Question

Explain why it is necessary to include the density of \(\mathrm{Hg}(1)\) and the value of the acceleration due to gravity, \(g,\) in a precise definition of a millimeter of mercury (page 194 ).

Short answer

The density of mercury and the acceleration due to gravity (g) are crucial in defining a millimeter of mercury because they determine the pressure exerted by a 1 millimeter high column of mercury, which is what a 'millimeter of mercury' is defined as.

Step-by-step solution

Understand the concept of a millimeter of mercury

A millimeter of mercury (mmHg) is a unit of pressure, defined as the additional pressure that would be generated by a column of mercury exactly 1 millimeter high. It's an important unit in various scientific fields.

Discuss the role of mercury density

The density of an object is mass divided by volume. Denser substances will exert a greater pressure at the same height, because they have more mass in the same volume. Hence, the density of mercury is critical to calculating the pressure it will exert in a column of a given height. If we use a different substance with a different density, the resulting pressure (and thus, the value of one 'millimeter' of this substance) would be different.

Discuss the role of gravity

Gravity pulls things downwards, exerting a force on them. This is what causes fluid in a column (like our mercury) to exert pressure at the bottom of the column. More gravity would lead to a higher pressure, as the fluid is pulled down harder. Hence, the acceleration due to gravity (g) is critical in calculating how much pressure the mercury will exert.

Conclude with the necessity of both density and gravity

The exact pressure a column of mercury will exert (and thus, the definition of a mmHg) is directly dependent on both the density of mercury and the force exerted by gravity. Without these two values, it would be impossible to precisely define what a millimeter of mercury means in terms of pressure.

Key concepts

Density of Mercury: A Key Player in Pressure Measurement

When we talk about measuring pressure using a millimeter of mercury, it is crucial to consider the density of mercury. Density is essentially

• the mass of an object divided by its volume.
• measured in units like kilograms per cubic meter (kg/m³).

Mercury is a dense substance, which means a lot of its mass is packed into a small volume. This density plays a huge role in determining the pressure exerted by a column of mercury. The denser the substance, the more pressure it will exert at the same height.

When we use mercury in our pressure measurements, we benefit from its high density. It allows us to create a measurable amount of pressure with a short column of fluid. If we used a less dense substance instead of mercury, the pressure exerted for the same column height would be less. Consequently, understanding how much mass is present in a given volume of mercury helps us understand the pressure it exerts.

Acceleration Due to Gravity: The Force Behind Pressure

Gravity is the invisible force that pulls objects towards the Earth. It's a key component in why a column of mercury will exert pressure at all. This force is quantified as the acceleration due to gravity, typically represented by the symbol \( g \). In this context, \( g \) usually has a value of about \( 9.81 \text{ m/s}^2 \).

When gravity pulls mercury downward, it increases the pressure the fluid exerts at its base. This is why the same column of mercury on Earth would exert more pressure than it would on a planet with less gravitational pull. The pressure in a mercury column depends closely on both the density of the mercury and the gravitational force acting on it.

Hence, understanding acceleration due to gravity is vital when working with pressure measurements. It helps explain the relationship between the mass of mercury and the force it applies due to being pulled downwards by gravity.

Understanding the Unit of Pressure

Pressure is a measure of the force applied over a unit area. In the case of a millimeter of mercury ( mmHg), this represents the pressure exerted by a mercury column 1 millimeter high. This unit is often used in fields like medicine to measure blood pressure.

The unit of pressure is commonly expressed in Pascals (Pa), where:

• 1 Pascal is one Newton per square meter (N/m²).
• Converting mmHg to Pascals relies on knowing both the force exerted by mercury's weight and the area over which it's applied.

In other words, a certain column height of mercury will exert a measurable pressure that we can quantify using standard units like mmHg or Pascals.

By understanding these measurements, we can relate different systems and scales of pressure. This standardized measurement allows for a consistent understanding and accurate calculations across various scientific and practical applications. It's vital for accurately conveying the level of force exerted by fluids in fields ranging from meteorology to medicine.