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HW Key _11 s problems part a

# HW Key _11 s problems part a - S37 Calculate W and Q for...

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S37. Calculate W and Q for each of the four steps of a Carnot cycle applied to 1 mole of an ideal gas. Divide the expression that you obtain for total W by Q 2 , and compare the result with - W / Q 2 = (T 2 -T 1 )/T 2 to show that temperatures on the ideal gas scale (those used in your calculation) are proportional to those on the thermodynamic scale (those used in the equation above). By choosing the size of the degree to be the same, the two scales become identical. The Carnot cycle consists of 4 steps. We need to calculate W and Q for each step and then sum them together to find the total heat and work. Step 1 Isothermal expansion W = nRT 2 ln V 2 V 1 Q = nRT 2 ln V 2 V 1 Step 2: Adiabatic expansion W = nC v Δ T = n 3 2 R ( T 1 T 2 ) Q = 0 Step 3: Isothermal compression W = nRT 1 ln V 4 V 3 Q = nRT 1 ln V 4 V 3 Step 4: Adiabatic compression W = nC v Δ T = n 3 2 R ( T 2 T 1 ) Q = 0 Because the process is cyclic, Δ U = 0 and W tot = Q tot . Adding up the work for each of the four steps gives W tot = RT 2 ln V 2 V 1 + 3 2 R ( T 1 T 2 ) RT 1 ln V 4 V 3 + 3 2 R ( T 2 T 1 ) = RT 2 ln V 2 V 1 RT 1 ln V 4 V 3 where we have used n = 1.

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