Question

Torricelli, who invented the barometer, used mercury in its construction because mercury has a very high density, which makes it possible to make a more compact barometer than one based on a less dense fluid. Calculate the density of mercury using the observation that the column of mercury is \(760 \mathrm{~mm}\) high when the atmospheric pressure is \(1.01 \times 10^{5} \mathrm{~Pa}\). Assume the tube containing the mercury is a cylinder with a constant cross-sectional area.

Short answer

The density of mercury is approximately \(13,536\text{ kg/m}^3\).

Step-by-step solution

Write down the given parameters

We are given the following values: - Height of mercury column, h: \(760\text{ mm}\) - Atmospheric pressure, P: \(1.01 \times 10^{5}\text{ Pa}\) - Acceleration due to gravity, g: \(9.8 \text{ m/s}^2\) Let's convert the height of the column to meters: \(h = 760\text{ mm} \times \frac{1 \text{ m}}{1000\text{ mm}} = 0.76\text{ m}\)

Express the force exerted by the atmosphere

Using the formula for pressure \(P = \frac{F}{A}\), we can express the force exerted by the atmosphere as: \(F_{atm} = P \times A = (1.01 \times 10^{5}\text{ Pa}) \times A\)

Express the weight of the mercury column

First, let's express the volume of the mercury column: \(V = A \times h = A \times 0.76\text{ m}\) Next, express the mass of the mercury column using its density, \(\rho\) (which we're trying to find): \(m = \rho \times V = \rho \times (A \times 0.76\text{ m})\) Now we can express the weight of the mercury column, W: \(W = m \times g = (\rho \times (A \times 0.76\text{ m})) \times 9.8 \text{ m/s}^2\)

Equate the force exerted by the atmosphere and the weight of the mercury column

As the mercury column is in equilibrium, the force exerted by the atmosphere is equal to the weight of the column: \(F_{atm} = W\) Substitute the expressions for \(F_{atm}\) and W: \((1.01 \times 10^{5}\text{ Pa}) \times A = (\rho \times (A \times 0.76\text{ m})) \times 9.8 \text{ m/s}^2\)

Solve for the density of mercury, \(\rho\)

Notice that the cross-sectional area A is present on both sides of the equation, so we can simplify the equation and solve for \(\rho\): \(\rho = \frac{(1.01 \times 10^{5}\text{ Pa})}{(0.76\text{ m} \times 9.8 \text{ m/s}^2)}\) Now, calculate the value of \(\rho\): \(\rho \approx 13,536 \text{ kg/m}^3\) The density of mercury is approximately \(13,536\text{ kg/m}^3\).

Key concepts

Barometer

A barometer is a scientific instrument used to measure atmospheric pressure. It provides vital data for meteorologists and scientists to understand weather patterns. The concept of a barometer was invented by Evangelista Torricelli, who employed mercury due to its high density, which allowed the design to be compact.

A traditional mercury barometer consists of a glass tube standing vertical with one end closed. The tube is filled with mercury, and then inverted into a mercury container, with the open end submerged. The atmospheric pressure pushes the mercury up the tube. Its height varies depending on the atmospheric pressure.

In our exercise, the height of the mercury column is observed to be 760 mm under standard atmospheric pressure. This height is directly related to the atmospheric pressure pushing on the surface of the mercury in the container.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the atmosphere above a given point. It is crucial for understanding weather and atmospheric conditions on Earth. Barometers help measure this pressure. Thunderstorms, for example, are associated with low atmospheric pressure, while fair weather is connected with high atmospheric pressure.

The standard atmospheric pressure, at sea level, is about 101,325 Pa (pascals). However, in our calculation, we rounded it to 1.01 x 10^5 Pa for simplicity. When using a barometer, this pressure pushes down on the mercury, causing it to rise in the tube.

Atmospheric pressure varies depending on altitude and weather conditions. At higher altitudes, pressure decreases because there is less atmosphere above, while low pressure systems typically bring storms and precipitation. Utilizing a barometer gives insights into predicting these environmental changes.

Gravity

Gravity is a fundamental force of nature pulling objects toward the center of the earth. It's indispensable in understanding how barometers work and for many calculations related to atmospheric sciences.

In a mercury barometer, gravity's role is seen in the mercury column's weight, which balances the atmospheric pressure. This balance is why equations for barometers equate atmospheric force to the weight of the mercury.

• Formula for pressure: \( P = \frac{F}{A} \)
• Weight of mercury: \( W = m \times g \)

In calculations, we use 9.8 m/s² as the standard gravitational acceleration on Earth's surface. This factor helps calculate the density of mercury using these equilibrium principles. Understanding gravity's influence on objects ensures precise readings and applications.