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A mixture of 1.78 kg of water and 262 g of ice at 0ºC is, in a reversible process, brought to a final equilibrium state where the water/ice ratio, by mass is 1:1 at 0ºC, (a) Calculate the entropy change of the system during this process. (b) The system is then returned to the first equilibrium state, but in an irreversible way (by using a Bunsen burner, for instance). Calculate the entropy change of the system during this process. (c) Show that your answer is consistent with the second law of thermodynamics.
The entropy change ΔS for a reversible isothermal process is defined as,
ΔS = Q/T
= -mL/T
Here m is the mass, L is the latent heat and T is the temperature.
(a) Mass of water = 1.78 kg
Mass of ice = 262 g
So the total mass of ice and water mixture will be,
Mass of ice-water mixture = (Mass of water) + (Mass of ice)
= (1.78 kg) + (262 g) = (1.78 kg) + (262 g×10^{-3} kg/1 g)
= 1.78 kg + 0.262 kg = 2.04 kg
If eventually the ice and water have the same mass, then the final state will have 1.02 kg (2.04 kg/2) of each.
Thus the mass of the water that changed into ice m will be the difference of mass of water m_{w } and mass of final state m_{s}.
So, m = m_{w} - m_{s}
To obtain mass of water that changed into ice m, substitute 1.78 kg for mass of water m_{w} and 1.02 kg for mass of final state m_{s} in the equation m = m_{w} - m_{s},
m = m_{w} - m_{s}
= 1.78 kg – 1.02 kg
= 0.76 kg
The change of water at 0^{°} C to ice at 0^{°} C is isothermal.
To obtain the change in entropy ΔS of the system during this process, substitute 0.76 kg for mass m, 333×10^{3} J/kg for heat of fusion of water L and 273 K for T in the equation ΔS = -mL/T,
ΔS = -mL/T
= -(0.76 kg) (333×10^{3} J/kg )/(273 K)
= -927 J/K
From the above observation we conclude that, the change in entropy ΔS of the system during this process will be -927 J/K.
So, ΔS = -(- 927 J/K)
= 927 J/K
From the above observation we conclude that, the change in entropy ΔS of the system would be 927 J/K.
The total change in entropy will be,
ΔS = (-927 J/K) + (927 J/K) = 0
From the above observation we conclude that, our answer is consistent with the second law of thermodynamics.
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Engine A, compared to engine B, produces, per cycle, five times the work but receives three times the heat input and exhausts out twice the heat. Determine the efficiency of each engine.
Work done in an engine is equal to the difference of input heat and output heat.
Efficiency ε_{A} of engine A is equal to,
ε_{A} = W_{A}/ Q_{i,A}
Here, W_{A} is the work done by engine A and Q_{i,A} is the input heat of engine A.
Efficiency ε_{B} of engine B is equal to,
ε_{B} = W_{B}/ Q_{i,B}
Here, W_{B} is the work done by engine B and Q_{i,B} is the input heat of engine B.
Let work done by the engine B is W_{B}. Since the engine A produces, per cycle, five times the work of B, so the work done W_{A} by the engine A will be,
W_{A} = 5 W_{B}
Let heat input of engine B is Q_{i,B}. Since engine A receives three times heat input of B, so the heat input Q_{i,A} of engine A will be,
Q_{i,A} = 3 Q_{i,B}
Let heat output of engine B is Q_{o,B}. Since engine A exhausts two times heat output of B, so the heat input Q_{o,A} of engine A will be,
Q_{o,A} = 2 Q_{o,B}
So, W_{A} = Q_{i,A} - Q_{o,A}
and
W_{B} = Q_{i,B} - Q_{o,B}
Q_{o,B} = Q_{i,B} -W_{B}
Substitute 5 W_{B} for W_{A}, 3 Q_{i,B} for Q_{i,A} and 2 Q_{o,B} for Q_{o,A} in the equation W_{A} = Q_{i,A} - Q_{o,A}, we get,
W_{A} = Q_{i,A} - Q_{o,A}
5 W_{B} = 3 Q_{i,B} - 2 Q_{o,B}
= 3 Q_{i,B} – 2(Q_{i,B} -W_{B}) (Since, Q_{o,B} = Q_{i,B} -W_{B})
3 W_{B} = Q_{i,B}
So, efficiency of engine B, ε_{B} will be,
= 1/3
And
efficiency of engine A, ε_{A} will be,
= 5 W_{B}/3 Q_{i,B }(Since, W_{A} = 5 W_{B} and Q_{i,A} = 3 Q_{i,B})
= (5/3) (1/3) (Since,W_{B}/Q_{i,B} = 1/3)
= 5/9
From the above observation we conclude that, the efficiency of engine A will be 5/9 and engine B will be 1/3.
