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--------------------- 1915 - Boltzmann’s Constant - Energy and Temperature
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- Boltzmann’s Constant relates the amount of thermal energy in a body to its temperature. In 1871 Ludwig Boltzmann, at the age of 27, proposed that the quantity of radiation, or energy intensity, was directly proportional to the 4th power of temperature.
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-Ludwig Boltzmann was born in 1844 in Vienna, Austria. He was working off Josef Stefan’s earlier proposal that stated an object emits energy at a rate proportional to its temperature in Kelvin^4. Josef Stefan was born in St. Peter, Austria, in 1835. Boltzmann was also a friend of James Clerk Maxwell who wrote the famous mathematical equations for electromagnetism and showed that light was an electromagnetic wave. Boltzmann applied his mathematics to deduce Stefan’s proposal regarding thermal energy and temperature using thermodynamic principles, entropy, and statistical mechanics. In other words, he did the math.
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- The total energy emitted from each square meter of a body’s surface each second is called energy flux. And energy flux = (Stefan-Boltzmann’s constant) * Kelvin^4.
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----------------- Energy = (5.67 * 10^-8 joules/m^2 * K^4*sec) * T^4
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----------------- Stefan-Boltzmann’s Constant is 5.67 * 10^-8 joules/m^2 * K^4*sec
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------------------ “T” is the temperature in degrees Kelvin.
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- Raising the temperature from 300K to 600K increases the radiated energy by 2^4, or 16 times. Raising the temperature to 6000K, the surface of the Sun, increases radiated energy to 20^4, or 160,000 times more energy. This is the energy emitted per square meter each second. If we know the surface area of the Sun we can calculate its total energy emitted per second.
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- Energy from the Sun per square meter = (5.67*10^-8) * 6000^4
- Energy from the Sun per square meter = 7.348 * 10^7 joules/m^2*sec
- The surface are of the Sun = 4*pi*r^2
- The radius of the Sun = 6.96*10^8 meters ( or, 432,500 miles )
- The surface are of the Sun = 6.09 * 10^18 square meters
- Energy from the Sun per second = 45 * 10^25 joules/second
- Energy from the Sun = 450,000,000,000,000,000,000,000,000 watts (note 1)
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- So, if we know the Temperature of the Sun we can calculate its energy, or luminosity, which is power, or, total energy emitted each second. Joules of energy / second is watts of power. But, how do we know the temperature of the Sun?
- We determine the temperature of the Sun by using Wien’s law which says that the energy intensity of each wavelength emitted by a blackbody is determined solely by its temperature. A blackbody is a body that absorbs or emits all wavelengths and is therefore perfectly black. The Sun by definition is a “blackbody“. The peak wavelength emitted is inversely proportional to the temperature.
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- Maximum wavelength = 2.9*10^-3 / T
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- So, we measure the intensity of the Sun at each wavelength of light. Then, we determine the wavelength where the intensity peaks. The wavelength peaks at 500 nanometers, or the color of blue-green light. The Sun looks yellow to us, or 600 nanometers wavelength, because we see the Sun through our atmosphere which filters out the shorter wavelengths.
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- T = 2.9*10^-3 / maximum wavelength
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- T = 2.9*10^-3 / 500 * 10^-9 meters
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- T = 5800 Kelvin
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- The surface temperature of the Sun is 5800 Kelvin. Temperature is inversely proportional to maximum wavelength allows us to measure the temperature of the stars. Reddish stars are 2000-3000 Kelvin. Orange stars are 3000-5000 Kelvin. Yellow stars are 5000-8000 Kelvin and white stars are 8000-12,000 Kelvin.
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- Wilhelm Wien was born in 1864 in Prussia. He received the Nobel Prize in 1911 for discovering this law. He died in 1928 at the age of 64. Ludwig Boltzmann died in 1906 at the age of 62. He committed suicide due to episodes of severe mental depression brought on by other scientists disagreeing with his theories.
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- Stefan-Boltzmann Constant = 5.67*10^-8 watts /meter^2 / Kelvin^4
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The Stefan Boltzmann Constant is the energy in watts per square meter per the 4th power of temperature.
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- The Stefan-Boltzmann Constant came from = 2*pi^5*k^4 / 15*h^3*c^2
- Where “h” is Planck’s Constant of Action
- Where “c” is the speed of light
- Where “ k” is Boltzmann’s other constant.
