Lecture 8-9: The Second Law

5.60 Spring 2008 Lecture #8-9 page 1 The Second Law ▲ First Law ΔU = q + w , showed the equivalence of work and heat ∫ dU = 0 for cyclic process ⇒ q = −w Suggests engine can run in a cycle and convert heat into useful work. ▲ Second Law • Puts restrictions on useful conversion of q to w • Follows from observation of a directionality to natural or spontaneous processes • Provides a set of principles for - determining the direction of spontaneous change - determining equilibrium state of system Heat reservoir Definition: A very large system of uniform T, which does not change regardless of the amount of heat added or withdrawn. Also called heat bath. Real systems can come close to this idealization. Different statements of the Second Law Kelvin: It is impossible for any system to operate in a cycle that takes heat from a hot reservoir and converts it to work in the surroundings without at the same time transferring some heat to a colder reservoir. q>0 w<0 -w=q T1 (hot) q T1 (hot) q1 -w q1>0 w<0 q2<0 -w q1=-w-q2 -q2 IMPOSSIBLE!! T2 (cold) OK!! 5.60 Spring 2008 Lecture #8-9 page 2 Clausius: It is impossible for any system to operate in a cycle that takes heat from a cold reservoir and transfers it to a hot reservoir without at the same time converting some work into heat. q2>0 w>0 -q1 q1<0 w -q1=w+q2 q2>0 T1 (hot) q1<0 -q1 -q1=q2 T1 (hot) q2 q2 IMPOSSIBLE!! T2 (cold) T2 (cold) OK!! Alternative Clausius statement: All spontaneous processes are irreversible. (e.g. heat flows from hot to cold spontaneously and irreversibly) Mathematical statement: ∫ đqrev T đqrev = 0 and T v∫ is a state function = ∫ dS → v∫ dS = đqirrev <0 T đqrev T S ≡ ENTROPY v∫ dS = 0 → ΔS = S2 − S1 = ∫ 2 1 2 đq đqrev irrev >∫ 1 T T irrev rev for cycle [1] ⎯⎯⎯ →[2] ⎯⎯⎯ →[1] 2 ∫1 2 ∫1 1 qrev q irrev q +∫ = ∫ irrev < 0 2 T T T 2 qirrev qirrev − ΔS < 0 ⇒ ΔS > ∫ 1 T T 5.60 Spring 2008 Lecture #8-9 page 3 Kelvin and Clausius statements are specialized to heat engines. Mathematical statement is very abstract. Link them through analytical treatment of a heat engine. The Carnot Cycle - a typical heat engine All paths are reversible T1 (hot) p 1 q1 isotherm (T1) 2 adiabat w adiabat 4 isotherm (T2) q2 3 T2 (cold) ΔU = q1 + w1 1→2 isothermal expansion at T1 (hot) 2→3 adiabatic expansion (q = 0) 3→4 isothermal compression at T2 (cold) ΔU = q2 + w2 4→1 adiabatic compression (q = 0) Efficiency = 1st Law ⇒ ∴ Kelvin: ΔU = w1′ ΔU = w2′ work output to surroundings − (w1 + w1′ + w2 + w2′ ) = heat in at T1 (hot) q1 v∫ dU = 0 ⇒ q1 + q2 = − (w1 + w1′ + w2 + w2′ ) Efficiency ≡ ε = q1 + q2 q =1+ 2 q1 q1 q2 < 0 → Efficiency ≡ ε < 1 (< 100%) -w = q1ε = work output 5.60 Spring 2008 Lecture #8-9 page 4 Note: if cycle were run in reverse, then q1 < 0, q2 > 0, w > 0. It’s a refrigerator! Carnot cycle for an ideal gas ⎛V2 ⎞ ⎟ ⎝V1 ⎠ 2 1→2 ΔU = 0; q1 = −w1 = ∫1 pdV = RT1 ln ⎜ 2→3 q = 0; w1′ = CV (T2 −T1 ) Rev. adiabat ⎛T2 ⎞ ⎛V2 ⎞ ⎜ ⎟=⎜ ⎟ ⎝T1 ⎠ ⎝V3 ⎠ ⇒ γ −1 ⎛V4 ⎞ ⎟ ⎝V3 ⎠ 4 3→4 ΔU = 0; q2 = −w2 = ∫3 pdV = RT2 ln ⎜ 4→1 q = 0; w2′ = CV (T1 −T2 ) Rev. adiabat ⎛T1 ⎞ ⎛V4 ⎞ ⎜ ⎟=⎜ ⎟ ⎝T2 ⎠ ⎝ V1 ⎠ ⇒ q2 T2 ln (V4 V3 ) = q1 T1 ln (V2 V1 ) γ −1 γ −1 ⎛ V1 ⎞ ⎜ ⎟ ⎝V4 ⎠ ⎛T ⎞ ⎛V ⎞ =⎜ 2⎟=⎜ 2⎟ ⎝T1 ⎠ ⎝V3 ⎠ or q1 q2 + =0 ⇒ T1 T2 ⇒ v∫ ⎛V4 ⎞ ⎛V1 ⎞ ⎜ ⎟=⎜ ⎟ ⎝V3 ⎠ ⎝V2 ⎠ đqrev T γ −1 ⇒ q2 T =− 2 q1 T1 =0 links heat engines to mathematical statement Efficiency ε =1+ q2 T =1− 2 q1 T1 → 100% as T2 → 0 K 5.60 Spring 2008 Lecture #8-9 For a heat engine (Kelvin): q1 > 0, w < 0, T2 < T1 ⎛T1 −T2 ⎞ ⎟ q1 ⎝ T1 ⎠ Total work out = −w = ε q1 = ⎜ page 5 ⇒ ( −w ) < q1 Note: In the limit T2 → 0 K, (-w) → q1, and ε → 100% conversion of heat into work. 3rd law will state that we can’t reach this limit! For a refrigerator (Clausius): Total work in But q1 q =− 2 T1 T2 q2 > 0, w > 0, T2 < T1 ⎛T −T ⎞ = w = ⎜ 2 1 ⎟ q1 ⎝ T1 ⎠ ⎛T −T ⎞ ⇒ w = ⎜ 1 2 ⎟ q2 ⎝ T2 ⎠ Note: In the limit T2 → 0 K, w → ∞. This means it takes an infinite amount of work to extract heat from a reservoir at 0 K ⇒ 0 K cannot be reached (3rd law).