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Chapter 4: Problem 25

The solar constant is the incident energy per unit of time on a unit area of a surface placed at right angles to a sunbeam just outside the earth's atmosphere. The value of the solar constant is \(1.37 \mathrm{~kW} / \mathrm{m}^{2}\). The area of a sphere with radius \(93,000,000\) miles is \(2.79 \times 10^{23} \mathrm{~m}^{2}\), and the surface area of the sun is \(6.09 \times 10^{18} \mathrm{~m}^{2}\). Assuming that the sun is a blackbody, calculate its surface temperature,

Short Answer

Expert verified
The surface temperature of the sun is approximately 5,777 K.

Step by step solution

01

Understand the Problem

The problem involves calculating the surface temperature of the sun given the solar constant, the area of a sphere at the distance of Earth from the sun, and the surface area of the sun.
02

Use the Inverse Square Law

The solar constant (\text{SC}) is related to the total power output of the sun (\text{P}) by the inverse square law:The formula is given by:\[ \text{SC} = \frac{\text{P}}{\text{A}_\text{EARTH}} \]where \text{A}_\text{EARTH} is the surface area of the sphere at Earth's distance.
03

Calculate the Total Power Output of the Sun

Rearrange the formula to solve for \text{P}:\[ \text{P} = \text{SC} \times \text{A}_\text{EARTH} \]Plug in the given values:\[ \text{P} = 1.37 \times 10^3 \frac{\text{kW}}{\text{m}^2} \times 2.79 \times 10^{23} \text{ m}^2 \]\[ \text{P} = 3.82 \times 10^{26} \text{ W} \]
04

Use the Stefan-Boltzmann Law

The Stefan-Boltzmann Law relates the power output (\text{P}) of a blackbody to its surface area (\text{A}_\text{SUN}) and its temperature (T):\[ \text{P} = \text{A}_\text{SUN} \times \text{σ} \times T^4 \]where \text{σ} (Stefan-Boltzmann constant) is approximately \[5.67 \times 10^{-8} \frac{\text{W}}{\text{m}^2 \text{K}^4} \]
05

Rearrange to Solve for the Temperature

Rearrange to solve for temperature (T):\[ T = \bigg(\frac{\text{P}}{\text{A}_\text{SUN} \times \text{σ}}\bigg)^{\frac{1}{4}} \]Substitute the known values:\[ T = \bigg(\frac{3.82 \times 10^{26}}{6.09 \times 10^{18} \times 5.67 \times 10^{-8}}\bigg)^{\frac{1}{4}} \]
06

Calculate the Temperature

Simplify the expression inside the parentheses and take the fourth root:\[ T = \bigg(\frac{3.82 \times 10^{26}}{3.45 \times 10^{11}}\bigg)^{\frac{1}{4}} \]This simplifies to:\[ T = (1.107 \times 10^{15})^{\frac{1}{4}} \]\[ T \to 5,777 \text{K} \]

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Key Concepts

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

Stefan-Boltzmann Law
The Stefan-Boltzmann Law is a fundamental principle in physics that describes how the power emitted by a blackbody relates to its temperature. A blackbody is an idealized object that absorbs all incident electromagnetic radiation and re-emits it based on its temperature. According to the Stefan-Boltzmann Law, the total power per unit area emitted by a blackbody is proportional to the fourth power of its absolute temperature (in Kelvin). The law is expressed mathematically as:
\[ P = \text{A} \times \text{σ} \times T^4 \]
where:
  • P = Total power emitted (W)
  • A = Surface area of the blackbody (m^2)
  • σ = Stefan-Boltzmann constant = 5.67 × 10^{-8} \frac{\text{W}}{\text{m}^2 \text{K}^4}
  • T = Absolute temperature (K)
By rearranging this formula, you can solve for the temperature given the power and surface area.
Blackbody Radiation
Blackbody radiation refers to the theoretical spectrum of radiation emitted by an object that absorbs all incident radiation, regardless of frequency or angle of incidence. Such an object is called a blackbody. The concept is crucial for understanding the thermal radiation emitted by stars, including our Sun. When considering a blackbody, it is essential to note:
  • A blackbody in thermal equilibrium emits radiation with a spectrum that only depends on its temperature.
  • The peak wavelength of the emitted radiation shifts to shorter wavelengths as the temperature increases, a principle known as Wien's Displacement Law.
  • The total energy emitted per unit area increases with the fourth power of the blackbody's temperature, as described by the Stefan-Boltzmann Law.
Understanding these concepts helps us deduce the properties of celestial bodies, such as calculating the Sun's surface temperature.
Inverse Square Law
The Inverse Square Law describes how a specified physical quantity (like the intensity of light or radiation) decreases in proportion to the square of the distance from the source of that physical quantity. It is given by the equation:
\[ \text{Intensity} \backsim \frac{1}{\text{distance}^2} \]
In the context of solar radiation, it means that the intensity of solar energy reaching a unit area decreases with the square of the distance from the Sun. To find the solar constant, we can apply this law:
  • The solar constant is the amount of solar energy received per unit area at a specific distance from the Sun (Earth's distance).
  • The total power output from the Sun (P) can be calculated by multiplying the solar constant with the surface area of a sphere at Earth's distance from the Sun:
    • \[ \text{P} = \text{SC} \times \text{A}_\text{EARTH} \]
This law is key to determining how energy spreads out from a point source like the Sun.
Surface Temperature Calculation
To calculate the surface temperature of the Sun, we use the principles from both Stefan-Boltzmann Law and the Inverse Square Law. Here's a simplified step-by-step guide:
  • First, use the Inverse Square Law to find the total power output of the Sun (P):
    • \[ \text{P} = \text{SC} \times \text{A}_\text{EARTH} \]
    Substitute the known values:
    \[ \text{P} = 1.37 \times 10^3 \frac{\text{kW}}{\text{m}^2} \times 2.79 \times 10^{23} \text{ m}^2 = 3.82 \times 10^{26} \text{ W} \]
  • Next, use the Stefan-Boltzmann Law to relate this power to the surface temperature (T):
    • \[ \text{P} = \text{A}_\text{SUN} \times \text{σ} \times T^4 \]
  • Rearrange the formula to solve for T:
    • \[ T = \bigg(\frac{\text{P}}{\text{A}_\text{SUN} \times \text{σ}}\bigg)^{\frac{1}{4}} \]
  • Substitute the known values to find the surface temperature:
    • \[ T = \bigg(\frac{3.82 \times 10^{26}}{6.09 \times 10^{18} \times 5.67 \times 10^{-8}}\bigg)^{\frac{1}{4}} \]
  • This simplifies further to:
    • \[ T \to 5,777 \text{K} \]
By following these steps, you can easily calculate the Sun's surface temperature, showcasing the interconnectedness of these fundamental principles.