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- Boltzmann’s other constant = 1.38065 * 10^-23 joules / Kelvin:
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- The Boltzmann Constant came from = the Molar Gas Constant / Avogadro’s Number
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- Amedeo Avogadro was born in Turin, Italy in 1776. He lived to age 80, to 1856. Avogadro invented the word “molecule” and separated it from atoms. He was studying the electrolysis of water separating it into hydrogen and oxygen. He learned that water was 2 parts hydrogen and one part oxygen. He also learned that equal volumes of all gases contain the same number of molecules, assuming that temperature and pressure is the same. Equal volumes of all gases contain the same number of molecules. What is that number of molecules? It is called Avogadro’s Number.
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- When hydrogen and oxygen are a gas they are molecules, with molecular weights , H=2 and O=32. The atomic weights are individual atoms of hydrogen and oxygen, H=1 and O=16. H2O has a molecular weight or 18, H=2, O=16. It is convenient for physicists to express molecular weights in grams. So, the gram-molecular weight of water is 18 grams, and this is referred to as a “mole”. A mole of water has a mass of 18 grams.
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- A mole of hydrogen gas (2 grams) at zero Centigrade has a volume of 22.4 liters (5.5 gallons). A mole of oxygen (32 grams) takes up the exact same volume, 22.4 liters. Each gas, hydrogen and oxygen, and every other gas has the same number of molecules per mole. Once you know how many molecules are in a known volume and known weight you can calculate the weights and diameters of each molecule, and therefore, of each atom.
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- Johan Loschmidt was born in Bohemia ( now Czechoslovakia) in 1821. As a businessman he went bankrupt and became a teacher in 1856 and began publishing papers. He was the first one to calculate Avogadro’s Number.
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- Avogadro’s Number = 6.0221367*10^23 molecules / mole
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- One mole of any gas has 602,213,670,000,000,000,000,000 molecules.
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- A mole of oxygen weighs 32 grams and contains 6.02*10^23 oxygen molecules. Therefore, each molecule of oxygen weighs 5.3*10^-23 grams.
An atom of oxygen weighs 1.67343*10^-24 grams.
The diameter of an oxygen atom, and any other atom is 10^-10 meters.
254,000,000 atoms placed side-by-side would make a line one inch long.
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- The Molar Gas Constant comes from a mole of gas where pressure * volume = Molar Gas Constant * Temperature. p*V = R*T. The Molar Gas Constant, R = 8.314510 joules / Kelvin / mole
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- Now, back to Boltzmann’s Constant:
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- Boltzmann’s other constant = 1.38*10^-23 joules / Kelvin:
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- The Boltzmann Constant came from = the Molar Gas Constant / Avogadro’s Number
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Boltzmann’s Constant = 8.3 joules / Kelvin / mole // 6.02*10^23 molecules / mole
Boltzmann’s Constant = 1.3*10^-23 joules / Kelvin.
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- So, in summary you need Avogadro’s Constant Number to calculate Boltzmann’s Constant. And Boltzmann’s Constant to calculate the Stefan-Boltzmann Constant which relates Energy to Temperature^4. Ludwig Boltzmann knew that thermal energy measured by temperature was created by atoms constantly flying around like tiny missiles.
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- Their individual velocities vary a great deal but their velocities can be described using statistics. For example, the mean velocity of an oxygen gas molecule at room temperature and pressure is 0.3 miles per second, or 1,080 miles per hour. It is the kinetic energy of these fast moving atoms that we measure as temperature and it is Boltzmann’s Constant that relates the two, temperature with energy. It amazes me that this was all figured out in Europe in the mid-1800’s, over a hundred fifty years ago, by some very smart people.
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- Note (1): More accurate answer is 3.827 * 10^26 watts because actual surface temperature of the Sun is 5800 Kelvin, by measuring the peak wavelength from above the Earth’s atmosphere.
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- Entropy is mathematical. It is the result of statistics. The is more possibilities of things to be disorderly than orderly. So, if things vibrate, fluctuate, change then they are eventually gravitate to “ disorderly’. That is Entropy. The same thing happens with atoms.
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- Joule - is a unit of Energy whereby one Newton of Force acts over one meter of work.
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- Newton - is a unit of Force that will accelerate a mass of 1 kilogram to 1 meter / second^2
------------------------------ F = m*a
----------------------------- Force = mass * acceleration
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