Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 9: Problem 26

If the ocean absorbs an additional \(3 \mathrm{~W} / \mathrm{m}^{2}\) of forcing, \({ }^{86}\) how long would it take to heat the \(\sim 4 \mathrm{~km}\) deep column of water directly under a particular square meter (thus \(4 \times 10^{6} \mathrm{~kg}\) ) by \(1^{\circ} \mathrm{C} ?^{87}\)

Short Answer

Expert verified
Answer: It would take approximately 64,444.4 days for the 4 km deep column of water to be heated by 1°C.

Step by step solution

01

Write down known variables

Heat input (P): \(3\, \mathrm{W/m}^2\) Mass of water column (m): \(4 × 10^6\, \mathrm{kg}\) Depth of water column: \(4\, \mathrm{km}\) Temperature change (ΔT): \(1^{\circ}\mathrm{C}\) Specific heat capacity of water (c): \(4.18 × 10^3\,\mathrm{J/\! kg\cdot K}\)
02

Calculate the heat required to raise the temperature of the water column by \(1^{\circ}\mathrm{C}\)

We will use the heat formula: \(Q = mc\Delta T\) \(Q = (4 \times 10^6\, \mathrm{kg}) \cdot (4.18 × 10^3\,\mathrm{J/\! kg\cdot K}) \cdot (1^{\circ}\mathrm{C})\)
03

Simplify the expression to find the value of Q

\(Q = 4 \times 10^6\, \mathrm{kg} \cdot 4.18 × 10^3\,\mathrm{J/\! kg\cdot K} \cdot 1^{\circ}\mathrm{C}\) \(Q = 16.72 \times 10^9\, \mathrm{J}\)
04

Calculate the time required to heat the water column by the given heat input

We will use the formula for time: \(t = Q / P\) \(t = (16.72 \times 10^9\, \mathrm{J}) / (3\, \mathrm{W/m}^2)\)
05

Simplify the expression to find the value of t

\(t = (16.72 \times 10^9\, \mathrm{J}) / (3\, \mathrm{W/m}^2)\) \(t = 5.57 \times 10^9\, \mathrm{s}\) (in seconds)
06

Convert the time to a more convenient unit of measurement

From seconds to days: \(t = (5.57 \times 10^9\, \mathrm{s}) \cdot (1\, \mathrm{day} / 86400\, \mathrm{s})\) \(t \approx 64444.4\,\mathrm{days}\)
07

Write the conclusion

It would take approximately \(64444.4\,\mathrm{days}\) to heat the \(4\,\mathrm{km}\) deep column of water directly under a particular square meter by \(1^{\circ}\mathrm{C}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Typical household electricity use in the U.S. is about 30 kilowatthours (kWh) per day. If the electricity is produced by a natural gas plant operating at \(40 \%\) efficiency, each household requires \(75 \mathrm{kWh}\) of energy from natural gas each day. Converting this to Joules, and using the intensity of \(49 \mathrm{~g} / \mathrm{MJ}\) from Table \(9.1\), how \(\mathrm{much} \mathrm{CO}_{2}\) is produced daily to supply a house with electricity from a natural gas source?

If carbon dioxide emission is a chief concern, switching from a coal-fired electricity plant to natural gas input is still a use of fossil fuels, but natural gas produces less \(\mathrm{CO}_{2}\) per energy unit than coal (Table 9.1). By what factor would \(\mathrm{CO}_{2}\) emissions be reduced if replacing all energy \(^{76}\) we now get from coal with energy from natural gas?

Treating Figure \(9.9\) somewhat literally, in which one out of every four infrared photons escapes without being absorbed by a greenhouse gas molecule, what would the effective upward radiation be if \(400 \mathrm{~W} / \mathrm{m}^{2}\) left the ground, and re-radiation of the absorbed fraction was split equally between upward radiation and radiation returning to the ground? \(^{83}\)

On a planet two-thirds covered in ocean, and \(1 \%\) covered by an ice sheet melting at a rate of 1 meter per year, how fast would sea level be rising?

If it were certain that the only way to provide a comfortable existence to future generations involved substantial cutbacks toward a lifestyle having far fewer energy and material comforts today, do you think humanity would voluntarily do so? Can we leave treats on the shelf, within easy reach? If so, is it only uncertainty that prevents us? If not, what do you see as the barriers?
See all solutions

Recommended explanations on Environmental Science Textbooks

Energy Resources

Read Explanation

Pollution

Read Explanation

Ecological Conservation

Read Explanation

Living Environment

Read Explanation

Environmental Policy

Read Explanation

Sustainability

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free