# Are all cyclical processes isothermal

## Heat and the first law of thermodynamics

### 15.1

• Object A has twice the mass of object B. If both objects absorb the same amount of heat, both objects have the same temperature change. What is the relationship between their specific heat capacities? a) \ (c _ {\ mathrm {A}} = 2 \, c _ {\ mathrm {B}} \), b) \ (2 \, c _ {\ mathrm {A}} = c _ {\ mathrm {B} } \), c) \ (c _ {\ mathrm {A}} = c _ {\ mathrm {B}} \), d) none of these relationships.

### 15.2

• In Joule's experiment, which showed the equivalence of heat and work, mechanical energy is converted into internal energy. Give some examples in which the internal energy of a system is converted into mechanical energy.

### 15.3

• Can a certain amount of gas absorb heat without changing its internal energy? If so, give an example. If not, give reasons for your answer.

### 15.4

• In the equation \ (Q = \ Updelta UW \) (a formulation of the First Law of Thermodynamics) the quantities \ (Q \) and \ (W \) stand for: a) the heat supplied to the system and that produced by it Work, b) the heat supplied to the system and the work done on it, c) the heat given off by the system and the work done by it, d) the heat given off by the system and the work done on it. - Which of the statements are correct, which are wrong?

### 15.5

• A certain gas consists of ions that repel each other. The gas experiences a free expansion, during which it neither absorbs heat nor does any work. How does the temperature change? Give reasons for your answer.

### 15.6

• A gas changes its state reversibly from A to C in the \ (p \) - \ (V \) diagram of Fig. 15.19. The work done by the gas is a) greatest for path A \ (\ rightarrow \) B \ (\ rightarrow \) C, b) smallest for path A \ (\ rightarrow \) C, c) largest for path A. \ (\ rightarrow \) D \ (\ rightarrow \) C, d) the same size for all three paths. - Which of these statements is true?

### 15.7

• The volume of a certain amount of a gas remains constant while its temperature and pressure change. Which of the following statements apply or apply to this? a) The internal energy of the gas remains unchanged. b) The gas does no work. c) The gas does not absorb any heat. d) The change in the internal energy of the gas corresponds to the net amount of heat absorbed by it. e) None of these statements apply.

### 15.8

•• Which metal, in your estimation, has the higher heat capacity per unit of mass: Lead or copper? Why? (Don't look up heat capacities before answering the question.)

### 15.9

•• An ideal gas goes through a process in which the product \ (P \ sqrt {V} \) is constant and the gas volume decreases. How does the temperature change during this? Explain the context.

### 15.10

• A power plant is to be built on a coast and seawater is to be used for cooling. The power plant should deliver an electrical output of 1.00 GW and have an efficiency of a third (which is a good value for modern power plants). The heat transfer to the cooling water is therefore 2.00 GW. According to the regulations, its temperature rise must not exceed 10 \ ({} ^ {\ circ} \) C. Estimate how high the cooling water throughput (in kg \ (/ \) s) must be.

### 15.11

•• A normal microwave oven consumes around 1200 W of electrical power. Estimate how long it takes to bring a cup of water to the boil if 50% of that power is used to heat the water. Does the calculated value correspond to your experience?

### 15.12

• A house heated with solar energy consists of, among other things. made of \ (1 {,} 00 \ cdot 10 ^ {5} \ text {\ leavevmode \ nobreak \ kg} \) concrete (specific heat capacity 1.00 \ (\ mathrm {kJ \, kg} ^ {- 1} \ , \ mathrm {K} ^ {- 1} \)). How much heat does this amount of concrete give off when it cools from 25.0 \ ({} ^ {\ circ} \) C to 20.0 \ ({} ^ {\ circ} \) C?

### 15.13

• How much heat has to be added to 60.0 g ice with \ (- 10 {,} 0 \, ^ {\ circ} \) C to 60.0 g water with 40.0 \ ({} ^ {\) circ} \) C to convert?

### 15.14

•• How much heat has to be dissipated if 0.100 kg water vapor is cooled at 150 \ ({} ^ {\ circ} \) C and converted to 0.100 kg ice with 0.00 \ ({} ^ {\ circ} \) C? become?