To make some ice, a freezer extracts 185 kJ of heat at -12.0ºC. The freezer has a coefficient of performance of 5.70. The room temperature is 26.0ºC, (a) How much heat is delivered to the room? (b) How much work is required to run the freezer?
A freezer would like to extract as much heat Q_{L} as possible from the low-temperature reservoir (“what you want”) for the least amount of work W (“what you pay for”). So the efficiency of a freezer is defined as,
K = (what you want)/(what you pay for)
= Q_{L}/W
and this is called coefficient of performance. The larger the value of K, the more efficient is the refrigerator.
Thus, W = Q_{L}/K
The first law of thermodynamics, applied to the working substance of the freezer, gives
W = Q_{H} – Q_{L}
Here Q_{H} is the exhausted heat.
Thus exhausted heat will be,
Q_{H} = W + Q_{L}
(a) To obtain the heat that is delivered to the room, first we have to find out the required work W to run the freezer.
To obtain the required work W to run the freezer, substitute 185 kJ for extracted heat Q_{L} and 5.70 for the coefficient of performance K in the equation W = Q_{L}/K,
W = Q_{L}/K
= 185 kJ/5.70
= 32.5 kJ
To obtain the heat that is delivered to the room, substitute 32.5 kJ for work W which is required to run the freezer and 185 kJ for extracted heat Q_{L} and in the equation Q_{H} = W + Q_{L},
=32.5 kJ + 185 kJ
= 217.5 kJ
Rounding off to three significant figures, the heat delivered to the room would be 218 kJ.
Therefore the required work W to run the freezer would be 32.5 kJ.
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How much work must be done to extract 10.0 J of heat (a) from a reservior at 7ºC and transfer it to one at 27ºC by means of a refrigerator using a Carnot cycle; (b) from one at -73ºC to one at 27ºC; (c) from one at -173ºC to one at 27ºC; and (d) from one at -223ºC to one at 27ºC?
Coefficient of performance K of a Carnot refrigerator is defined as,
K = T_{L} / T_{H} - T_{L} …… (1)
Here T_{L} is the lower temperature of sink and T_{H} is the higher temperature of source.
A refrigerator would like to extract as much heat Q_{L} as possible from the low-temperature reservoir (“what you want”) for the least amount of work W (“what you pay for”). So the efficiency of a refrigerator is defined as,
Thus, W = Q_{L}/K …… (2)
Substitute the value of K from equation (1) in the equation W = Q_{L}/K,
= Q_{L}/( T_{L} / T_{H} - T_{L})
= Q_{L} (T_{H}/ T_{L} – 1)
(a) To obtain work W, substitute 10.0 J for Q_{L}, 27^{°} C for T_{H} and 7^{°} C for T_{L} in the equation W = Q_{L} (T_{H}/ T_{L} – 1),
W = Q_{L} (T_{H}/ T_{L} – 1)
= 10.0 J (27^{°} C/ 7^{°} C -1)
= 10.0 J ((27+273) K /(7+273) K -1)
= 10.0 J (300 K/280 K – 1)
= 0.714 J
Therefore the work done would be 0.714 J.
