Question 1

A heat pump is used to heat a house in Minnesota in the winter and has half the ideal coefficient of performance for a reversible heat pump. If the outside air temperature is −15°C and the indoor temperature desired is 20°C, what is its coefficient of performance under these conditions?

Accepted Answer

The answer of A heat pump is used to heat...

Question 2

A friend tells you that he knows of a situation in which heat is passed from one body to another body with a higher temperature. He explains that he thinks the 2nd law of thermodynamics must have been violated. Is he correct?

Accepted Answer

The 2nd law of thermodynamics states that heat cannot spontaneously flow from a cooler body to a hotter body without external work being done on the system. This principle allows for situations where heat is transferred from a cooler to a hotter body if work is applied, thus not violating the 2nd law.

Question 3

100 coins are tossed randomly onto a table. The macrostate is defined as the number of heads that appear after the toss. A microstate is a specification of the outcome of each individual coin. What is the value for the number of heads that has the greatest number of possible microstates?

Accepted Answer

The total number of possible microstates is 2^100, which is a very large number. The number of possible microstates for a given macrostate (i.e. the number of heads) is given by the binomial coefficient (100 choose X) where X is the number of heads. The graph of binomial coefficients peaks at the midpoint, which in this case is X=50. Therefore, the value for the number of heads that has the greatest number of possible microstates is 50.

Question 4

An engineer uses a river (minimum temperature of 4°C) to cool his electrical plant, which heats water with a boiler operating at 220°C. His boiler uses 420 kW of energy to heat water to steam. What is the maximum power output under ideal circumstances?

Accepted Answer

The maximum power output under ideal circumstances can be calculated using the Carnot efficiency formula, which is $ \eta = 1 - \frac{T_c}{T_h} $, where $ \eta $ is the efficiency, $ T_c $ is the cold reservoir temperature in Kelvin, and $ T_h $ is the hot reservoir temperature in Kelvin. Converting the given temperatures to Kelvin, we have $ T_c = 4°C + 273.15 = 277.15K $ and $ T_h = 220°C + 273.15 = 493.15K $. Thus, the efficiency is $ \eta = 1 - \frac{277.15}{493.15} \approx 0.438 $ or 43.8%. The maximum power output is then $ 0.438 \times 420 kW = 184 kW $.

Question 5

A system in a certain macrostate has 1.2 × 10³ microstates. Through some process, the system changes the number of microstates to 3 × 10⁴. What is the change in entropy for the process?

Accepted Answer

The change in entropy ΔS is given by:
ΔS = k ln(Nf/Ni)
where k is the Boltzmann constant, Ni is the initial number of microstates, and Nf is the final number of microstates.

Plugging in the values given in the problem, we get:
ΔS = k ln(3 × 10^14/1.2 × 10^13) 
ΔS = k ln(25) 
ΔS = k × 3.21887 
ΔS = 4.4 × 10^-23 J/K

Therefore, the correct answer is C.

Question 6

An ideal, reversible refrigerator keeps food in the freezer portion at −15°C and dumps excess heat into the kitchen, whose temperature is 20°C. What is the refrigerator's coefficient of performance under these conditions?

Accepted Answer

The coefficient of performance (COP) for a refrigerator is given by the formula COP = T_c / (T_h - T_c), where T_c is the absolute temperature of the cold reservoir and T_h is the absolute temperature of the hot reservoir. Converting the temperatures to Kelvin, T_c = 258 K and T_h = 293 K. Thus, COP = 258 / (293 - 258) = 7.4.

Question 7

An engineer wants to use a river (minimum temperature of 4°C) to cool his electrical plant, which heats water with a boiler operating at 220°C. If he wishes to produce 185 kW of power using this plant, what is the minimum power he must supply to his boiler if it is maximally efficient?

Accepted Answer

The answer of An engineer wants to use a river...