### 15.15

•• Assume that during his various participations in the Tour de France, the cyclist Lance Armstrong produced an average output of 400 W for 5.0 hours a day for 20 days each time. What amount of water with an initial temperature of 24 \ ({} ^ {\ circ} \) C would have to be heated to the boiling point if the total energy generated by Armstrong during a tour could be made usable?

### 15.16

•• A piece of ice with a mass of 200 g and a temperature of 0.0 \ ({} ^ {\ circ} \) C is placed in 500 g of water with 20 \ ({} ^ {\ circ} \) C. The system is a thermally insulated container with negligible heat capacity. a) How high is the equilibrium temperature of the system in the end? b) How much ice has melted then?

### 15.17

•• A well-insulated container with negligible heat capacity contains 150 g of ice with a temperature of 0.0 \ ({} ^ {\ circ} \) C. a) What equilibrium temperature does the system reach after 20 g of steam at 100 \ ({} ^ {\ circ} \) C have been injected into it? b) Is there still ice when the system is back in equilibrium?

### 15.18

•• A calorimeter with negligible mass contains 1.00 kg of water with 303 K. 50.0 g of ice with 273 K are added. a) What is the final temperature after a while? b) What is the final temperature with an amount of ice of 500 g?

### 15.19

• A certain amount of a diatomic gas does 300 J work and absorbs 2.50 kJ heat. What is the change in internal energy?

### 15.20

• A certain amount of a gas absorbs 1.67 MJ of heat while doing 800 J of work. What is the change in internal energy?

### 15.21

•• A lead bullet with an initial temperature of 30 \ ({} ^ {\ circ} \) C was just melting when it hit a plate inelastically. Assume that the entire kinetic energy of the projectile was converted into its internal energy on impact, thereby causing the temperature increase that led to the melting. What was the speed of the projectile on impact?

### 15.22

• 1.00 mol of an ideal gas has the following initial state: \ (P_ {1} = \ text {3.00 \, bar} \), \ (V_ {1} = \ text {1.00 \, l} \ ) and \ (U_ {1} = \ text {456 \, J} \). The final state is \ (P_ {2} = \ text {2.00 \, bar} \), \ (V_ {2} = \ text {3.00 \, l} \) and \ (U_ {2} = \ text {912 \, J} \). The gas expands at constant pressure to the specified final volume. Then it is cooled at constant volume until it has reached the specified final pressure. a) Create the \ (p \) - \ (V \) diagram for this process and calculate the work that the gas does. b) What amount of heat is added during the process?

### 15.23

•• 1.00 mol of an ideal gas initially has a pressure of 1.00 bar and a volume of 25.0 l. The gas is slowly heated, for which the \ (p \) - \ (V \) diagram shows a straight line to the final state with a pressure of 3.00 bar and a volume of 75.0 l. How much work does the gas do and how much heat does it absorb?

### 15.24

• A certain amount of an ideal gas takes up a volume of 5.00 l at 2.00 bar. It is cooled at constant pressure until the volume is only 3.00 l. What work is being done on the gas?

### 15.25

•• A certain amount of a diatomic gas is at the pressure \ (P_ {0} \) in a closed container with the constant volume \ (V \). What amount of heat \ (Q \) has to be added to the gas in order to triple the pressure?

### 15.26

•• A certain amount of carbon dioxide (\ (\ mathrm {CO_ {2}} \)) sublimes at a pressure of 1.00 bar and a temperature of \ (- 78 {,} 5 \, ^ {\ circ} \) C. So it goes directly from the solid to the gaseous state without passing through the liquid phase. What is the change in the molar heat capacity (at constant pressure) during sublimation? Is the change positive or negative? Assume that the gas molecules can rotate but not oscillate. The structure of the \ (\ mathrm {CO_ {2}} \) molecule is shown in Fig. 15.20.

### 15.27

• Dulong-Petit’s rule was originally used to determine the molar mass of a metallic substance sample from its heat capacity. The specific heat capacity of a certain solid was measured as \ (0 {,} 447 \, \ mathrm {kJ \, kg ^ {- 1} \, K} ^ {- 1} \). a) What is its molar mass? b) Which element can it be?