= 10.0 J (27^{°} C/ (-73^{°} C) -1)
= 10.0 J ((27+273) K /(-73+273) K -1)
= 10.0 J (300 K/200 K – 1)
= 5.00 J
Therefore the work done would be 5.00 J.
= 10.0 J (27^{°} C/ (-173^{°} C) -1)
= 10.0 J ((27+273) K /(-173+273) K -1)
= 10.0 J (300 K/100 K – 1)
= 20.0 J
Therefore the work done would be 20.0 J.
= 10.0 J (27^{°} C/ (-223^{°} C) -1)
= 10.0 J ((27+273) K /(-223+273) K -1)
= 10.0 J (300 K/50 K – 1)
= 50.0 J
Therefore the work done would be 50.0 J.
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The motor in a refrigerator has a power output of 210 W. The freezing compartment is at -3.0ºC and the outside air is at 26ºC. Assuming that the efficiency is 85%of the ideal, calculate the amount of heat that can be extracted from the freezing compartment in 15 min.
K = T_{L} / T_{H} - T_{L}
Since here the efficiency is 85% of the ideal, therefore the coefficient of performance K of the refrigerator will be,
K = 0.85 (T_{L} / T_{H} - T_{L})
From the above equation K = Q_{L}/W, Q_{L} will be,
Q_{L} = (K) (W)
Work done (W) is equal to the product of power (P) and time (t).
W = (P) (t)
First we have to find out the coefficient of performance K and work done W.
To obtain coefficient of performance K, substitute 270 K for T_{L} and 299 K for T_{H} in the equation K = 0.85 (T_{L} / T_{H} - T_{L}),
= 0.85 (270 K / 299 K - 270 K)
= 7.91
To obtain work done W, substitute 210 W for power P and 15 min for time t in the equation W = (P) (t),
= (210 W) (15 min)
= (210 W) (15 min) (60 s/1 min)
= (210 W) (900 s)
= (1.89×10^{5} Ws) (1 J/1 Ws)
= 1.89×10^{5} J
To obtain the amount of heat Q_{L} that can be extracted from the freezing, substitute 7.91 for coefficient performance K and 1.89×10^{5} J for work done W in the equation
Q_{L} = (K) (W),
= (7.91) (1.89×10^{5} J)
= 1.50×10^{6} J
From the above observation we conclude that, the amount of heat Q_{L} that can be extracted from the freezing compartment in 15 min would be 1.50×10^{6} J.
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Apparatus that liquefies helium is in a laboratory at 296 K. The helium in the apparatus is at 4.0 K. If 150 mJ of heat is transferred from the helium, find the minimum amount of heat delivered to the laboratory.
= Q_{L} (T_{H}/ T_{L} – 1) …… (3)
The first law of thermodynamics, applied to the working substance of the refrigerator, gives,
Q_{H} = W + Q_{L} …… (4)
Substitute the value of W from equation (3) in the equation Q_{H} = W + Q_{L},
= Q_{L} (T_{H}/ T_{L} – 1) + Q_{L}
= Q_{L} (T_{H}/ T_{L})
To obtain the minimum amount of heat delivered to the laboratory, substitute 150 mJ for Q_{L}, 296 K for T_{H} and 4.0 K for T_{L} in the equation Q_{H} = Q_{L} (T_{H}/ T_{L}),
Q_{H} = Q_{L} (T_{H}/ T_{L})
= ((150 mJ) (10^{-3} J/1 mJ)) (296 K/4.0 K)
= 11 J
From the above observation we conclude that, the minimum amount of heat delivered to the laboratory would be 11 J.
_____________________________________________________________________________________
A refigerator does 153 J of work to transfer 568 J of heat from its cold compartment. (a) Calculate the refrigerator’s coefficient of performance, (b) How much heat is exhausted to the kitchen?
The first law of thermodynamics, applied to the working substance of the refrigerator, gives
(a) To obtain the coefficient of performance K of refrigerator, substitute 568 J for extracted heat Q_{L} and 153 J for the work W in the equation K = Q_{L}/W,
K = Q_{L}/W
= 568 J/153 J
= 3.71
Therefore, the coefficient of performance K of refrigerator would be 3.71.
