Entropy

Problem:

An ice tray contains 500 g of water.  Calculate the change in entropy of the water as it freezes completely and slowly at 0 oC.
latent heat of fusion for water/ice = 333000 J/kg

Solution:

Problem:

A cup contains 0.2 kg of tea and is heated to 90 oC.  The specific heat capacity of tea is ~ 1100 cal/(kg K).  If the tea is allowed to cool to 20 oC calculate the change in entropy (J/K).  Also calculate the entropy change in the room (Troom = 20 oC) and show that the total change in entropy is positive.

Solution:

Problem:

100 g of water at 20 oC is mixed with 100 g of water at 80 oC.  What is the net change in entropy?

Solution:

Problem:

The surface of the Sun is approximately at 5700 K, and the temperature of the Earth's surface is approximately 290 K.  What entropy changes occur when 1000 J of thermal energy is transferred from the Sun to the Earth?

Solution:

Problem:

A certain amount of water of heat capacity C is at a temperature of 0°C.  It is placed in contact with a heat reservoir at 100°C and the two come into thermal equilibrium.
(a)  What is the entropy change of the universe?
(b)  The process is now divided into two stages: first the water is brought into contact with a heat reservoir at 50°C and comes into thermal equilibrium; then it is placed in contact with the heat reservoir at 100°C.  What is the entropy change of the universe?
(c)  What is the entropy change of the universe in the limit of infinitely many stages with infinitely small temperature differences?

Solution:

Problem:

Consider two identical blocks of material with heat capacity C, which is temperature independent. The "hot" one is initially at temperature TH and the "cold" one at temperature TC < TH.   The two blocks act as reservoirs for a ideal heat engine.  As the engine works, the reservoirs gradually equilibrate and reach a common temperature Tf
(a)  What is the temperature Tf?
(b)  How much work does the engine do?
(c)  Compute the change in entropy ΔS for each of the two blocks.

Solution:

Problem:

An ideal heat engine is powered by two reservoirs of equal heat capacity C, which is temperature independent.  As the engine works, the reservoirs gradually equilibrate. 
(a)  Find the overall efficiency of the engine from the starting point where the reservoirs are at temperatures T1 = 90 oC and T2 = 30 oC to the moment of complete equilibration.
(b)  Now assume that a real heat engine is powered by the same reservoir with initial temperatures T1 = 90 oC and T2 = 30 oC.  Let C = 105 J/oC.  The real engine's overall efficiency is 20% of the overall efficiency of the ideal engine.  Find the change in entropy of the system once the reservoirs have equilibrated.

Solution:

Problem:

A Carnot heat engine has the following entropy-temperature diagram.

image

(a)  Describe the cycle.  For each segment identify the process, say whether work is done by the working system or on it and whether heat is added to the system or extracted from it.
(b)  How much work is done by the system?