Energy per/Area \(\frac{P}{A} = \frac{2\pi^5 k^4}{15h^3c^2}T^4\) k = boltzmans constant h = plancks constant c = speed of light T = temperature in kelvin That's the non-integrated simple form that I use, like many equations there are different forms to suit the different needs. If you know the energy of the body at one point, and the size of the object. It should be simple to find the total energy...say a sphere. \(A = 4\pi r^2\) So... \(P = \frac{8r^2\pi^7 k^4}{15h^3c^2}T^4\)
Would you like equations or not? I'll leave them out for now. Basically, in thermodynamics, internal energy is the total kinetic energy of the constituent particles in a substance, including energy from linear motion, rotation, vibration etc. If we are talking about a gas temperature is more or less related only to the kinetic energy due to linear motion. Ok maybe I will get out one equation: \( 1/T = dS/dU \) T is temperature, S is entropy, U is internal energy. This is the definition of temperature for a system at constant volume in thermal equilibrium. This basically tells you that if a system has more degrees of freedom (for instance if you have a gas of molecules rather than single atoms then the molecules can rotate and vibrate in various ways which the single atom can't) then you have to pump in more energy to make the temperature increase. A simple, more concrete example is \( U = 3/2kT \) where k is the Boltzman constant. This is the same equation as above, basically, applied to a monatomic ideal gas. It's kind of a famous formula. In this case internal energy and temperature are pretty much the same thing, just with a bit of scaling. Actually it is possible that for all ideal gases it turns out pretty similar to this, with a different scaling to account for the different degrees of freedom.
Utter nonsense. The left hand side, pressure/area, has units of (force/area)/area, or mass/length[sup]3[/sup]/time[sup]2[/sup]. The right hand side has units of power, or mass*length[sup]2[/sup]/time[sup]3[/sup]. An "equation" that does not have consistent units throughout is ill-formed. Another name for such an "equation" is utter nonsense.
Presumably that's the Stefan-Boltzmann law for the power emitted by a black body so the left hand side is power/area not pressure/area.
I'm sure you know the convention of using "P" for power. I've never heard of "pressure per area" since pressure is already the measurement of "force per area". \(\frac{J}{t}\frac{1}{A}=\frac{Kg m^2}{s^2}\frac{1}{s}\frac{1}{m^2}=\frac{Kg}{s^3}\) You've assumed that area is in the right hand side, I divided it out to get energy per unit area - which is the way I (personally) typically measure. Brutish comments don't make the world a better place. Please Register or Log in to view the hidden image!
Temperature is a measure of the average internal kinetic energy of atoms or molecules in a substance. The more the atoms rotate, vibrate or move about within the substance, the more kinetic energy they have and the higher the temperature of the object.
We are skirting the issue raised by the original poster, who (implicitly) asked why do we need temperature at all? Why not use internal energy and be done with it? Internal energy is a rather difficult quantity to measure for a real substance. Temperature is much more readily measurable. Another issue with internal energy is that for a real substance the relation between internal energy and temperature is a lot more complex than it is for a monatomic ideal gas (\(U=3/2 kT\), see post #3). Consider the case of gaseous hydrogen. Monatomic gases have three translational degrees of freedom, so the constant volume specific heat (C[sub]V[/sub]) should be \(3/2R\). Hydrogen is diatomic, so it should have at least two more degrees of freedom. A molecule of a diatomic gases such as hydrogen can rotate about either of the two axes normal to the line connecting the atoms that form the molecule. That's two additional degrees of freedom, suggesting that C[sub]V[/sub] should be \(5/2R\) rather than \(3/2R\) for a diatomic gas. The atoms in a molecule are not rigidly connected. Some of the energy can go into vibrations in the molecule itself. That's another degree of freedom, suggesting that C[sub]V[/sub] should be \(6/2R\). So which is it? In low temperature regimes (up to about 80 K or so), gaseous hydrogen has a constant volume specific heat of about \(\frac 3 2 R\). At low temperatures, gaseous hydrogen acts like monatomic ideal gas. It's as if the rotational and vibrational degrees of freedom are not available at low tempatures. Beyond 80K, C[sub]V[/sub] starts rising, eventually plateauing at \(\frac 5 2 R\) between 250K and 750K. The rotational degrees of freedom slowly become available as temperature rises above 80K. Beyond 750K, C[sub]V[/sub] starts rising again, coming close to \(\frac 7 2 R\) at around 3200K, which is when molecular hydrogen dissociates. Bottom line: How internal energy changes and how that relates to temperature are very important concepts from a thermodynamics perspective. Total internal energy is not that important a concept.
