„Adiabatikus kitevő” változatai közötti eltérés

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In the preceding, it may not be obvious how <math>C_P</math> is involved because during the expansion and subsequent heating, the pressure does not remain constant. Another way of understanding the difference between <math>C_P</math> and <math>C_V</math> is to consider the difference between adding heat to the gas with a locked piston, and adding heat with a piston free to move, so that pressure remains constant. In that case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. The heat that is added to the gas goes only partly into heating the gas; while the rest is transformed into the mechanical work performed by the piston. In the constant-volume case (locked piston) there is no external motion, and thus no mechanical work is done on the atmosphere. Thus the amount of heat required to raise the gas temperature (the heat capacity) is higher in the constant pressure case.
 
== Ideal gas relations ==
 
For an ideal gas, the heat capacity is constant with temperature. Accordingly we can express the [[enthalpy]] as <math>H = C_P T</math> and the [[internal energy]] as <math>U = C_V T</math>. Thus, it can also be said that the heat capacity ratio is the ratio between the enthalpy to the internal energy:
:<math> \gamma = \frac{H}{U}</math>
 
Furthermore, the heat capacities can be expressed in terms of heat capacity ratio ( <math>\gamma</math> ) and the [[gas constant]] ( <math>R</math> ):
:<math> C_P = \frac{\gamma R}{\gamma - 1} \qquad \mbox{and} \qquad C_V = \frac{R}{\gamma - 1}</math>
 
It can be rather difficult to find tabulated information for <math>C_V</math>, since <math>C_P</math> is more commonly tabulated. The following relation, can be used to determine <math>C_V</math>:
:<math>C_V = C_P - R</math>
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Általában nehéz a <math>c_v</math> állandó térfogat mellett mért fajhőre az irodalomban konkrét értékeket találni, ezért értékét célszerű az alábbi összefüggésből számítani:
:<math>c_v = c_p - R \,</math>
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=== RelationKapcsolata witha degrees of freedomszabadságfokokkal ===
Az adiabatikus kitevő ideális gázok esetén kifejezhető a gázmolekulák fizikai-kémiai <math> f \, </math> szabadságfokával:
The heat capacity ratio ( <math>\gamma</math> ) for an ideal gas can be related to the [[degrees of freedom (physics and chemistry)|degrees of freedom]] ( <math>f</math> ) of a molecule by:
:<math> \gammakappa = \frac{f+2}{f} \,</math>
 
Thus we observe that for a [[monatomic]] gas, with three degrees of freedom:
Így egyatomos gázokra
:<math> \gamma\ = \frac{5}{3} = 1.67</math>,
 
while for a [[diatomic]] gas, with five degrees of freedom (at room temperature):
:<math> \gammakappa\ = \frac{75}{53} = 1.4,67</math>.,
kétatomos gázokra pedig (szobahőmérsékleten):
:<math> \gammakappa = \frac{H7}{U5} = 1,4</math> .
 
Például a [[levegő]] nagyrészt kétatomos gázokból áll, ~78% [[nitrogén]]ből (N<sub>2</sub>) és ~21% [[oxigén]]ből (O<sub>2</sub>) és standard viszonyok mellett gyakorlatilag ideális gáznak tekinthető. A kétatomo gáz molekuláinak öt szabadságfoka van (három transzlációs és két rotációs), a vibrációs szabadságfok csak magas hőmérsékletek esetén jön számításba. Ez
: <math> C_P\kappa = \frac{\gamma5 R}{\gamma+ - 12} \qquad \mbox{and5} \qquad C_V = \frac{R7}{\gamma5} -= 1},4</math>
értéket ad ki, amely jól egyezik a mérések 1,403 eredményével.
 
E.g.: The terrestrial [[air]] is primarily made up of [[diatomic]] gasses (~78% [[nitrogen]] (N<sub>2</sub>) and ~21% [[oxygen]] (O<sub>2</sub>)) and, at standard conditions it can be considered to be an ideal gas. A diatomic molecule has five degrees of freedom (three translational and two rotational degrees of freedom, the vibrational degree of freedom is not involved except at high temperatures). This results in a value of
: <math>\gamma = \frac{5 + 2}{5} = \frac{7}{5} = 1.4</math>.
This is consistent with the measured adiabatic index of approximately 1.403 (listed above in the table).
 
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== Referenciák ==
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