Consider a reversible Carnot cycle with 2.5 mol of an ideal gas with Cv= 3/2R as
ID: 2307014 • Letter: C
Question
Consider a reversible Carnot cycle with 2.5 mol of an ideal gas with Cv= 3/2R as the working substance. The initial isothermal expansion occurs at the hot reservoir temperature of T(hot) = 750. K from an initial volume of 31.1775 L (Va) to a volume of 62.3550 L (Vb). The system then undergoes an adiabatic expansion until the temperature falls to Tcold = 345. K at point c. The system then undergoes an isothermal compression to point d and a subsequent adiabatic compression until the initial state described by Ta = 750 K and Va = 31.1775 is reached.
Calculate w, q, Delta U and Delta H for each step in the cycle and the total cycle. What is the efficiency, epsilon for this heat engine?Explanation / Answer
Number of moles n = 2.5
Hot reservoir temprature T = 750 K
Cold reservoir temprature T ' = 345 K
Initial volume V = 31.1775 L
= 31.1775 x10 -3 m 3
Final volume in isothermal expansion V ' = 62.355 L = 62.355 x10 -3 m 3
Isothermal expansion :
Work done W = nRT ln(V ' / V ) Where R = Gas constant = 8.314 J / mol K
= 2.5 x 8.314 x750 x ln(62.355/31.1775)
= 2.5 x8.314 x 750 xln(2)
= 10800 J
Change in internal energy U = 0
Heat Q = U + W
=10800 J
Adiabatic expansion:
Heat Q ' = 0
Work done W ' = [nR/(r-1)][T-T'] Where r = ratio of specific heats = 1.67
=[(2.5 x8.314)/(1.67 -1)][750-345]
= 12564 J
Change in internal energy U ' = Q ' - W '
= -12564 J
Volume at point C is V " = ?
In adiabatic process T V r-1 = constant
Apply this equation to b and c points,
T V ' r-1 = T ' V" r-1
(V"/V) r-1 = T / T ' = 750/345
= 2.173
V " / V ' = (2.173) 1/(r-1)
= 3.186 Since r = 1.67
V " = 3.186 V '
= 3.186 x 62.355 x10 -3 m 3
= 0.1987 m 3
Isothermal compression :
We don't know the pressure or volume values after isothermal compression.
If you know any one value you find V"' i.e., volume of the gas after isothermal compression
Work done W "= nRT ' ln(V "' / V '' ) Where R = Gas constant = 8.314 J / mol K
From this you find W " .
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