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THERMODYNAMICS Entropy: Entropy is defined as a quantitative measure of disorder or randomness in a system. The heat change, dq and the temperature T are thermodynamic quantities. A thermodynamic function whose change (dq/T) is independent of the path of the system is called entropy

푑푆 = 푑푞 푇 1. Entropy is a state function whose magnitude depends only on the parameters of the system and can be expressed in terms of (P,V,T) 2. dS is a perfect differential. Its value depends only on the initial and final states of the system 3. Absorption of heat increases entropy of the system. In a reversible adiabatic change dq=0, the entropy change is zero 4. For carnot cycle ∮ 푑푆 = 0 5. The net entropy change in a reversible process is zero,

∆푆푢푛푖푣푒푟푠푒 = ∆푆푠푦푠푡푒푚 + ∆푆푠푢푟푟표푢푛푑푖푛푔 = 0

In irreversible expansion ∆푆푢푛푖푣 = +푣푒

cyclic processs, ∆푆푢푛푖푣 > 0 All natural process will take place in a direction in which the entropy would increase. A thermodynamically irreversible process is always accompanied by an increase in the entropy of the system and its surroundings taken together while in a thermodynamically reversible process, the entropy of the system and its surroundings taken together remains constant.

Entropy change in isothermal and reversible expansion of ideal gas: In isothermal expansion of an ideal gas carried out reversibly, there will be no change in internal energy, ∆U=0, hence from the first law q = -W

In such case, the work done in the expansion of n moles of a gas from volume V1 to V2 at constant temperature T is given by

– 푊 = 푛푅푇푙푛(푉2) 푉1

푉2 푞푟푒푣 = −푊 = 푛푅푇푙푛( ) 푉1 Hence 푑푆 = 푑푞 = 1 푛푅푇푙푛 (푉2) = 푛푅푙푛 (푉2) 푇 푇 푉1 푉1 (Problem: 5 moles of ideal gas expand reversibly from volume of 8 dm3 to 80 dm3 at a temperature of 27oC. Calculate the change in entropy)

Physical significance of Entropy: 1. Entropy as a measure of disorder of the system: All spontaneous processes, such as flow of heat from hot end to cold end, expansion of a gas in vacuum, diffusion of a solute from a concentrayed to a dilute solution, are accompanied by increase in disorder of the system. Spontaneous processes are acoompanied by increase in entropy as well as increase in disorder of the system. Thus entropy is regarded as a measure of the disorder of a system. 2. Entropy as a measure of probability: All spontaneous processes leads to increase in entropy. When a process is spontaneous, that means a system undergoes a change from less probable to a more probable state. It appears therefore that there is a close relationship between entropy and thermodynamic probability W. This relationship was expressed by S= k lnW (k is boltzman constant)

Work and free energy function: The Helmholtz free energy or work function, A=U-TS

For Isothermal reversible change, ∆AT=∆U-T∆S = ∆U- q = Wmax The work function A is a thermodynamic property of the system. Its decrease gives the maximum amount of work available during isothermal transformation. If the change is not Isothermal, A=U-TS dA = du-TdS-SdT dA= dU-dq-SdT dA= -PdV-SdT

For Isochoric process, (푑퐴) = −푆 푑푇 푉 For isothermal process, (푑퐴) = −푃 푑푉 푇

For isothermal reversible change: G = H-TS = U+PV-TS ∆G = ∆U+P∆V+V∆P-T∆S-S∆T

∆GP,T = ∆U+P∆V-T∆S = ∆U+P∆V+V∆P- q = P∆V –W P∆V is the mechanical work involved in the system itself during transformation and W denotes the maximum output. So, W-P∆V is the amount of work received for any external use, exclusive of the mechanical work during transformation of the system Gibbs free energy is a property whose decrease is the measure of external work available is the measure of external work available during the transformation of the system G= H-TS G=U+PV-TS dG=Du+PdV+VdP-TdS-SdT in reversible process, dU+PdV=TdS So, dG=VdP-SdT

For an isobaric process, (푑퐺) = −푆 푑푇 푃 For an isothermal process, (푑퐺) = 푉 푑푃 푇