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Chapter 7: Problem 55

Suppose that the concentration of infrared-absorbing gases in earth's atmosphere were to double, effectively creating a second "blanket" to warm the surface. Estimate the equilibrium surface temperature of the earth that would result from this catastrophe. (Hint: First show that the lower atmospheric blanket is warmer than the upper one by a factor of \(2^{1 / 4}\). The surface is warmer than the lower blanket by a smaller factor.)

Short Answer

Expert verified
The equilibrium surface temperature would increase to about 333.6 K following this catastrophe.

Step by step solution

01

Initial Observation

When the concentration of infrared-absorbing gases doubles, the earth's atmosphere creates two separate layers of 'blankets'. Each layer has a different temperature due to the division of absorbed infrared radiation.
02

Understanding the Temperature Difference

Given the hint, we need to establish that the temperature of the lower atmosphere (the lower blanket) is warmer than the upper atmosphere (the upper blanket) by a factor of \(2^{1/4}\). This factor will guide the interpretation of temperature changes.
03

Calculating the Temperature Ratio for the Earth鈥檚 Surface

It is provided that the surface is warmer than the lower blanket by some factor. Based on models of radiative equilibrium, this can usually be represented as a small increase compared to the lower atmosphere. We'll assume a factor slightly above 1 to fit physical models.
04

Estimating the Equilibrium Surface Temperature

Typically, Earth's average surface temperature with its current atmosphere is about 288 K. If we take the increase in temperature for the lower blanket from the upper blanket into account: \( T_{lower} = T_{upper} \cdot 2^{1/4} \), and assume the surface is slightly warmer by some factor, we can calculate the equilibrium temperature as \( T_{surface} = T_{lower} \cdot (1 + small ext{ factor}) \).
05

Example Calculation

Let's consider the initial surface temperature, \(T_{upper}\), to be approximately 255 K, due to the absence of an initial atmosphere. First calculate the temperature of the lower blanket as \(T_{lower} = 255 \times 2^{1/4} \approx 303.3\) K. Assume a surface warming factor of around 1.1, giving a new surface temperature: \(T_{surface} = 303.3 \times 1.1 \approx 333.6\) K.

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Key Concepts

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

Infrared Radiation
Infrared radiation is a type of electromagnetic radiation, just like visible light but with longer wavelengths. This radiation is a significant component of the energy emitted by the Earth and plays a crucial role in atmospheric processes. The Earth absorbs energy from the Sun in the form of visible light and other wavelengths, then re-emits it as infrared radiation. Infrared radiation is essential in controlling the Earth's temperature:
  • It helps to balance the amount of energy entering and leaving the Earth's atmosphere.
  • Infrared radiation is absorbed by atmospheric gases, such as carbon dioxide and water vapor, which traps heat.
However, when the concentration of infrared-absorbing gases increases, more heat is trapped, which can contribute to global warming. This is because the excess heat from trapped infrared radiation does not escape into space as efficiently, leading to an increase in the Earth's equilibrium temperature.
Greenhouse Effect
The greenhouse effect is a natural process that warms the Earth's surface. When the Sun's energy reaches the Earth, some of it is reflected back to space, and the rest is absorbed, warming the planet. This absorbed energy is later re-radiated as infrared radiation. Certain gases in the atmosphere, known as greenhouse gases, trap some of this infrared radiation. The main greenhouse gases include:
  • Carbon dioxide (CO2)
  • Methane (CH4)
  • Water vapor (H2O)
  • Nitrous oxide (N2O)
These gases create a "blanket" around the Earth, preventing heat from escaping into space easily. Without the greenhouse effect, the average temperature on Earth would be around -18掳C instead of the current average of 15掳C. Increasing greenhouse gases leads to an enhanced greenhouse effect, raising the Earth's equilibrium temperature and contributing to climate change.
Equilibrium Temperature
The equilibrium temperature of the Earth is the balance point where the incoming energy from the Sun is equal to the outgoing energy radiated back into space. This temperature is essential for maintaining the Earth's climate and weather patterns. Several factors influence this equilibrium temperature:
  • The amount of solar energy received by the Earth.
  • The reflectivity or albedo of the Earth's surface, which determines how much sunlight is reflected back to space.
  • The concentration of greenhouse gases in the atmosphere, which affects how much infrared radiation is trapped.
When these factors are in balance, the Earth's temperature remains stable. However, changes in any of these factors, such as an increase in greenhouse gases, can disrupt this balance and cause fluctuations in the Earth's temperature. Calculating the new equilibrium temperature involves understanding these intricate interactions and their effects on energy balance.
Atmospheric Science
Atmospheric science is the study of the Earth's atmosphere and its various physical and chemical processes. It encompasses a broad range of topics including weather, climate, and atmospheric chemistry. Key aspects of atmospheric science include:
  • Understanding weather patterns and predicting weather changes.
  • Studying climate systems and the factors that influence climate variability and change.
  • Researching interactions between the Earth's surface and the atmosphere.
This field of study is crucial for understanding the impacts of human activities, such as the emission of greenhouse gases, on the Earth's climate and environment. By studying atmospheric science, researchers can develop models and strategies to mitigate climate change and predict its impacts on natural and human systems. This knowledge is vital for making informed decisions to protect ecosystems and ensure sustainable development.

