360 M. CLAPEYRON ON THE MOTIVE POWER OF HEAT. 
This equation shows also that when a gas varies in volume without 
change of temperature, the quantities of heat absorbed or disengaged by 
this gas are in arithmetical progression, if the increments or reductions 
of volume are in geometrical progression. M. Carnot enunciates this 
result in the work already cited. 
The equation 
Q— Q’=RC log (5) 
expresses a more general law; it includes all the circumstances by 
which the phenomenon can be affected, such as the pressure, the 
volume, and the temperature. 
In fact, since 
we have 
Q — Qi = 26744 © Iog v 
pv 
This equation exhibits the influence of the pressure; it shows that 
equal volumes of all the gases, taken at the same temperature, being com- 
pressed or expanded by the same fraction of their volume, disengage or 
absorb quantities of heat proportionate to the pressure. 
This result explains why the sudden entrance of the air into the va- 
cuum of the air-pump does not disengage a sensible quantity of heat. 
The vacuum of the air-pump is nothing but a volume of gas v, of 
which the pressure p is very small; if atmospheric air be admitted, its 
pressure p will suddenly become equal to the pressure of p! of the at- 
mosphere, its volume v will be reduced to v', and the expression of the 
heat disengaged will be 
ee Toe ee Ce ae 
C log a C cer ae 108 
The heat disengaged by the reentrance of atmospheric air into the 
vacuum will therefore be what this expression becomes when p is there 
U 
made very small; it is then found that log“ becomes very great, 
Pp 
but the product of p by log = is not the less small on that account ; 
in fact we have 
- plog © = pilog p! — p log p = p (log p! — log p), 
a quantity which converges towards zero when p diminishes. ; 
The quantity of heat disengaged will therefore be small in propor- 
tion to the feebleness of pressure in the recipient, and it will be re- 
duced to zero when the vacuum is perfect. 