= 153 J + 568 J
= 721 J
Therefore, exhausted heat to the kitchen by refrigerator would be 721 J.
____________________________________________________________________________
An invertor claims to have created a heat pump that draws heat from a lake at 3.0ºC and delivers heat at a rate of 20 kW to a building at 35ºC, while using only 1.9 kWof electrical power. How would you judge the claim?
Coefficient of performance (K) of a heat pump is defined as,
K = T_{L }/ (T_{H} - T_{L})
Where, T_{L} is the lower temperature and T_{H }is the higher temperature of the reservoir.
Again, coefficient of performance (K) of a heat engine is defined as,
= (Q_{H} – W) /W
Here Q_{H} heat at higher temperature and W is the work done.
We can write above equation K =(Q_{H} – W) /W,
K =(Q_{H} – W) /W
=(Q_{H}/t – W/t) /(W/t)
= (P_{H} – P) /P (Since Power (P) = W/t)
To obtain the Coefficient of performance (K) of a heat pump, substitute 3.0 ^{°} C for T_{L} and 35 ^{°} C for T_{H} in the equation K = T_{L }/ (T_{H} - T_{L}),
= 3.0 ^{°} C/(35 ^{°} C-3.0 ^{°} C)
= (3+273) K/((35+273) K- (3+273) K)
= 276 K/(308 K-276 K)
= 8.62
To find out the coefficient of performance K of the machine which the inventor claims, substitute 20 kW for P_{H} and 1.9 kW for P in the equation K = (P_{H} – P) /P,
K = (P_{H} – P) /P
= (20 kW-1.9 kW)/1.9 kW
= 9.53
Since the coefficient of performance K of the machine which the inventor claims is greater than the coefficient of performance (K) of a heat pump, therefore it cannot be done.
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The motor in a refrigerator has a power output of 210 W. The freezing compartment is at -3.0ºC and the outside air is at 26ºC. Assuming that the efficiency is 85% of the ideal, calculate the amount of heat that can be extracted from freezing compartment in 15 min.
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An ideal gas undergoes an isothermal expansion at 77ºC increasing its volume from 1.3 to 3.4 L. The entropy change of the gas is 24 J/K. How many moles of gas are present?
The entropy ΔS of an ideal gas undergoes an isothermal expansion is defined as,
= n R ln V_{f}/V_{i}
Here Q is the added heat, T is the temperature, n is the number of moles, R is the gas constant, V_{f} is the final volume and V_{i} is the initial volume.
From the equation ΔS = nR ln V_{f}/V_{i}, the number of moles n will be,
n = ΔS/ (R ln V_{f}/V_{i})
To obtain the number of moles n are present in the gas, substitute 24 J/K for ΔS, 8.31 J/mol. K for R, 3.4 L for V_{f} and 1.3 L for V_{i} in the equation,
= (24 J/K) / (8.31 J/mol. K) (ln 3.4 L/1.3 L)
= (24 J/K) / (8.31 J/mol. K) (ln (3.4/1.3))
= 3.00 mol
From the above observation we conclude that, the number of moles n are present in the gas would be 3.00 mol.
Water standing in the open at 32ºC evaporates because of the escape of some of the surface molecules. The heat of vaporization is approximately eqaul to εn, where ε is the average energy of the escaping molecules and n is the number of molecules per kilogram. (a) Find ε. (b) What is the ratio of ε to the average kinetric energy of H_{2}O molecules, assuming that the kinetic energy is related to temperature in the same way as it is for gases?
Number of molecules n is equal to the Avogadro’s number N_{A} divided by molecular weight M.
So, n = N_{A}/M
The average kinetic energy E_{av} of a diatomic molecule like H_{2}O is defined as,
E_{av} = 3/2 kT
Here k is the Boltzmann constant and T is the temperature.