Question 8

A refrigerator keeps food in the freezer portion at −15°C and dumps excess heat into the kitchen, whose temperature is 22°C. If the refrigerator has half the coefficient of performance of an ideal reversible refrigerator, and if its motor consumes 250 W of power, what is the rate of heat flow out of the freezer portion of the refrigerator?

Accepted Answer

The answer of A refrigerator keeps food in the freezer...

Question 9

For a system undergoing an adiabatic process,

Accepted Answer

In an adiabatic process, there is no heat exchange between the system and its surroundings, so Q = 0. From the first law of thermodynamics, ΔU = Q - W, and if Q = 0, then ΔU = -W. Therefore, the correct answer is D.

Question 10

A friend tells you that he knows of a situation in which bowl of ice sitting by itself on a table in a room whose air temperature is 20°C gives up heat to the surrounding air. What's wrong with this concept?

Accepted Answer

The 2nd law of thermodynamics states that heat naturally flows from a hotter object to a cooler one, not the other way around. A bowl of ice cannot give up heat to the surrounding warmer air.

Question 11

Four coins are tossed randomly onto a table. The macrostate is defined as the number of heads showing, which can be 0, 1, 2, 3, or 4. A microstate is a specification of the outcome of each individual coin. What value for the number of heads has the greatest number of possible microstates?

Accepted Answer

To determine which macrostate has the greatest number of possible microstates, we need to consider the number of ways we can arrange the outcomes of each individual coin toss to achieve that macrostate. 
For example, if we are interested in the macrostate of 2 heads, there are 6 possible microstates: HTTH, THHT, THTH, HHTT, THHH, and HTHH. 
Using this method, we can find the following number of microstates for each macrostate:

0 heads: 1 microstate (TTTT)
1 head: 4 microstates (HTTT, THTT, TTHT, TTTH)
2 heads: 6 microstates (HHTT, HTHT, THHT, TTHH, HTTH, THTH)
3 heads: 4 microstates (HHHT, HTHH, THHH, HTHH)
4 heads: 1 microstate (HHHH)

Therefore, the macrostate with the greatest number of possible microstates is C) 2 heads.

Question 12

An engineer uses a river (which averages 4°C) to cool his electrical plant, which heats water with a boiler operating at 750°C. What's the maximal performance (in terms of efficiency) theoretically possible for this electrical plant?

Accepted Answer

The maximal theoretical efficiency of a heat engine is given by the Carnot efficiency formula, which is (Th - Tc)/Th, where Th is the absolute temperature of the hot reservoir (in Kelvin) and Tc is the absolute temperature of the cold reservoir (in Kelvin). In this case, Th = 750°C + 273.15 = 1023.15 K and Tc = 4°C + 273.15 = 277.15 K. Therefore, the maximal theoretical efficiency is (1023.15 - 277.15)/1023.15 = 73%. Therefore, the correct answer is D.

Question 13

For an ideal gas undergoing an isothermal expansion, the work done on the gas is

Accepted Answer

During an isothermal expansion, the temperature of the gas remains constant. The work done on the gas is given by the formula W = nRT ln(V2/V1), where n is the number of moles of the gas, R is the gas constant, T is the temperature, V1 is the initial volume of the gas, and V2 is the final volume of the gas. Since the gas is expanding, V2 is greater than V1, which means that ln(V2/V1) is positive. Therefore, the work done on the gas is negative, which means that the correct choice is D.

Question 14

On a PV diagram, for a curve representing an isothermal compression of an ideal gas, what is the relationship between the initial P value and the final P value?

Accepted Answer

During an isothermal compression of an ideal gas, the volume decreases while the temperature remains constant. According to Boyle's Law (P1V1 = P2V2 for a given mass of gas at constant temperature), if the volume decreases (V2 < V1), the pressure must increase (P2 > P1) to maintain the equation's equality. Therefore, the initial pressure is less than the final pressure.

