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Adiabatic process



The change in internal energy of a system, measured from state 1 to state 2, is equal to

\text{(1)} \qquad \delta U = \alpha R n_2T_2 - \alpha R n_1T_1 = \alpha R (n_2T_2 - n_1T_1)

At the same time, the work done by the pressure-volume changes as a result from this process, is equal to

\text{(2)} \qquad \delta W = P_2V_2 - P_1V_1

Since we require the process to be adiabatic, the following equation needs to be true

\text{(3)} \qquad \delta U + \delta W = 0

Substituting (1) and (2) in (3) leads to

\alpha R (n_2T_2 - n_1T_1) + (P_2V_2 - P_1V_1) = 0 \qquad \qquad \qquad

or

\frac {(P_2V_2 - P_1V_1)} {-(n_2T_2 - n_1T_1)} = \alpha R \qquad \qquad \qquad

If it's further assumed that there are no changes in molar quantity (as often in practical cases), the formula is simplified to this one:

\frac {(P_2V_2 - P_1V_1)} {-(T_2 - T_1)} = \alpha n R \qquad \qquad \qquad

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Graphing adiabats

An adiabat is a curve of constant entropy on the P-V diagram. Properties of adiabats on a P-V diagram are:

  1. Every adiabat asymptotically approaches both the V axis and the P axis (just like isotherms).
  2. Each adiabat intersects each isotherm exactly once.
  3. An adiabat looks similar to an isotherm, except that during an expansion, an adiabat loses more pressure than an isotherm, so it has a steeper inclination (more vertical).
  4. If isotherms are concave towards the "north-east" direction (45 °), then adiabats are concave towards the "east north-east" (31 °).
  5. If adiabats and isotherms are graphed severally at regular changes of entropy and temperature, respectively (like altitude on a contour map), then as the eye moves towards the axes (towards the south-west), it sees the density of isotherms stay constant, but it sees the density of adiabats grow. The exception is very near absolute zero, where the density of adiabats drops sharply and they become rare (see Nernst's theorem).

The following diagram is a P-V diagram with a superposition of adiabats and isotherms:

Image:Entropyandtemp.PNG

The isotherms are the red curves and the adiabats are the black curves. The adiabats are isentropic. Volume is the horizontal axis and pressure is the vertical axis.

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See also

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References

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