosphere, then the height of the liquid can change, as seen in the diagram below.

Both the unknown gas and the atmosphere are exerting pressure on the liquid, but now the pressure is different, so the height of the water on the sides of the tube has a difference, .

If is higher than , then it means there is less pressure pushing down on the water; thus, is less than : when and vice versa.

Letβs look at some examples.

### Example 1: Liquid Column Manometer Unknown Gas Comparison

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, and ?

### Answer

is the pressure on the right side, coming from the gas reservoir. is the pressure on the left side, coming from the atmosphere.

If was larger than , then the height of the liquid in the manometer would be pushed up to the left. If was instead larger than , then it would be pushed up to the right.

Since the height is equal, then it means the pressure on both sides is equal. The correct answer is C, .

### Example 2: Liquid Column Manometer Uneven Unknown Gas Comparison

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, and ?

### Answer

is the pressure on the right side, coming from the gas reservoir. is the pressure on the left side, coming from the atmosphere.

Since the liquid column heights are different in the tube, we know that cannot be equal to . They are different, so one must be greater than the other.

If was larger, the liquid in the column would be pushed up to the right, but the liquid there is not higher; it is lower. Therefore, must be greater than .

The correct answer is thus B, .

The proportion by which the pressure is different is related to the height difference. A larger difference in height means a larger difference in pressure. We can use a modified version of the equation for pressure in a column to express this.

### Equation: Change in Pressure of a Liquid Column Manometer

The equation used to relate the change in pressure to change in height
in a manometer is
where is the difference in pressure,
is the density of the fluid, is the force
due to gravity
(9.81 m/s^{2} for
Earth), and is the difference in height.

We can see in this equation that is proportional to . Twice the change in height means twice the change in pressure.

Letβs solve a problem that uses this relation. A liquid column manometer
contains water, the density of which we will take as
997 kg/m^{3}. The left side is connected to a gas of unknown pressure, and the right
side is open to the atmosphere at sea level, which we will take as
101.3 kPa. We can also take gravity as
9.81 m/s^{2}.

We are looking for the unknown gas pressure on the left side. Letβs take a look at the equation for change in pressure of a column:

It is important to note that we only care about the magnitude of change in height, so this cannot be negative.

Similarly, the difference in pressure is also a magnitude. Expressed by itself, it could look like or

We know that, due to its liquid height being lower, . This means that we can take the second form without the absolute value, since know it will be positive.

This makes the equation look like

We want on one side, so letβs add to both sides to isolate it:

is the atmosphere at sea level,
101.3 kPa. The other
values are as
997 kg/m^{3}, as
9.81 m/s^{2}, and
as
10 cm. Before putting in
, letβs convert it into metres, remembering
that there are 100 cm in
a metre:

So, 10 cm is 0.1 m. Putting these into the equation gives

The right side has the metres partially cancel each other out to become the
1/m^{3} given by density. Substituting the numbers gives

We need to have the units on the right side match before we can add them together. Recall that pascals are newtons per square metre Β and that newtons are kilogram-metres per second squared Β . We can convert the units we have to newtons per square metre as follows: making our equation

Letβs make the units of pascal the same. There are 1βββ000 Pa in 1 kPa:

Adding them together thus gives

Rounding, the unknown pressure on the left side is 102.3 kPa, a small difference. Sometimes, denser fluids are used in order to observe more dramatic changes in pressure.

Letβs look at an example.

### Example 3: Liquid Column Manometer Gas Pressure Calculation

The diagram shows a liquid column manometer connected at one end to a gas reservoir
and at the opposite end to the atmosphere. The U-shaped tube contains mercury that has
a density of 13βββ595 kg/m^{3}. The top of the mercury column in contact
with the atmosphere is vertically below the top of the mercury column in contact
with the gas reservoir. The vertical distance between the column tops
. Find the pressure of the gas in the reservoir. Use a value of
for atmospheric
pressure.

- 33 kPa
- 68 kPa
- 101 kPa
- 105 kPa
- 135 kPa

### Answer

To find the pressure of the gas in the reservoir, we can relate the change in height to the change in pressure between the reservoir and the side open to atmosphere.

Letβs look a the change in pressure in a fluid column equation:

We have to solve for the pressure of the gas reservoir, henceforth , and we already know . Looking at the liquid heights, the one open to the atmosphere is lower, so . Thus, the change in pressure will look like

Putting this back into the equation,

We want on one side, so letβs add it to both sides:

We can then subtract from both sides to isolate :

We know the density,
13βββ595 kg/m^{3}, gravity,
9.81 m/s^{2}, and
,
25 cm. Letβs first convert the
centimetres to
metres so all the units are the same:

We can then put our known values into the equation:

Multiplying together gives

The units for this recent calculation are converted to
pascals, or
Pa (N/m^{2}), as follows:
making the equation

The pressure for the left side, , was given in kilopascals, so letβs convert the value on the right side to that as well:

is given as 101.3 kPa. We put this in and solve to get

The pressure of the gas reservoir is thus B, 68 kPa.

Sometimes, it is not an unknown pressure that needs to be found, but an unknown height. If the pressures are given, we can find the height.

Looking at just the equation

We can isolate by dividing both sides by : canceling the right side to give

Just like change in pressure, we then need to determine how to order the heights so as not to have a negative change. In the diagram below, for example, it can clearly be seen that is greater than .

The change in height would thus look like though typically only the change in height is needed.

Letβs look at an example.

### Example 4: Liquid Column Manometer Height Difference Calculation

The diagram shows a liquid column manometer connected at opposite ends to two gas
reservoirs. The pressures of the gas reservoirs are
and . The U-shaped tube contains an oil with a density of
1βββ080 kg/m^{3}. What is the vertical distance between the tops of the oil columns? Give your answer to two decimal places.

- 22.05 m
- 12.22 m
- 11.65 m
- 1.25 m
- 0.62 m

### Answer

Letβs start by manipulating the liquid column manometer equation to isolate :

We can get by itself by dividing both sides by : which becomes

The value of is the difference between the pressures and . We want it to be positive, so since , the change in pressure looks like

is 123.3 kPa and is 110.1 kPa, meaning that the change in pressure is

Since we will have a final answer in terms of metres, we want kilopascals, kPa to be in pascals, Pa. There are 1βββ000 Pa in a kilopascal:

Now, we have all the variables we need to find the change in height. Density
is 1βββ080 kg/m^{3}
and is 9.81 m/s^{2}. The equation thus looks like

We need to relate the units of pascal to kilograms, metres, and seconds. The units of pascal are newtons per square metre Β , and newtons are kilogram-metres per second squared. Together, the conversion looks like

Putting all these units together gives

By canceling out the units in the numerator with the units in the denominator, the units greatly simplify to just metres:

We then simplify to give

Given to two decimal places, the answer is thus D, 1.25 m.

Letβs summarize what we have learned in this explainer.

### Key Points

- The equation used to relate change in pressure and change in height of a liquid column manometer is where is the difference in pressure, is the density of the fluid, is the force due to gravity, and is the difference in height.
- In a liquid column manometer, the ratio of change in height is equal to the ratio of the change in pressure.