Question 15

An ideal reversible heat pump is used to heat a house in Minnesota in the winter. If the outside air temperature is −15°C and the indoor temperature desired is 20°C, what is the expected coefficient of performance for the heat pump under these conditions?

Accepted Answer

The coefficient of performance (COP) is given by:

COP = Qh/W

where Qh is the heat supplied to the house and W is the work input to the heat pump.

In an ideal reversible heat pump, the COP is given by:

COP = Th/(Th-Tc)

where Th is the temperature of the heat source (the outdoor air in this case) and Tc is the temperature of the cold sink (the indoor air in this case).

Using the given temperatures, we have:

COP = (−15°C + 273.15 K)/(−15°C + 273.15 K − 20°C + 273.15 K)

COP = 288.15 K/46.3 K

COP = 6.22

However, this assumes an ideal reversible heat pump. In reality, there will be losses due to irreversible processes, such as friction and heat transfer to the environment. The actual COP will be lower than this value.

Therefore, the closest option is B) 8.4.

Question 16

A system in a certain macrostate has 2 × 10²³ microstates. Through some process, the system changes the number of microstates to 3 × 10²³. What is the change in entropy for the process?

Accepted Answer

The change in entropy for a process is given by: ΔS = kBln(Wf/Wi) where kB is the Boltzmann constant and Wf and Wi are the final and initial number of microstates, respectively. Plugging in the given values, we get: ΔS = kBln(3 × 10²³ / 2 × 10²³) ΔS = kBln(1.5) ΔS ≈ 5.6 × 10⁻²⁴ J/K Therefore, the correct answer is D.

Question 17

A heat pump is used to heat a house in Minnesota in the winter and has half the ideal coefficient of performance for a reversible heat pump. If the outside air temperature is −15°C and the indoor temperature desired is 20°C, and if it inputs work at a rate of 1,250 W, what is the rate of heat flow into the house?

Accepted Answer

The coefficient of performance (COP) for a heat pump is given by $COP = \frac{Q_H}{W}$, where $Q_H$ is the heat delivered to the house and $W$ is the work input. For a reversible heat pump operating between two temperatures, the ideal COP is given by $COP_{ideal} = \frac{T_H}{T_H - T_L}$, where $T_H$ and $T_L$ are the indoor and outdoor temperatures in Kelvin, respectively. Given that the heat pump has half the ideal COP, we first calculate the ideal COP with $T_H = 293K$ (20°C) and $T_L = 258K$ (-15°C), then divide by 2 for the actual COP, and finally solve for $Q_H$.$$COP_{ideal} = \frac{293}{293 - 258} = \frac{293}{35}$$$$COP_{actual} = \frac{1}{2} \times COP_{ideal} = \frac{1}{2} \times \frac{293}{35}$$Given $W = 1250W$,$$Q_H = COP_{actual} \times W = \frac{1}{2} \times \frac{293}{35} \times 1250$$$$Q_H \approx 5230W$$Therefore, the rate of heat flow into the house is approximately 5,230 W.

Question 18

Two coins are tossed randomly onto a table. The macrostate is defined as the number of heads showing, which can be 0, 1, or 2. A microstate is a specification of the outcome of each individual coin. What value for the number of heads has the greatest number of possible microstates?

Accepted Answer

When one coin shows heads, there are two possible microstates (H, T) and (T, H). When no coins show heads, there is only one possible microstate (T, T). When both coins show heads, there is also only one possible microstate (H, H). Therefore, the macrostate of 1 head has the greatest number of possible microstates, which is 2.

Question 19

For a system undergoing an isothermal process,

Accepted Answer

The answer of For a system undergoing an isothermal process,
...

Question 20

A refrigerator keeps food in the freezer portion at −15°C and dumps excess heat into the kitchen, whose temperature is 20°C. If the refrigerator has half the coefficient of performance of an ideal reversible refrigerator, what is the refrigerator's coefficient of performance under these conditions?

Accepted Answer