Question

If the atmosphere can support a column of mercury 760 mm high at sea level, what height of a hypothetical liquid whose density is 1.40 times the density of mercury could be supported?

Short answer

542.86 mm

Step-by-step solution

Understand the Problem

We need to find the height of a liquid with a density 1.40 times that of mercury which can be supported by the atmospheric pressure at sea level.

Write Down Known Values

Let the density of mercury be denoted as \( \rho_{Hg} \), and the height of the mercury column is 760 mm.

Define the Density of the Hypothetical Liquid

The density of the hypothetical liquid would be \( 1.40 \rho_{Hg} \).

Use the Pressure Equation

We know that the pressure is given by the product of density, gravitational acceleration, and height (P = \rho g h). Since atmospheric pressure supports both the mercury column and the hypothetical liquid column, we set the pressures equal to each other: \[ \rho_{Hg} g \times 760 \text{ mm} = 1.40 \rho_{Hg} g \times h \]

Solve for the Height

We cancel out \(\rho_{Hg} \) and g from both sides and solve for \( h \): \[ 760 \text{ mm} = 1.40 \times h \] \[ h = \frac{760 \text{ mm}}{1.40} \] \[ h = 542.86 \text{ mm} \]

Key concepts

density

Density is a measure of how much mass is contained within a certain volume. It's crucial to understand density to solve problems involving fluids and pressure. The formula for density is given by \(\rho = \frac{mass}{volume}\). This means that if you know the mass and volume of an object, you can calculate its density.

In this exercise, the densities of mercury and the hypothetical liquid are compared. Since the density of the liquid is 1.40 times that of mercury, it shows that for the same volume, the liquid has more mass compared to mercury. This difference in density is key to understanding how the height of the liquid column is affected.

atmospheric pressure

Atmospheric pressure is the force exerted by the weight of the atmosphere. It is crucial in problems involving liquid columns, particularly when calculating the height a column can reach.

At sea level, atmospheric pressure is typically sufficient to support a column of mercury 760 mm high. This is often used as a reference point in pressure calculations. In the given problem, we are using this atmospheric pressure to determine the height of a liquid with a different density. The concept is that the pressure exerted by the liquid column must equal the atmospheric pressure to be supported.

liquid columns

Liquid columns are used to measure pressure and they rely on the principles of fluid statics. The height of a liquid column is related to the pressure at the base of the column, the density of the liquid, and the acceleration due to gravity, with the relationship given by the equation \(\text{P} = \rho g h\).

In our problem, we compare the heights of two liquid columns supported by the same atmospheric pressure. Since the atmospheric pressure is constant, any changes in the liquid density will affect the height of the column that can be supported. For the liquid which is denser than mercury (1.40 times the density), we derived that the height is shorter than mercury’s when balancing pressures, which we calculated to be 542.86 mm.