Criteria for spontaneity
Consider a system in thermal equilibrium with its surroundings at temperature T. When change in the system occurs and there is a transfer of energy as heat between the system and the surroundings the Clausius inequality (dS ≥ dq/Tsurr) reads:
dS - dU/Tsurr ≥ 0 (2.0a)
The importance of the inequality in this form is that it expresses the criterion for spontaneuos change solely in terms of the state functions of the system. The inequality is easily rearranged to
TdS ≥ dU (constant V, no additional work) (2.0b)
At either constant internal energy (dU = 0) or constant entropy (dS = 0), this expression becomes, respectively
dSU,V ≥ 0 dUS,V ≤ 0 (2.0c)
where the subscripts indicate the constant conditions. These relations express the criteria for spontaneuos change in terms of properties relating to system. The first inequality states that, in a system at constant volume and constant internal energy (such as an isolated system), the entropy increases in a spontaneuos change. That statement is essentially the content of the Second Law and corresponds to the tendency of an isolated system to collapse into its most probable distribution and never into a less probable one. The second inequality is less obvious, for it says that, if the entropy and volume of the system are constant, then the internal energy must decrease in a spontaneuos change. Do not interpret this criterion as a tendency of the system to sink to lower energy. It is a disguised statement about entropy, and should be interpreted as implying that, if the entropy of the system is unchanged, then there must be an incerase in entropy of the surroundings, which can be achieved only if the energy of the system decreases as energy flows out as heat.
Example of dUS,V ≤ 0: Rigid container with partition between two chambers of gas at two different temperatures. Container initially insulated. Two step process. Step 1, remove partition and allow gases to equilibrate adiabatically, so entropy increases. Step 2, remove insulation and place container in contact with an ideal reservoir at a low enough temperature so that the entropy decrease in step 2 cancels the entropy increase from step 1.
When energy is transferred as heat at constant pressure, and there is no work other than expansion work, we can write dqp = dH and obtain
TdS ≥ dH (constant P, no additional work) (2.1)
At either constant enthalpy or constant entropy this inequality becomes, respectively,
dSH,P ≥ 0 dHS,P ≤ 0
The interpretations of these inequalities are similar to those of eqn. The entropy of the system at constant pressure must increase if the enthalpy remains constant (for there can then be no change in entropy of the surroundings). Alternatively, the enthalpy must decrease in the entropy of the system is constant, for then it is essential to have an increase in entropy of the surrounding.
Because eqns and have the forms dU - TdS ≤ 0 and dH - TdS ≤ 0 respectively, they can be expressed more simply dy introducing two more thermodynamic quantities. One is the Helmotz energy, A, which is defined as
A ≡ U - TS
The other function we introduce is the Gibbs energy, G
G ≡ H - TS
When the state of the system changes at constant temperature, the two properties change as follows:
dA = dU - TdS dG = dH -TdS
When we introduce eqns. 2.0b and 2.1a, repsectively, we obtain the criteria of spontaneuos change as
dAT,V ≤ 0 dGT,P ≤ 0
These inequalities are the most important conclusion from thermodynamics for chemistry. They are developed in subsequent sections and chapters.
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