Question

Explain why the height of mercury in a barometer is independent of the cross- sectional area of the tube.

Short answer

The height of mercury is independent of the tube's cross-sectional area due to the area canceling out in the pressure equation.

Step-by-step solution

Understand the Principle of a Barometer

A barometer measures atmospheric pressure using a column of mercury. The height of the mercury column corresponds to the atmospheric pressure since the weight of the mercury opposes the atmospheric force.

Analyze the Forces Involved

The force that balances the weight of the mercury column is the atmospheric pressure exerted on the mercury in the reservoir. This force exerted by the atmosphere equals the weight of the mercury column in the tube.

Examine the Equation of Pressure

Pressure is defined as force per unit area \[ P = \frac{F}{A} \]where \( F \) is the force and \( A \) is the area. Thus, with constant atmospheric pressure, the force is dependent on the weight of the mercury column rising in the tube.

Relate Weight of Mercury to Volume and Area

Weight is related to volume and density: \[ W = \rho \cdot V \cdot g \]where \( \rho \) is the density, \( V \) is the volume, and \( g \) is the acceleration due to gravity. The volume of mercury is the product of the cross-sectional area \( A \) and height \( h \): \( V = A \cdot h \). Thus:\[ W = \rho \cdot A \cdot h \cdot g \].

Explain Why Height is Independent of Area

Since the pressure is constant (atmospheric pressure), by substituting the expression for weight:\[ P = \frac{\rho \cdot A \cdot h \cdot g}{A} = \rho \cdot h \cdot g \] The area cancels out, showing that the height \( h \) depends only on the density \( \rho \), gravity \( g \), and the atmospheric pressure but not on the cross-sectional area of the tube.

Key concepts

Barometer

A barometer is a fascinating instrument primarily used for measuring the atmospheric pressure. Its design often involves a long glass tube that is closed at one end and filled with mercury. This mercury-filled tube is then inverted into a mercury reservoir. The atmospheric pressure pushes down on the mercury in the reservoir, and because there is a vacuum in the top of the tube, this force causes the mercury to rise.
In simpler terms, the height of the mercury column in a barometer reflects the atmospheric pressure. When the atmospheric pressure increases, it pushes the mercury higher in the tube. Conversely, when the pressure decreases, the mercury level drops. This principle helps scientists, meteorologists, and meteorology enthusiasts to predict weather changes and understand atmospheric behaviors. Barometers are vital for atmospheric studies and form the basis for many weather forecasting techniques.

Mercury Column

The mercury column is a key component in a mercury barometer. This column is essentially the standing height of mercury in the tube and gives a direct measurement of atmospheric pressure. Why mercury? Mercury is chosen because of its high density, allowing the column to be a manageable height for measuring purposes.
It's interesting to note how the mercury column works: it balances the atmospheric pressure outside with the pressure inside the column itself. This equilibrium is represented by the column's height regardless of the tube's size. Since the density of mercury is consistent, any change in the column's height is directly due to the change in atmospheric pressure, making it an accurate gauge for such measurements. The independence of the mercury column's height from the tube's cross-sectional area is an essential feature of barometric function.

Pressure Equation

The pressure equation is foundational to understanding how barometers function. It explains the relationship between force exerted on a surface and the area over which that force is distributed. The basic equation for pressure is given by \[ P = \frac{F}{A} \]where \( P \) represents pressure, \( F \) is force, and \( A \) is the area. This tells us that pressure is the force applied per unit area.
In a barometer context, the force \( F \) is due to the weight of the mercury column, which the atmospheric pressure must counterbalance. The interesting outcome of this setup is that the height of the mercury, which indicates the pressure, remains constant regardless of the cross-sectional area of the tube. This is because, mathematically, the area term \( A \) cancels out when setting up the equation for the weight \( W \) of the mercury as \( \rho \cdot A \cdot h \cdot g \). This cancellation leaves us with the dependency of height on density and gravitational force, but not on the tube's size.

Atmospheric Force

Atmospheric force is the force exerted by the weight of air in the atmosphere pushing down on a surface. This is a crucial concept as it directly influences how barometers work. Every inch of the planet is covered by a column of air stretching from the surface to the edge of space. This column exerts a measurable pressure on Earth's surface, which is what the barometer measures.
In a barometer, atmospheric force pushes on the mercury in the reservoir, allowing us to measure this invisible force with the visible height of mercury in a tube. Atmospheric force, thus, becomes tangible through our reading of the mercury height. This physical setup helps in various scientific applications, such as weather prediction and studies related to changes in atmospheric pressure. It's fascinating how a simple principle facilitates the study of such a significant environmental force.