Question

What is standard atmospheric pressure (in \(\mathrm{cm}\) ) measured by a barometer filled with water (given that mercury is 13.6 times more dense than water)?

Short answer

The standard atmospheric pressure measured by a water barometer is 1033.6 cm.

Step-by-step solution

Understanding the Problem Statement

Standard atmospheric pressure is typically given as 76 cm of mercury (Hg) in a mercury barometer. We need to find the equivalent height when using a water barometer, considering that mercury is 13.6 times denser than water.

Setting Up the Equation

The formula to relate heights and densities in barometers is \( h_1 \times \rho_1 = h_2 \times \rho_2 \), where \( h \) is height and \( \rho \) is density. We know \( h_1 = 76 \) cm for mercury and \( \rho_2 = 1 \) for water, and \( \rho_1 = 13.6 \) for mercury.

Solving for Water Height

Substitute the known values into the equation \( h_1 \times \rho_1 = h_2 \times \rho_2 \), resulting in \( 76 \times 13.6 = h_2 \times 1 \). Therefore, \( h_2 = 76 \times 13.6 \).

Calculating the Value

Calculate \( 76 \times 13.6 \) to find \( h_2 \). The calculation yields \( h_2 = 1033.6 \) cm.

Key concepts

Barometer

A barometer is an instrument used to measure atmospheric pressure. It operates based on the principle of balancing the weight of a fluid column against the atmospheric pressure. Essentially, it helps us understand how much "weight" the air carries. This is crucial for weather forecasting and understanding weather patterns. Barometers are commonly filled with either mercury or water. An important aspect of using barometers is recognizing that different fluids will require different heights to balance the same atmospheric pressure due to their varying densities.

Density

Density is a fundamental concept in science that describes how much mass is contained in a given volume. It is often expressed in units like grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³).

• Higher density means more mass per unit volume.
• In our context, mercury is denser than water, meaning that it has more mass in the same volume compared to water.

Understanding density is key when comparing how different substances behave under the same conditions. This notion is central in the concept of barometers, where dense liquids like mercury create shorter columns than less dense liquids like water.

Mercury and Water Comparison

Comparing mercury and water is fundamental in understanding how barometers function differently when filled with these fluids. Mercury is 13.6 times denser than water. This means that a column of mercury doesn't need to be as tall as a column of water to exert the same pressure at its base.

• A mercury column of 76 cm is equivalent in pressure to a much taller column of water, due to this difference in density.
• The mercury barometer is more compact but provides the same pressure measurement as a much larger water barometer.

This comparison highlights why mercury is traditionally used in barometers, as it requires less space due to its higher density.

Pressure Measurement

Pressure measurement is a crucial aspect of many scientific and practical applications, including meteorology. Atmospheric pressure, for instance, is typically measured in millibars or centimeters of mercury (cmHg).

• With a mercury barometer, atmospheric pressure is measured by the height of the mercury column.
• For water-filled barometers, because of their lower density, the column height will be much greater for the same pressure.

Accurately measuring atmospheric pressure allows for predictions of weather changes and understanding of atmospheric conditions.

Scientific Calculation

Scientific calculations are essential for deriving meaningful information from measurements. In the case of converting the pressure measurement from a mercury barometer to a water barometer:

• We use the equation: \( h_1 \times \rho_1 = h_2 \times \rho_2 \)
• This equation helps us relate the height and density of different fluids to the same pressure exerted by the atmosphere.
• For our exercise, with mercury's density being 13.6 times greater than water's, the atmospheric pressure (76 cmHg) translates to 1033.6 cm of water.

Such calculations enable clear and accurate comparisons and conversions between different measuring systems and substances.