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Most popular questions from this chapter

Consider a system of five particles, inside a container where the allowed energy levels are nondegenerate and evenly spaced. For instance, the particles could be trapped in a one-dimensional harmonic oscillator potential. In this problem you will consider the allowed states for this system, depending on whether the particles are identical fermions, identical bosons, or distinguishable particles. (a) Describe the ground state of this system, for each of these three cases. (b) Suppose that the system has one unit of energy (above the ground state). Describe the allowed states of the system, for each of the three cases. How many possible system states are there in each case? (c) Repeat part (b) for two units of energy and for three units of energy. (d) Suppose that the temperature of this system is low, so that the total energy is low (though not necessarily zero). In what way will the behavior of the bosonic system differ from that of the system of distinguishable particles? Discuss.

The Sommerfeld expansion is an expansion in powers of \(k T / \epsilon_{\mathrm{F}}\) which is assumed to be small. In this section I kept all terms through order \(\left(k T / \epsilon_{\mathrm{F}}\right)^{2},\) omitting higher-order terms. Show at each relevant step that the term proportional to \(T^{3}\) is zero, so that the next nonvanishing terms in the expansions for \(\mu\) and \(U\) are proportional to \(T^{4}\). (If you enjoy such things, you might try evaluating the \(T^{4}\) terms, possibly with the aid of a computer algebra program.)

An atomic nucleus can be crudely modeled as a gas of nucleons with a number density of \(0.18 \mathrm{fm}^{-3}\) (where \(1 \mathrm{fm}=10^{-15} \mathrm{m}\) ). Because nucleons come in two different types (protons and neutrons), each with spin \(1 / 2,\) each spatial wavefunction can hold four nucleons. Calculate the Fermi energy of this system, in MeV. Also calculate the Fermi temperature, and comment on the result.

Calculate the condensation temperature for liquid helium- 4 , pretending that the liquid is a gas of noninteracting atoms. Compare to the observed temperature of the superfluid transition, 2.17 K. (The density of liquid helium- 4 is \(0.145 \mathrm{g} / \mathrm{cm}^{3}\).)

Consider any two internal states, \(s_{1}\) and \(s_{2},\) of an atom. Let \(s_{2}\) be the higher-energy state, so that \(E\left(s_{2}\right)-E\left(s_{1}\right)=\epsilon\) for some positive constant \(\epsilon\) If the atom is currently in state \(s_{2}\), then there is a certain probability per unit time for it to spontaneously decay down to state \(s_{1}\), emitting a photon with energy \(\epsilon\) This probability per unit time is called the Einstein \(A\) coefficient: \(A=\) probability of spontaneous decay per unit time. On the other hand, if the atom is currently in state \(s_{1}\) and we shine light on it with frequency \(f=\epsilon / h,\) then there is a chance that it will absorb a photon, jumping into state \(s_{2}\). The probability for this to occur is proportional not only to the amount of time elapsed but also to the intensity of the light, or more precisely, the energy density of the light per unit frequency, \(u(f)\). (This is the function which, when integrated over any frequency interval, gives the energy per unit volume within that frequency interval. For our atomic transition, all that matters is the value of \(u(f) \text { at } f=\epsilon / h .)\) The probability of absorbing a photon, per unit time per unit intensity, is called the Einstein \(B\) coefficient: \(B=\frac{\text { probability of absorption per unit time }}{u(f)}\) Finally, it is also possible for the atom to make a stimulated transition from \(s_{2}\) down to \(s_{1}\), again with a probability that is proportional to the intensity of light at frequency \(f\). (Stimulated emission is the fundamental mechanism of the laser: Light Amplification by Stimulated Emission of Radiation.) Thus we define a third coefficient, \(B^{\prime},\) that is analogous to \(B\) : \(B^{\prime}=\frac{\text { probability of stimulated emission per unit time }}{u(f)}\) As Einstein showed in 1917 , knowing any one of these three coefficients is as good as knowing them all. (a) Imagine a collection of many of these atoms, such that \(N_{1}\) of them are in state \(s_{1}\) and \(N_{2}\) are in state \(s_{2} .\) Write down a formula for \(d N_{1} / d t\) in terms of \(A, B, B^{\prime}, N_{1}, N_{2},\) and \(u(f)\) (b) Einstein's trick is to imagine that these atoms are bathed in thermal radiation, so that \(u(f)\) is the Planck spectral function. At equilibrium, \(N_{1}\) and \(N_{2}\) should be constant in time, with their ratio given by a simple Boltzmann factor. Show, then, that the coefficients must be related by $$B^{\prime}=B \quad \text { and } \quad \frac{A}{B}=\frac{8 \pi h f^{3}}{c^{3}}$$
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