(a) As heat of vaporization L_{v} is approximately equal to εn, where ε is the average energy of the escaping molecules and n is the number of molecules per kilogram, so,
L_{v} = εn
Or,
ε = L_{v}/n
= L_{v}/( N_{A}/M)
= L_{v}M / N_{A}
To obtain the average energy ε, substitute 2256×10^{3} J/kg for latent heat of vaporization of water molecule (H_{2}O) L_{v}, 0.018 kg/mol for M and 6.03×10^{23}/mol for N_{A} in the equation ε = L_{v}M / N_{A}, we get,
ε = L_{v}M / N_{A}
= (2256×10^{3} J/kg)( 0.018 kg/mol)/( 6.03×10^{23})
= 6.75×10^{-20} J
From the above observation we conclude that, the value of average energy ε would be 6.75×10^{-20} J.
(b) As, E_{av} = 3/2 kT, therefore the ratio of ε to the average kinetic energy E_{av} of H_{2}O molecule will be,
ε / E_{av} = ε / (3/2 kT)
= 2 ε / 3 kT
To obtain the ratio of ε to the average kinetic energy E_{av} of H_{2}O molecule, substitute 6.75×10^{-20} J for ε , 1.38×10^{-23} J/K for k and 32° C for T in the equation ε / E_{av}= 2 ε / 3 kT, we get,
ε / E_{av}= 2 ε / 3 kT
= 2(6.75×10^{-20} J)/3(1.38×10^{-23} J/K)( 32° C)
= 2(6.75×10^{-20} J)/3(1.38×10^{-23} J/K)( 32+273) K)
=2(6.75×10^{-20} J)/3(1.38×10^{-23} J/K)( 305K)
= 10.7
From the above observation we conclude that, the ratio of ε to the average kinetic energy E_{av} of H_{2}O molecule would be 10.7.
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(a) Calculate the rate at which body heat flows out through the clothing of a skier, given the following data: the body surfave area is 1.8 m^{2} and the clothing is 1.2 cm thick; skin surface temperature is 33ºC, whereas the outer surface of the clothing is at 1.0ºC; the thermal conductivity of the clothing is 0.040 W/m.K. (b) How would the answer change if, after a fall, the skier’s clothes become soaked with water? Asssume that the thermal conductivity of water is 0.60 W/m.K.
The rate H at which heat flows out through the clothing of a skier,
(a)directly proportional to the surface area (A) of the body.
(b) inversely proportional to the thickness of the clothing Δx.
(c) directly proportional to the temperature difference ΔT.
So, H = kA ΔT/ Δx
Where k is the proportionality constant and is called thermal conductivity of the material.
(a) Surface area of the body,
A = 1.8 m^{2}
Temperature difference,
ΔT = (skin surface temperature) – (outer surface temperature of clothing)
= 33^{°} C – 1° C = (32+273) K = 305 K
Thickness,
Δx = 1.2 cm
= (1.2 cm) (1 m/100 cm) = 0.012 m
To find out the rate H at which heat flows out through the clothing of a skier, substitute 1.8 m^{2} for the area A, 305 K for temperature difference ΔT and 0.012 m for thickness Δx and 0.040 W/m. K for the thermal conductivity k in the equation H = kA ΔT/ Δx,
H = kA ΔT/ Δx
= (0.040 W/m. K) (1.8 m^{2} ) (305 K) / (0.012 m)
= 1830 W
From the above observation we conclude that, the rate H at which heat flows out through the clothing of a skier would be 1830 W.
(b) From the equation H = kA ΔT/ Δx, we observed that rate H at which heat flows out through the clothing of a skier is directly proportional to the thermal conductivity k of clothing.
Now the thermal conductivity of water is changed to 0.60 W/m. K.
Thus thermal conductivity k is increased by a factor of,
= (0.60 W/m. K) / (0.04 W/m. K)
= 15
Since that rate H at which heat flows out through the clothing of a skier is directly proportional to the thermal conductivity k of clothing, therefore the rate H would increase by a factor of 15.
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