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Chapter 11: Problem 91

Mercury is poured into a tall glass. Ethyl alcohol is then poured on top of the mercury until the height of the ethyl alcohol itself is \(110 \mathrm{~cm}\). The two fluids do not mix, and the air pressure at the top of the ethyl alcohol is one atmosphere. What is the absolute pressure at a point that is \(7.10 \mathrm{~cm}\) below the ethyl alcohol-mercury interface?

Short Answer

Expert verified
The absolute pressure at the point is approximately 119.3 kPa.

Step by step solution

01

Understand the Problem

We need to calculate the absolute pressure at a point 7.10 cm below the interface of ethyl alcohol and mercury in a container with air pressure at the top of the ethyl alcohol being one atmosphere (1 atm).
02

Determine Atmospheric Pressure

The atmospheric pressure at the top of the ethyl alcohol is 1 atm, which is equivalent to \(101,325\) Pascals (Pa). This will contribute to the total pressure at the point of interest.
03

Calculate Pressure Due to Ethyl Alcohol

Ethyl alcohol has a density of approximately \(789\, \text{kg/m}^3\). The height of ethyl alcohol is 110 cm, which is 1.10 m in SI units. Use the hydrostatic pressure formula to calculate the pressure due to the column of ethyl alcohol: \[P_{ ext{ethyl}} = \rho_{ ext{ethyl}} \cdot g \cdot h_{ ext{ethyl}} = 789 \cdot 9.81 \cdot 1.10\] This simplifies to \(P_{ ext{ethyl}} \approx 8,521.33\, \text{Pa}\).
04

Calculate Pressure Due to Mercury

Now, consider the pressure exerted by the mercury column below the ethyl alcohol. Mercury has a density of approximately \(13,600\, \text{kg/m}^3\). The depth of interest (7.10 cm below the interface) is converted to meters, which is 0.071 m. Calculate the pressure using the formula:\[P_{ ext{mercury}} = \rho_{ ext{mercury}} \cdot g \cdot h_{ ext{mercury}} = 13,600 \cdot 9.81 \cdot 0.071\]This simplifies to \(P_{ ext{mercury}} \approx 9,454.56\, \text{Pa}\).
05

Calculate Total Absolute Pressure

The absolute pressure at 7.10 cm below the interface is the sum of the atmospheric pressure, the pressure exerted by the ethyl alcohol, and the pressure exerted by the mercury:\[P_{ ext{total}} = P_{ ext{atm}} + P_{ ext{ethyl}} + P_{ ext{mercury}} = 101,325 + 8,521.33 + 9,454.56\]This yields \(P_{ ext{total}} \approx 119,300.89\, \text{Pa}\) or approximately 119.3 kPa.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hydrostatic Pressure
Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It increases with the depth of the fluid.
When calculating hydrostatic pressure, you use the formula:
  • \( P = \rho \cdot g \cdot h \)
  • \( \rho \) is the fluid's density (mass per unit volume)
  • \( g \) is the acceleration due to gravity (approximately \(9.81 \, \mathrm{m/s^2}\) on Earth)
  • \( h \) is the height or depth of the fluid column above the point you are measuring
The hydrostatic pressure increases as you go deeper into a fluid.
In our exercise, both the ethyl alcohol and the mercury exert hydrostatic pressure on the point 7.10 cm below their interface.
Density
Density is a measure of mass per unit volume and it plays a critical role in determining hydrostatic pressure.
It is denoted by the Greek letter \( \rho \) and is usually expressed in kilograms per cubic meter (kg/m鲁).
Different substances have different densities, which influence the pressure exerted by fluid columns.For instance, in our exercise, ethyl alcohol has a density of approximately \(789 \, \text{kg/m}^3\), while mercury has a much higher density of approximately \(13,600 \, \text{kg/m}^3\).
This means that for the same depth, mercury will exert more pressure than ethyl alcohol due to its greater density.
Absolute Pressure
Absolute pressure is the total pressure exerted by a fluid, including atmospheric pressure and hydrostatic pressure.
It is an important concept because it tells us the true pressure at a point within a fluid. To find absolute pressure, you sum up:
  • Atmospheric pressure, which is the pressure exerted by the weight of the atmosphere above the fluid
  • Hydrostatic pressure of any fluid column above the point of interest
In our example, to find the absolute pressure 7.10 cm below the interface, we consider the pressures from both the ethyl alcohol and mercury columns, as well as the atmospheric pressure above the ethyl alcohol.
Atmospheric Pressure
Atmospheric pressure is the pressure exerted by the Earth's atmosphere at any given point.
It is a crucial part of understanding total pressure in fluid systems since every place on Earth below sea level experiences this pressure in a fluid system. At sea level, atmospheric pressure is approximately 101,325 Pascals or 1 atmosphere (atm).
This pressure varies with altitude and weather, but for most calculations, it's assumed to be constant.
In our exercise, this atmospheric pressure acts on top of the ethyl alcohol column, contributing to the total pressure at the point of interest below the fluid interface.

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Most popular questions from this chapter

An airtight box has a removable lid of area \(1.3 \times 10^{-2} \mathrm{~m}^{2}\) and negligible weight. The box is taken up a mountain where the air pressure outside the box is \(0.85 \times 10^{5} \mathrm{~Pa}\). The inside of the box is completely evacuated. What is the magnitude of the force required to pull the lid off the box?

The vertical surface of a reservoir dam that is in contact with the water is \(120 \mathrm{~m}\) wide and \(12 \mathrm{~m}\) high. The air pressure is one atmosphere. Find the magnitude of the total force acting on this surface in a completely filled reservoir. (Hint: The pressure varies linearly with depth, so you must use an average pressure.)

(a) The mass and radius of the sun are \(1.99 \times 10^{30} \mathrm{~kg}\) and \(6.96 \times 10^{8} \mathrm{~m}\). What is its density? (b) If a solid object is made from a material that has the same density as the sun, would it sink or float in water? Why? (c) Would a solid object sink or float in water if it were made from a material whose density was the same as that of the planet Saturn (mass \(=5.7 \times 10^{26} \mathrm{~kg},\) radius \(\left.=6.0 \times 10^{7} \mathrm{~m}\right)\) ? Provide a reason for your answer.

A suitcase (mass \(m=16 \mathrm{~kg}\) ) is resting on the floor of an elevator. The part of the suitcase in contact with the floor measures \(0.50 \mathrm{~m}\) by \(0.15 \mathrm{~m} .\) The elevator is moving upward, the magnitude of its acceleration being \(1.5 \mathrm{~m} / \mathrm{s}^{2}\). What pressure (in excess of atmospheric pressure) is applied to the floor beneath the suitcase?

A hot-air balloon is accelerating upward under the influence of two forces, its weight and the buoyant force. For simplicity, consider the weight to be only that of the hot air within the balloon, thus ignoring the balloon fabric and the basket. (a) How is the weight of the hot air determined from a knowledge of its density \(\rho\) hot air and the volume \(V\) of the balloon? (b) For a given volume \(V\) of the balloon, does the buoyant force depend on the density of the hot air inside the balloon, the density of the cool air outside the balloon, or both? Provide a reason for your answer. (c) Draw a free-body diagram for the balloon, showing the forces that act on it. How is the upward acceleration of the balloon related to these forces and to its mass?
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