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Chapter 14: Problem 25

In one observation, the column in a mercury barometer (as is shown in Fig. \(14-5 a\) ) has a measured height \(h\) of \(740.35 \mathrm{~mm}\). The temperature is \(-5.0^{\circ} \mathrm{C}\), at which temperature the density of mercury \(\rho\) is \(1.3608 \times 10^{4} \mathrm{~kg} / \mathrm{m}^{3}\). The free-fall acceleration \(g\) at the site of the barometer is \(9.7835 \mathrm{~m} / \mathrm{s}^{2} .\) What is the atmospheric pressure at that site in pascals and in torr (which is the common unit for barometer readings)?

Short Answer

Expert verified
The atmospheric pressure is 9879.55 Pa or approximately 74.1 torr.

Step by step solution

01

Convert Height to Meters

First, we need to convert the height of the mercury column from millimeters to meters for use in standard SI units. Given: \[ h = 740.35 \text{ mm} \]Convert millimeters to meters:\[ h = \frac{740.35}{1000} \text{ m} = 0.74035 \text{ m} \]
02

Calculate Atmospheric Pressure in Pascals

Using the formula for pressure due to a column of fluid:\[ P = \rho gh \]where:- \( \rho = 1.3608 \times 10^{4} \text{ kg/m}^3 \)- \( g = 9.7835 \text{ m/s}^2 \)- \( h = 0.74035 \text{ m} \)Substitute these values:\[ P = (1.3608 \times 10^{4} \text{ kg/m}^3) (9.7835 \text{ m/s}^2) (0.74035 \text{ m}) \]Calculate the pressure:\[ P = 9879.55 \text{ Pa} \]
03

Convert Pressure to Torr

Atmospheric pressure measured in torr can be calculated using the conversion:\[ 1 \text{ atm} = 760 \text{ torr} = 101325 \text{ Pa} \]Convert the pressure from pascals to torr:\[ P = \frac{9879.55 \text{ Pa}}{101325 \text{ Pa/atm}} \times 760 \text{ torr} \]Calculate the value:\[ P \approx 74.1 \text{ torr} \]

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Key Concepts

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

Mercury Barometer
A mercury barometer is a scientific instrument used to measure atmospheric pressure. It consists of an empty glass tube standing upside-down in a vessel of mercury. Atmospheric pressure pushes the mercury up into the glass tube. The height of the mercury column is related to the atmospheric pressure, with a higher column indicating higher pressure.
  • History: Invented by Evangelista Torricelli in the 17th century.
  • Function: Atmospheric pressure balances against the weight of the mercury column.
  • Units: Often measured in millimeters of mercury (mmHg) or torr.
A mercury barometer is responsive to temperature changes, affecting the mercury's density. Thus, temperature adjustments are sometimes necessary for precise measurements.
Fluid Pressure
Fluid pressure is the force exerted by a fluid per unit area within containers or when in contact with surfaces. It's a fundamental concept in understanding pressure in fluids like water, oil, air, or mercury. Atmospheric pressure itself is a form of fluid pressure due to the Earth's air pushing down on surfaces.This concept is governed by the formula: \[ P = \rho gh \]Where:
  • \(P\) is the pressure exerted by the fluid.
  • \(\rho\) is the fluid's density.
  • \(g\) is the acceleration due to gravity.
  • \(h\) is the height of the fluid column.
This equation shows that pressure depends directly on the density, gravitational acceleration, and height of the column of fluid. For a mercury barometer, higher atmospheric pressure will push the mercury column higher.
Unit Conversion
Unit conversion is crucial when calculating measurements in different units. In scientific problems involving atmospheric pressure, converting between units like millimeters (mm), meters (m), pascals (Pa), and torr is often necessary. Here's a quick guide:
  • Millimeters to meters: Divide by 1000.
  • Pascals to torr: Use the conversion factor 鈥� 1 atm = 101325 Pa = 760 torr.
Proper unit conversion ensures accuracy and consistency in calculations, particularly in physics where precise measurements are crucial.
Density and Temperature
Both density and temperature significantly affect fluid properties and measurements in experiments. Density:
  • The mass per unit volume of a substance, expressed in kg/m鲁 for mercury.
  • Higher density means a heavier fluid, thus exerting more pressure.
Temperature:
  • Affects the density of fluids. For instance, water expands and becomes less dense when heated.
  • In barometers, temperature changes can affect the height of the mercury column by altering its density.
Therefore, considering both factors is crucial when interpreting barometer readings and calculating exact pressures.

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Most popular questions from this chapter

A \(5.00 \mathrm{~kg}\) object is released from rest while fully submerged in a liquid. The liquid displaced by the submerged object has a mass of \(3.00 \mathrm{~kg}\). How far and in what direction does the object move in \(0.200 \mathrm{~s}\), assuming that it moves freely and that the drag force on it from the liquid is negligible?

About one-third of the body of a person floating in the Dead Sea will be above the waterline. Assuming that the human body density is \(0.98 \mathrm{~g} / \mathrm{cm}^{3}\), find the density of the water in the Dead Sea. (Why is it so much greater than \(1.0 \mathrm{~g} / \mathrm{cm}^{3} ?\) )

(a) If this longnecked, gigantic sauropod had a head height of \(21 \mathrm{~m}\) and a heart height of \(9.0 \mathrm{~m}\), what (hydrostatic) gauge pressure in its blood was required at the heart such that the blood pressure at the brain was 80 torr (just enough to perfuse the brain with blood)? Assume the blood had a density of \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). (b) What was the blood pressure (in torr or \(\mathrm{mm} \mathrm{Hg}\) ) at the feet?

When researchers find a reasonably complete fossil of a dinosaur, they can determine the mass and weight of the living dinosaur with a scale model sculpted from plastic and based on the dimensions of the fossil bones. The scale of the model is \(1 / 20 ;\) that is, lengths are \(1 / 20\) actual length, areas are \((1 / 20)^{2}\) actual areas, and volumes are \((1 / 20)^{3}\) actual volumes. First, the model is suspended from one arm of a balance and weights are added to the other arm until equilibrium is reached. Then the model is fully submerged in water and enough weights are removed from the second arm to reestablish equilibrium (Fig. 14-42). For a model of a particular T. rex fossil, \(637.76 \mathrm{~g}\) had to be removed to reestablish equilibrium. What was the volume of (a) the model and (b) the actual \(T\), rex? \((\mathrm{c})\) If the density of \(T\). rex was approximately the density of water, what was its mass?

Models of torpedoes are sometimes tested in a horizontal pipe of flowing water, much as a wind tunnel is used to test model airplanes. Consider a circular pipe of internal diameter \(25.0 \mathrm{~cm}\) and a torpedo model aligned along the long axis of the pipe. The model has a \(5.00 \mathrm{~cm}\) diameter and is to be tested with water flowing past it at \(2.50 \mathrm{~m} / \mathrm{s}\). (a) With what speed must the water flow in the part of the pipe that is unconstricted by the model? (b) What will the pressure difference be between the constricted and unconstricted parts of the pipe?
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