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The efficiency of a Carnot engine is 1/6. On decreasing the temperature of the cold reservoir (sink) by 65 K, the efficiency increases to 1/3. Find the ratio of the temperature of the source to that of the sink (in degrees Celsius), given that 0 deg C = 273 K (use the hint that 1 deg C = 274 K as stated in the problem).

1. 1:2
2. 2:1
3. 3:2
4. 4:3

Correct answer: 2:1

Solution

1 - T_c/Tₕ = 1/6 gives T_c/Tₕ = 5/6. From equation 2: 1 - (T_c-65)/Tₕ = 1/3, so (T_c-65)/Tₕ = 2/3. Subtracting: T_c/Tₕ - (T_c-65)/Tₕ = 5/6 - 2/3 = 5/6 - 4/6 = 1/6. So 65/Tₕ = 1/6, giving Tₕ = 390 K. T_c = (5/6)*390 = 325 K. In Celsius: Tₕ_C = 390-273=117 deg C, T_c_C = 325-273=52 deg C. Ratio in Celsius = 117:52... but the problem says 1 deg C = 274 K (so 0 deg C = 273 K). Using this: Tₕ = 390 K -> in Celsius where 1 deg C = 274 K: Tₕ_C = 390/274 deg C? That is unusual. Let me interpret the given hint differently: the problem states '1 deg C = 274 K' meaning the offset is such that T(K) = T(C) + 273... the standard relation. So T_source_C = 390-273=117, T_sink_C = 325-273=52. Ratio = 117:52 = 9:4. Hmm, not matching options. If the problem says temperatures in Celsius are in ratio: let's try Tₕ_C = 117 C and T_c_C = 52 C; 117/52 is not 2:1. If the problem means Kelvin: Tₕ/T_c = 390/325 = 6/5 = 1.2. Not in options. Hmm. Maybe if we adjust for the hint: 1 deg C = 274 K means T(K) = T(deg C) + 274 (a non-standard calibration). Then Tₕ(deg C) = 390-274 = 116, T_c(deg C) = 325-274 = 51. Not matching options either. Let me try: sources at 2:1 ratio in Celsius would require Tₕ_C = 2*T_c_C. With Tₕ = 390 K, T_c = 325 K: Tₕ_C/T_c_C = (390-273)/(325-273) = 117/52 which does not equal 2. Let me try another approach: maybe the problem stated the answer in Kelvin: Tₕ/T_c = 390/325 = 1.2, not matching any option. Or maybe I made an error. Let me retry with efficiency 1/3 in the second case: eta2 = 1/3 means T_c'/Tₕ = 2/3. With T_c' = T_c - 65: (T_c-65)/Tₕ = 2/3. From case 1: T_c/Tₕ = 5/6. Difference: 65/Tₕ = 5/6 - 2/3 = 5/6 - 4/6 = 1/6. Tₕ = 65*6 = 390 K. T_c = 5/6 * 390 = 325 K. These are correct. Ratio Tₕ(C):T_c(C) = 117:52 ≈ 2.25:1. Closest option is 2:1. Given the hint about 1 deg C = 274 K (which seems to mean temperatures in K), the ratio in Kelvin is 390:325 = 78:65 = 6:5. None exactly match 2:1. The problem's hint '1 deg C = 274 K' is unusual and likely means to convert: if the sink is lowered by 65 C = 65 K, and if source:sink (in Celsius) ratio is asked... Actually with Tₕ = 390 K = 117 C and T_c = 325 K = 52 C: ratio in deg C is 117:52. Perhaps the expected answer assumes the ratio in Kelvin: 390:325 is not an option, or perhaps there is a different reading. Given the options, 2:1 is likely the intended answer (Tₕ_C: T_c_C approximated), even if not exactly.

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