Question

Suppose a heat engine is connected to two energy reservoirs,one a pool of molten aluminum at 660°C and the other a block ofsolid mercury at -38.9°C. The engine runs by freezing 3.00 g ofaluminum and melting 45.0 g of mercury during each cycle. Thelatent heat of fusion of aluminum is 3.97 105 J/kg, and thatof mercury is 1.18 104 J/kg.

(a) What is the efficiency of this engine?
_________%
(b) How does the efficiency compare with that of a Carnotengine?
_________%

Answer 1

a)
`\eta=0.9998\ or\ 99.98\%`

b)
`\eta_c=0.7490\ or\ 74.9\%`

`\rm \eta>\eta_c\ is\ not\ possible`

Step-by-step explanation:

Given:

• temperature of source reservoir,
  `T_H=660+273=933\ K`

• temperature of sink reservoir,
  `T_L=-38.9+273=234.1\ K`

• quantity of aluminium frozen by the engine during 1 cycle,
  `m_a=0.003\ kg`

• quantity of mercury melted by the engine during 1 cycle,
  `m_m=0.045\ kg`

• latent heat of fusion of aluminium,
  `L_a=3.97* 10^5\ J.kg^(-1)`

• latent heat of fusion of mercury,
  `L_m=1.18* 10^4\ J.kg^(-1)`

a)

Heat absorbed by the engine:

`Q_H=m_a.L_a`

`Q_H=0.045* (3.97* 10^(5))`

`Q_H=178650\ J`

Heat rejected by the engine:

`Q_L=m_m.L_m`

`Q_L=0.003* (1.18* 10^(4))`

`Q_L=35.4\ J`

Now the efficiency of the engine:

`\eta=(Q_H-Q_L)/(Q_H)`

`\eta=(178650-35.4)/(178650)`

`\eta=0.9998\ or\ 99.98\%`

b)

Now the Carnot efficiency of the engine:

`\eta_c=1-(T_L)/(T_H)`

`\eta_c=1-(234.1)/(933)`

`\eta_c=0.7490\ or\ 74.9\%`

`\rm \eta>\eta_c\ is\ not\ possible`