Question

Mercury is 13.5 times denser (heavier) than water. If you built a barometer using water rather than mercury, how tall (in inches) would it have to be to record standard sea-level pressure?

Short answer

The water barometer would be approximately 404.92 inches tall.

Step-by-step solution

Understand Standard Sea-Level Pressure

Standard sea-level pressure is typically measured using a mercury barometer, with the column of mercury being about 29.92 inches tall. This is the pressure exerted by the atmosphere at sea level.

Calculate Density Ratio

Mercury is 13.5 times denser than water. This means that to exert the same pressure, a column of water would need to be 13.5 times taller than a column of mercury.

Compute Water Barometer Height

To find the height of the water column, we multiply the height of the mercury column by the density ratio: \[ \text{Height of water column} = 29.92 \times 13.5 \] Perform the multiplication to find the water column height in inches.

Calculate the Final Height

Perform the multiplication: \[ 29.92 \times 13.5 = 404.92 \]The water barometer would need to be approximately 404.92 inches tall to record standard sea-level pressure.

Key concepts

Mercury Barometer

A mercury barometer is a simple yet effective device used to measure atmospheric pressure. It consists of a long glass tube that is filled with mercury and then inverted into a dish of mercury. The atmospheric pressure pushes on the open dish of mercury, causing the mercury inside the tube to rise. The height of the mercury column accurately represents the atmospheric pressure. Typically, at sea level, this mercury column stands at about 29.92 inches high. This height corresponds to the standard sea-level atmospheric pressure. The use of mercury is crucial because its high density allows for a manageable column height.

Density

Density is a measure of how much mass is contained in a given volume of a substance. In the context of barometers, the density of mercury as compared to water plays a significant role. Mercury is 13.5 times denser than water, meaning mercury has more mass within the same volume that water does. This higher density is why we use mercury in barometers. A denser liquid like mercury means the barometer is conveniently sized. If the liquid were less dense, such as water, the column would need to be much taller to exert the same pressure at the base.

Sea-Level Pressure

Sea-level pressure refers to the atmospheric pressure at sea level, which is a standard reference point for meteorologists. The pressure is generally measured as 1013.25 hPa (hectopascals) or 29.92 inches of mercury (inHg). This value is used as a baseline to compare atmospheric pressures at various altitudes and conditions. At sea level, the weight of the atmosphere above is greatest, hence the established standard pressure. This benchmark allows for easier communication and understanding of weather patterns and forecasts across various regions.

Barometer Height Calculation

To determine the height of a water column in a barometer under standard sea-level pressure, it's crucial to understand the period of replacement. Since mercury is much denser, a similar effect must be achieved by adjusting the column height. Since mercury is 13.5 times denser than water, we benefit from its density making the system more compact. When designing a barometer with water, you must extend the height to compensate for the lower density, thus performing a basic calculation:

• Multiply the mercury height (29.92 inches) by the density ratio (13.5).
• The result, a much taller column of 404.92 inches, provides the necessary pressure.

This method showcases the relationship between density and height in barometric measurements, highlighting why denser liquids are preferable where size is a constraint.