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Most popular questions from this chapter

A container of volume \(V\) contains \(n\) moles of gas at high pressure. Connected to the container is a capillary tube through which the gas may leak slowly out to the atmosphere, where the pressure is \(P_{0}\). Surrounding the container and capillary is a water bath, in which is immersed an electrical resistor. The gas is allowed to leak slowly through the capillary into the atmosphere while electrical energy is dissipated in the resistor at such a rate that the temperature of the gas, the container, the capillary, and the water is kept equal to that of the outside air. Show that, after as much gas as possible has leaked out during time interval \(t\), the change in internal energy is $$ \Delta U=\mathcal{E} I t-P_{0}\left(n v_{0}-V\right) $$ where \(v_{0}\) is the molar volume of the gas at atmospheric pressure, \(\mathcal{S}\) is the potential difference across the resistor, and \(I\) is the current in the resistor.

The molar heat capacity at constant volume of a metal at low temperatures varies with the temperature according to the equation $$ \frac{C_{V}}{n}=\left(\frac{124.8}{\Theta}\right)^{3} T^{3}+\gamma T $$ where \(\Theta\) is the Debye temperature, \(\gamma\) is a constant, and \(C_{V} / n\) is measured in units of \(\mathrm{mJ} / \mathrm{mol} \cdot \mathrm{K}\). The first term on the left is the contribution attributable to lattice vibrations and the second term is due to the contribution of free electrons. For copper, \(\Theta\) is \(343 \mathrm{~K}\) and \(\gamma\) is \(0.688 \mathrm{~mJ} / \mathrm{mol} \cdot \mathrm{K}^{2}\). How much heat per mole is transferred during a process in which the temperature changes from 2 to \(3 \mathrm{~K}\) ?

A thick-walled insulated metal chamber contains \(n_{i}\) moles of helium at high pressure \(P_{i}\). It is connected through a valve with a large, almost empty gasholder in which the pressure is maintained at a constant value \(P^{\prime}\), very nearly atmospheric. The valve is opened slightly, and the helium flows slowly and adiabatically into the gasholder until the pressure on the two sides of the valve is equalized. Prove that $$ \frac{n_{f}}{n_{i}}=\frac{h^{\prime}-u_{i}}{h^{\prime}-u_{f}} $$ where \(n_{f}=\) number of moles of helium left in the chamber, \(u_{i}=\) initial molar internal energy of helium in the chamber, \(u_{f}=\) final molar internal energy of helium in the chamber, and \(h^{\prime}=u^{\prime}+P^{\prime} v\) (where \(u^{\prime}=\) molar internal energy of helium in the gasholder; \(v^{\prime}=\) molar volume of helium in the gasholder).

A liquid is irregularly stirred in a well-insulated container and thereby experiences a rise in temperature. If the system is the liquid: (a) Has heat been transferred? (b) Has work been done? (c) What is the sign of \(\Delta U\) ?

One mole of a gas obeys the van der Waals equation of state: $$ \left(P+\frac{a}{v^{2}}\right)(v-b)=R T $$ and its molar internal energy is given by $$ u=c T-\frac{a}{v}, $$ where \(a, b, c\), and \(R\) are constants. Calculate the molar heat capacities \(c_{V}\) and \(c_{P}\).
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