Exam 15: The Laws of Thermodynamics

A real (non-Carnot) heat engine, operating between heat reservoirs at temperatures of 450 $\mathrm { K }$ and $270 \mathrm {~K}$ , performs $3.3 \mathrm {~kJ}$ of net work, and rejects $8.2 \mathrm {~kJ}$ of heat in a single cycle. (a) What is the thermal efficiency of this heat engine? (b) What is the maximum efficiency it could possibly have?

A heat engine with an efficiency of $30 \%$ performs $2500 \mathrm {~J}$ of work. How much heat is discharged to the lower temperature reservoir?

A container of ideal gas at STP undergoes an isothermal expansion and its entropy changes by $3.7$ $\mathrm { J } / \mathrm { K }$ . How much work does it do?

What is the efficiency of an ideal Carnot engine operating between a reservoir in which ice and water coexist, and a reservoir in which water and steam coexist? The pressure is constant at $1.0$ atm for both reservoirs.

A certain heat engine extracts $1.30 \mathrm {~kJ}$ of heat from a hot temperature reservoir and discharges $0.70$ $\mathrm { kJ }$ of heat to a cold temperature reservoir. What is the efficiency of this engine?

One of the most efficient engines built so far has the following characteristics: The combustion chamber temperature is $1900 ^ { \circ } \mathrm { C }$ , the exhaust temperature $= 430 ^ { \circ } \mathrm { C } , 7.0 \times 10 ^ { 9 }$ cal of fuel produces $1.4 \times 1010 \mathrm {~J}$ of work in one hour. ( $1 \mathrm { cal } = 4.186 \mathrm {~J}$ ) (a) What is the actual efficiency of this engine? (b) What is the power output of this engine? (c) What would be the maximum possible efficiency for an engine using the same temperature extremes?

An ideal gas undergoes the process $a \rightarrow b \rightarrow c \rightarrow a$ shown in the $p V$ diagram. In this figure, $P _ { a } = P _ { c } =$ $3.60 \times 10 ^ { 5 } \mathrm {~Pa} , V _ { b } = V _ { c } = 68.00 \mathrm {~L} , V _ { a } = 35 \mathrm {~L}$ , and $P _ { b } = 5.60 \times 10 ^ { 5 } \mathrm {~Pa}$ . How much work is done by the system in this process?

A certain gas is compressed adiabatically. The amount of work done on the gas is $800 \mathrm {~J}$ . What is the change in the internal (thermal) energy of the gas?

A gas expands from an initial volume of $30.0 \mathrm {~L}$ to a final volume of $65.0 \mathrm {~L}$ at a constant pressure of $110 \mathrm { kPa }$ . How much work is done by the gas during this expansion?

A cyclic process is carried out on an ideal gas such that it returns to its initial state at the end of a cycle, as shown in the $p V$ diagram in the figure. If the process is carried out in a clockwise sense around the enclosed area, as shown on the figure, then the magnitude of the enclosed area represents

What is the change in entropy when $15.0 \mathrm {~g}$ of water at $100 ^ { \circ } \mathrm { C }$ are turned into steam at $100 ^ { \circ } \mathrm { C }$ ? The latent heat of vaporization of water is $22.6 \times 10 ^ { 5 } \mathrm {~J} / \mathrm { kg }$ .

When $0.50 \mathrm {~kg}$ of water at $0 ^ { \circ } \mathrm { C }$ freezes, what is the change in entropy of the water? The latent heat of fusion of water is $33,400 \mathrm {~J} / \mathrm { kg }$ .

The ocean thermal energy conversion project uses the surface water near tropical islands with a temperature of $20 ^ { \circ } \mathrm { C }$ as the hot temperature reservoir, and the water at some depth, with a temperature of $5.0 ^ { \circ } \mathrm { C }$ , as the cold temperature reservoir for a heat engine. What is the maximum possible efficiency of an engine running between those two temperatures?

A monatomic ideal gas undergoes an isothermal expansion at $300 \mathrm {~K}$ , as the volume increased from $0.010 \mathrm {~m} ^ { 3 }$ to $0.040 \mathrm {~m} ^ { 3 }$ . The final pressure is $130 \mathrm { kPa }$ . What is the change in the internal (thermal) energy of the gas during this process? $( R = 8.31 \mathrm {~J} / \mathrm { mol } \cdot \mathrm { K } )$

During each cycle, a refrigerator removes $20.0 \mathrm {~kJ}$ of heat from the freezing compartment and ejects $24.0 \mathrm {~kJ}$ into a room. (a) How much work per cycle is required each cycle to run this refrigerator? (b) What is the coefficient of performance of this refrigerator?

Which of the following is a false statement?

An ideal Carnot engine is operated between a hot and a cold reservoir. The temperature difference between the two reservoirs is $284 ^ { \circ } \mathrm { C }$ . If the efficiency of this ideal engine is $24.0 \%$ , find the temperature of the cold reservoir in degrees Celsius.

An inventor tries to sell you his new heat engine that takes in $40 \mathrm {~J}$ of heat at $87 ^ { \circ } \mathrm { C }$ on each cycle, expels $30 \mathrm {~J}$ at $27 ^ { \circ } \mathrm { C }$ , and does $10 \mathrm {~J}$ of work. Would it be wise to invest in this engine? Back up your conclusion with numerical calculations.

The figure shows a $p V$ diagram for $0.98$ mol of ideal gas that undergoes the process $1 \rightarrow 2$ . The gas then undergoes an isochoric heating from point 2 until the pressure is restored to the value it had at point 1. What is the final temperature of the gas? $( R = 8.31 \mathrm {~J} / \mathrm { mol } \cdot \mathrm { K } )$ .

A coal-fired plant generates $600 \mathrm { MW }$ of electric power. The plant uses $4.8 \times 10 ^ { 6 } \mathrm {~kg}$ of coal each day, and the heat of combustion of coal is $3.3 \times 10 ^ { 7 } \mathrm {~J} / \mathrm { kg }$ . The steam that drives the turbines is at a temperature of $300 ^ { \circ } \mathrm { C }$ , and the exhaust water is at $37 ^ { \circ } \mathrm { C }$ . (a) What is the overall efficiency of the plant for generating electric power? (b) How much thermal energy is exhausted each day? (c) Using the same heat reservoirs, what is the maximum possible efficiency for a heat engine?