### 15.28

••• Dulong-Petit’s rule assumes that every atom in the solid has six degrees of freedom, each of which absorbs part of the energy. a) Which six degrees of freedom are there that play a role here? b) Some solids with strong bonds, such as diamond, actually have a significantly lower heat capacity than the heat capacity predicted by Dulong-Petit’s rule. The rule also fails at very low temperatures. Explain.

### 15.29

•• 0.500 mol of a monatomic ideal gas with a pressure of 400 kPa and a temperature of 300 K expand reversibly until the pressure has dropped to 160 kPa. Determine the final temperature, the final volume, the net added work and the net amount of heat absorbed if the expansion is a) isothermal or if it is b) adiabatic.

### 15.30

• Repeat the previous exercise for a diatomic gas.

### 15.31

•• 1.00 mol of a diatomic ideal gas can expand in such a way that the straight line from state 1 to state 2 is traversed in the \ (p \) - \ (V \) diagram (Fig. 15.21). Then they are isothermally compressed from state 2 to state 1, traversing the curved line. Calculate the total work done in this cycle.

### 15.32

••• At point D in Fig. 15.22, 2.00 mol of a monatomic ideal gas have a pressure of 2.00 bar and a temperature of 360 K. At point B in the \ (p \) - \ (V \) diagram the volume of the gas is three times as large as at point D, and its pressure is twice as large as at point C. The paths AB and CD correspond to isothermal processes. The gas goes through a complete cycle along the DABCD path. Determine the net work added to the gas and the net amount of heat added to it in each individual step.

### 15.33

•• A thermally insulated system consists of 1.00 mol of a diatomic ideal gas with a temperature of 100 K and 2.00 mol of a solid with a temperature of 200 K, which are separated from one another by a solid, insulating wall. Determine the equilibrium temperature the system will reach after the wall is removed. Assume that the equation of state for ideal gases or Dulong-Petit’s rule apply.

### 15.34

•• If a certain amount of an ideal gas undergoes a temperature change at constant volume, its internal energy changes by \ (\ Delta U = \ tilde {n} \, C_ {V} \, {\ mskip 2.0mu \ mathrm { d}} T \). a) Explain why this equation gives correct results for an ideal gas even if the volume changes. b) Using this relation and the first law of thermodynamics, show that for an ideal gas the following applies: \ (C_ {P} = C_ {V} + R \).

### 15.35

•• According to Einstein’s model for a crystalline solid, the following applies to its molar internal energy
\$\$ \ begin {aligned} \ displaystyle U _ {\ mathrm {Mol}} = \ frac {3 \, n _ {\ mathrm {A}} \, k _ {\ mathrm {B}} \, {\ varTheta} _ { \ mathrm {E}}} {\ mathrm {e} ^ {{\ varTheta} _ {\ mathrm {E}} / T} -1} \,. \ end {aligned} \$\$
Use this equation to determine the molar internal energy of diamond (\ ({\ varTheta} _ {\ mathrm {E}} = 1060 \ text {\ leavevmode \ nobreak \ K} \)) at 300 K and at 600 K as well as from this the increase in internal energy when 1.00 mol of diamond is heated from 300 K to 600 K.

### 15.36

••• According to Einstein’s model for a crystalline solid, the following applies to its molar internal energy
\$\$ \ begin {aligned} \ displaystyle U _ {\ mathrm {Mol}} = \ frac {3 \, n _ {\ mathrm {A}} \, k _ {\ mathrm {B}} \, {\ varTheta} _ { \ mathrm {E}}} {\ mathrm {e} ^ {{\ varTheta} _ {\ mathrm {E}} / T} -1} \,. \ end {aligned} \$\$
Here \ ({\ varTheta} _ {\ mathrm {E}} \) is the Einstein temperature and \ (T \) is the temperature of the solid to be used in Kelvin. indexSolid! Einstein temperature Use this relationship to show that the molar heat capacity of the crystalline solid at constant volume is:
\$\$ \ begin {aligned} \ displaystyle C_ {V} = 3 \, R \ left (\ frac {{\ varTheta} _ {\ mathrm {E}}} {T} \ right) ^ {\! 2} \ frac {\ mathrm {e} ^ {{\ varTheta} _ {\ mathrm {E}} / T}} {\ left (\ mathrm {e} ^ {{\ varTheta} _ {\ mathrm {E}} / T } -1 \ right) ^ {2}} \,. \ End {aligned} \$\$