Problems
Isothermal Stirring of Mercury
2 litre of mercury confined in a container in thermal contact to an environment at 15 ◦C are stirred with a 200 W stirrer. How much entropy is created in 20 minutes of stirring? Does the entropy of the mercury change? What happens to the entropy created?
Adiabatic Stirring of Mercury
2 litre of mercury confined in an isolated container are stirred with a 200 W stirrer. When the mercury was at 15 ◦C initially, what is its temperature after 20 minutes of stirring? How much entropy is created in the process? What happens to the entropy created?
Kneading of Pizza Dough
2 kg of dough (c = 2.73 kJ ) confined in a container are kneaded with a 350 W kitchen mixer.
1. The container is in thermal contact to an environment at 25 ◦C so that the temperature of the dough is 25 ◦C at all times. How much entropy is created in 10 minutes of kneading?
2. The container is thermally isolated. When the dough was at 25 ◦C initially, how long does it take until the temperature is 40 ◦C? How much entropy is created in the process?
3. Both processes are irreversible, hence entropy is created. Explain where the produced entropy goes.
Stirring of Petroleum
4 litre of petroleum (ρ = 640 kg , cp = 2.0 kJ ) confined in an isolated rigid container are stirred by an electric motor which consumes 50 W of electrical power.
1. How long does it take until the temperature of the petroleum is raised by 5 ◦C?
2. What is the relation between entropy generation and power? Is the process reversible or irreversible?
Industrial Stirrer
During manufacture, 2 tons of polyethylene (incompressible liquid, specific heat c = 2.9308 kJ ) are stirred in a well-insulated container for 20 minutes. It is observed that the temperature rises from 42 ◦C to 49 ◦C. Ignoring kinetic and potential energies, determine the power demand of the stirrer, and the entropy generated during the process.
A Brick Falls
A 2 t brick cube falls to the ground on a planet without atmosphere. The gravitational acceleration is 1 m . The cube crashes on the ground and comes to rest. From what height must the cube fall to increase its temperature by 10 K? When the brick’s initial temperature was 200 K, how much entropy is created in the process? How much work could be obtained in a reversible process? Brick: ρ = 1922 kg , c = 0.79 kJ
A Bad Accident
A 2t truck running at a speed of 120 km/ h crashes against a concrete wall and comes to rest. Assume that the truck is made of steel (ρ = 7830 kg , c = 0.5 kJ ), and that all energy stays in the truck.
1. By what amount will the average temperature of the truck change?
2. How much entropy is created in the process? Assume initial temperature is T0 = 20 ◦C.
3. How much work could have been obtained in a reversible process, e.g., by electromagnetic brakes that charge a battery? Compare the possible work to T0Sgen.
Dissipation of Kinetic Energy
In Sec. 4.9 it was shown that in an isolated system kinetic energy will vanish in equilibrium. Repeat the proof for a non-adiabatic system.
Tank and Contents
A well-insulated steel tank of mass 10 kg contains 5 litre of liquid water. Initially, the temperature of the tank is 7 ◦C, and the temperature of the water is 90 ◦C. Specific heats: csteel = 0.5 kJ , cwater = 4.18 kJ
1. What is the temperature of the system after equilibrium is established?
2. Compute the change of entropy of the tank.
3. Compute the change of entropy of the water.
4. How much entropy is created in the process? Is the process reversible or irreversible?
Property Change in Argon
The state of argon (ideal gas with constant specific heats) is changed by heating and compression from initial state p1 = 1 bar, T1 = 230 K to the final state p2 = 20 bar, T2 = 400 K. Compute the change of internal energy and the change of entropy of the gas. Do you have enough information to compute the heat and work exchanged? Why not?
Work and Heat
Krypton (Kr) gas at T1 = 230 ◦C is confined in a piston-cylinder system. The gas undergoes a reversible process where the pressure changes according to the relation p = p1(V1/V )2. The initial and final volumes are V1 = 0.2 m3 and V2 = 0.1 m3 and the initial pressure is p1 = 4 bar.
As all monatomic gases, krypton behaves as an ideal gas with constant specific heat; its molar mass is M = 83.8 kg . Determine:
1. The mass of Kr in the system.
2. Pressure p2 and temperature T2 at the end of the process.
3. The total work for the process.
4. The total heat exchange.
5. The change of entropy of the gas.
Irreversible Expansion of Xenon
Xenon (ideal gas with constant specific heats) is confined in one half of a 2.5 litre container. The other half of the container is evacuated, and the container is well-insulated. When the partition is removed, the gas expands irreversibly to fill the whole container. Initially, the xenon is at p1 = 20 bar, T1 = 400 K. Compute the final state p2, T2 and the entropy generated.
Irreversible Expansion of Neon
Neon (ideal gas with constant specific heats) is confined in a 1 litre gas container at p1 = 13 bar, T1 = 500 K. This container is enclosed in an evacuated rigid container of unknown volume, which is well-insulated. The inner container becomes defect, and the neon expands irreversibly to fill the accessible volume. The final pressure is measured as 4 bar. From the first and second law determine the final temperature T2, the volume of the bigger container, and the entropy generated.
Ideal Gas with Non-constant Specific Heat
1. To your table, add a column for the entropy at standard pressure p0, defined as s0 (T ) = [ T cp (T t ) dT + s for temperatures in the range (300 K, 900 K). As reference value chose s0 (300 K) = 7.14 kJ .
2. 3 kg of air are heated in a reversible isochoric process (constant volume) from 320 K, 2 bar to 800 K. By means of your table, determine the work W12, the heat supply Q12, and the change in entropy, S2 − S1.
3. Redo the calculation of 2. under the assumption that the specific heat can
be approximated by its value at 300 K (so that it is constant). Determine the relative errors for heat, work and entropy difference.
Equilibrium State I
N blocks of different metals with masses mi, specific heats ci and temperatures Ti are enclosed in an adiabatic rigid chamber. All blocks are brought into thermal contact. Use the first and second law to show that in equilibrium all blocks must have the same temperature.
Hint: In equilibrium entropy must be a maximum. Since energy is con- served, entropy must be maximized under the constraint of given energy. The most elegant way to solve the problem is using the method of Lagrange multipliers to take care of the constraint.
Equilibrium State II
An insulated container holds the mass m0 = [ ρdV of an ideal gas, and the overall energy is fixed at E0 = [ ρ (u (T ) + 1 V2 + gz) dV . Note that in general density ρ, temperature T and velocity V depend on location −→r .
Show that in equilibrium temperature is homogeneous, density follows the barometric law, and velocity vanishes.
Hint: Here you have to maximize total entropy S = [ ρs (T, ρ) dV under constraints of given mass and energy. Use Lagrange multipliers and Euler’s equation of variational calculus.
Equilibrium State III
An insulated room contains a rigid shelf on which rests a metal ball (mass m, specific heat c, initial temperature T ). The shelf is at height H above the floor. By using first and second law, answer the following questions: Is the system in a thermodynamic equilibrium state, and if so, why? If not, what is the system’s thermodynamic equilibrium state, and why? Does your answer depend on whether the room is evacuated, or filled with air? If you find the system is not in thermodynamic equilibrium, why do we find it in the unstable configuration?