Download Isothermal Compression A process is called isothermal, if the

Isothermal Compression
A process is called isothermal, if the temperature of the system does not change. It is therefore
possible to use the following simplification of the main equation (equation 1):
−p · dV = −T · dS + cV · dT = −T · dS
since dT = 0 by definition
Substituting p = n R T V −1 (ideal gas law) results in:
−p dV =
−n R T
dV = −n R T d ln V = −T dS
V
n R d ln V = dS
To obtain Boyle’s law from this equation dS needs to be replaced by an expression containing
dp. Such an expression can be found by exploiting the corresponding reciprocity relation 1 and
applying to it the ideal gas law.
nR
dV
d nRT
dS
=−
=−
=−
dp T
dT p
dT
p
p
dS = −
nR
dp = −n R d ln p
p
Substituting dS in the main equation yields
−pdV = nRT d ln p
resp.
− nR d ln V = nR d ln p
and by subsequent integration Boyle’s law is finally obtained.
Z p
Z V
V
V0
p
= − ln
= ln
d ln p = −
d ln V
ln
p
V
V
0
0
p0
V0
1
V0
p
=
p0
V
All reciprocity relations can be summarized in a single mathematical operation called turning of a differential
quotient. G. Job; Z. Naturforsch. 25a 1970 1502 – 1508.