If a liquid has twice the density of mercury, what will be the height of the liquid column in a barometer?

In a barometer, the height of the liquid column is determined by the balance between the weight of the liquid column and the atmospheric pressure. The relationship between the height of the liquid column and the density of the liquid is given by the following equation:

$$gh$$

Where:

• $P$ is the atmospheric pressure,
• ρ\rho is the density of the liquid,
• g is the acceleration due to gravity,
• $h$ is the height of the liquid column.

Given:

• The density of the liquid is twice the density of mercury.
• The height of the mercury column in a standard mercury barometer is approximately 760 mm at sea level.
• Let the density of mercury be ρHg.

Step 1: Relating the heights of the two columns

If the liquid has twice the density of mercury, then the new liquid’s density is ρnew=2⋅ρHg.

From Pascal’s law, the pressure exerted by the liquid column is:

P=ρHgghHg=ρnewghnew​

Since the atmospheric pressure $P$ is the same in both cases, we can set the pressures equal:

ρHgghHg=ρnewghnew​

Step 2: Solving for the new height hnew​

Substitute ρnew=2⋅ρHg​ into the equation:

ρHgghHg=2⋅ρHgghnew​

Cancel out ρHg and $g$ from both sides:

hHg=2⋅hnew​

Now solve for hnew:

hnew=hHg/2h​​

Step 3: Substituting the height of the mercury column

We know the height of the mercury column is approximately 760 mm:

hnew=760 mm/2=380 mm

Conclusion:

If the liquid has twice the density of mercury, the height of the liquid column in the barometer will be 380 mm. This is because a denser liquid exerts more pressure for the same height, so the column height decreases when using a liquid that is denser than mercury.