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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.