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# A black hole is a blackbody if ever there was one, so it ISBN: 9780201380279 40

## Solution for problem 53P Chapter 7

An Introduction to Thermal Physics | 1st Edition

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Problem 53P

A black hole is a blackbody if ever there was one, so it should emit blackbody radiation, called Hawking radiation. A black hole of mass M has a total energy of Mc2, a surface area of 16πG2M2/c4, and a temperature of hc3/16π2kGM (as shown in Problem 1).

(a) Estimate the typical wavelength or the Hawking radiation emitted by a one-solar-mass (2 × 1030 kg) black hole. Compare your answer to the size of the black hole.

(b) Calculate the total power radiated by a one-solar-mass black hole.

(c) Imagine a black hole in empty space, where it emits radiation but absorbs nothing. As it loses energy, its mass must, decrease; one could say it “evaporates.” Derive a differential equation for the mass as a function of time, and solve this equation to obtain an expression for the lifetime of a black hole in terms of its initial mass.

(d) Calculate the lifetime of a one-solar-mass black hole, and compare to the estimated age of the known universe (1010 years) .

(e) Suppose that a black hole that was created early in the history of the universe finishes evaporating today. What was its initial mass? In what part of the electromagnetic spectrum would most or its radiation have been emitted?

Problem 1:

Use the result of below Problem 2 to calculate the temperature of a black hole, in terms of its mass M. (The energy is Mc2.) Evaluate the resulting expression for a one-solar-mass black hole. Also sketch the entropy as a function of energy, and discuss the implications of the shape of the graph.

Problem 2:

A black hole is a region of space where gravity is so strong that nothing, not even light, can escape. Throwing something into a black hole is therefore an irreversible process, at least in the everyday sense of the word. In fact, it is irreversible in the thermodynamic sense as well: Adding mass to a black hole increases the black hole’s entropy. It turns out that there’s no way to tell (at least from outside) what kind of matter has gone into making a black hole. Therefore, the entropy of a black hole must be greater than the entropy of any conceivable type of matter that could have been used to create it. Knowing this, it’s not hard to estimate the entropy of a black hole.

a) Use dimensional analysis to show that a black hole of mass M should have a radius of order GM/c2, where G is Newton’s gravitational constant and c is the speed of light. Calculate the approximate radius of a one-solar-mass black hole (M = 2 × 1030 kg).

b)  In the spirit of below Problem 3, explain why the entropy of a black hole, in fundamental units, should be of the order of the maximum number of particles that could have been used to make it.

c) To make a black hole out of the maximum possible number of particles, you should use particles with the lowest possible energy: long-wavelength photons (or other mass less particles). But the wavelength can’t be any longer than the size of the black hole. By setting the total energy of the photons equal to Mc2, estimate the maximum number of photons that could be used to make a black hole of mass M. Aside from a factor of 8π2, your result should agree with the exact formula for the entropy of a black hole, obtained through a much more difficult calculation. d) Calculate the entropy of a one-solar-mass black hole, and comment on the result.

Problem 3:

For either a monatomic ideal gas or a high-temperature Einstein solid, the entropy is given by Nk times some logarithm. The logarithm is never large, so if all you want is an order-of-magnitude estimate, you can neglect it and just say S ~ Nk. That is, tire entropy in fundamental units is of the order of the number of particles in the system. This conclusion turns out to be true for most systems (with some important exceptions at low temperatures where the particles are behaving in an orderly way). So just for fun, make a very rough estimate of the entropy of each of the following: this book (a kilogram of carbon compounds); a moose (400 kg of water); the sun (2 × 1030 kg of ionized hydrogen).

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Thermodynamics, Entropy, Free Energy, and the Direction of Chemical Reactions Chapter 20 beings with an introduction to the 2 ndLaw of Thermodynamics, but don not forget that the first law of thermodynamics is essentially energy cannot be created or destroyed, but rather is conserved. I. The Second Law of Thermodynamics: Predicting Spontaneous Change ...

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##### ISBN: 9780201380279

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