Ch 22 #5, 8, 13, 27, 31, 35, 43, 59, train

Ch 22.1 #5

A multicylinder gasoline engine in an airplane, operating at 2500 rev/min, takes in energy 7890 J and exhausts 4580 J for each revolution of the crankshaft. (a) How many liters of fuel does it consume in 1.00 h of operation if the heat of combustion is 4.03 ´ 10⁷ J/L? (b) What is the mechanical power output of the engine? Ignore friction and express the answer in horsepower. (c) What is the torque exerted by the crankshaft on the load? (d) What power must the exhaust and cooling system transfer out of the engine?

(a) 2500 rev/min (60 min/ 1 hr) (7890 J / rev) (1 / (4.03 ´ 10^7) L/J) = 29.4 L/hr
(b) Qh = Weng + Qc Qh /t = Weng / t + Qc / t Weng/t = Qh / t - Qc /t Power = Work / time Peng = Qh / t - Qc /t Peng = (7890-4580)J/rev(2500 rpm) Peng = 8.28x10^6 J/min (1 min/60 s) Peng = 138,000 W (1hp / 746W) Peng = 185 hp	(c) Peng = τ ω τ = Peng / ω τ = (7890-4580)J/rev (1rev/2π) τ = 527 J (d) Pexhaust = Qc /t Pexhaust = (4580)J/rev(2500 rpm) Pexhaust = 11.5x10^6 J/min (1 min/60 s) Pexhaust = 191,000 W (1hp / 746W) Pexhaust = 256 hp

Ch 22.2 #8

A refrigerator has a coefficient of performance of 3.00. The ice tray compartment is at –20.0°C, and the room temperature is 22.0°C. The refrigerator can convert 30.0 g of water at 22.0°C to 30.0 g of ice at –20.0°C each minute. What input power is required? Give your answer in watts.

Heat removed each minute, Qc/t Qc/t = (mciceΔT + m Lf-ice + mcwaterΔT) / minute Qc/t = (30(½)20 + 30(80) + 30(1)22 ) / minute Qc/t = 3360 cal / minute Qc/t = 3360 cal / min (4.186J/cal)(1 min/60s) Qc/t = 234 J/s (or Watts)

COP = Qc/W Work = Qc/COP Work = Qc/3 Power = Work / time Power = 234 J/s / 3 Power = 78 W

Ch 22.3 #13

An ideal gas is taken through a Carnot cycle. The isothermal expansion occurs at 250°C, and the isothermal compression takes place at 50.0°C. The gas takes in 1200 J of energy from the hot reservoir during the isothermal expansion. Find (a) the energy expelled to the cold reservoir in each cycle and (b) the net work done by the gas in each cycle.

(a) |Qc| = |Qh| Tc/Th |Qc| = 1200 (273+50)/(273+250) |Qc| = 741 J

(b) Weng = |Qh| - |Qc| Weng = 1200 - 741 Weng = 459 J

Ch 22.4 #27

How much work does an ideal Carnot refrigerator require to remove 1.00 J of energy from helium at 4.00 K and reject this energy to a room-temperature (293-K) environment?

(COP)Carnot refrig = Tc / ∆T COP = 4 / (293-4) COP = 0.0138

Work = |Qc| / COP Work = 1 J / 0.0138 Work = 72.2 Joules

Ch 22.5 #31

In a cylinder of an automobile engine, just after combustion, the gas is confined to a volume of 50.0 cm³ and has an initial pressure of 3.00 x 106 Pa. The piston moves outward to a final volume of 300 cm³ and the gas expands without energy loss by heat. (a) If γ = 1.40 for the gas, what is the final pressure? (b) How much work is done by the gas in expanding?

(a) Pi Vi^γ = P V^γ P = Pi (Vi / V)^γ P = 3x106 (50/ 300)^γ P = 0.244 x 106 Pa or 244 kPa

(b) W = P ∫dV W = Pi ∫ (Vi/V)^γ dV W = Pi Vi^γ ∫ V^-^γ dV W = Pi Vi^γ [ ∆V^–^γ^+1 ] / (-γ+1) W = 3x10^6(5x10^-5)^1.4 [(5x10^-5)^-.4 - (30x10^-5)^-.4] / (-1.4+1) W = 192 Joules

Ch 22.6 #35

An ice tray contains 500 g of liquid water at 0°C. Calculate the change in entropy of the water as it freezes slowly and completely at 0°C.

∆S = ∆Q / T

∆S = m L / T

∆S = -(1/2kg)(3.33x105 J/kg) / 273K

∆S = -610 J/K

Ch 22.7 #43

How fast are you personally making the entropy of the Universe increase right now? Compute an order-of-magnitude estimate, stating what quantities you take as data and the values you measure or estimate for them.

Entropy is measure of disorder. We eat the food, break it down and expend it as non-reuseable form of body heat. The average person eats 2500 kcal per day. 2,500,000 cal/day (4.186 J / cal) (1 day / 24 hr) (1 hr / 3600 sec) = 120 J/s = 120 Watts
∆S = ∆Q/T ∆S/∆t = ∆Q/T /∆t ∆S/∆t = ∆Q/∆t /T	∆S/∆t = ∆Q/∆t /T ∆S/∆t = 120W / (273+20) ∆S/∆t = 0.4 W/K
It asks for NOW…how many of you are driving a car right NOW? Ans: none But if you include driving cars, sitting in air conditioning, etc…you are contributing MUCH more than ≈ 1 W/K

Ch 22 #59

A power plant, having a Carnot efficiency, produces 1000 MW of electrical power from turbines that take in steam at 500 K and reject water at 300 K into a flowing river. The water downstream is 6.00 K warmer due to the output of the power plant. Determine the flow rate of the river.

eff = 1 – Tc/Th eff = (Th – Tc) /Th eff = Weng / Qh eff = Weng/∆t/ Qh/∆t (Th – Tc) /Th = Weng/∆t/ Qh/∆t (Th – Tc) /Th = Power / Qh/∆t Qh/∆t = PTh/(Th–Tc)

Weng = |Qh| - |Qc| Weng/∆t = |Qh|/∆t - |Qc|/∆t P = PTh/(Th–Tc) - mc∆T/∆t mc∆T/∆t = PTh/(Th–Tc) - P mc∆T/∆t = PTh/(Th–Tc) – P [(Th–Tc)/(Th–Tc)] (m/∆t) c∆T = PTc/(Th–Tc) m/∆t = P Tc / [(Th - Tc) c ∆T] m/∆t = 10^9(300) / [(500-300)4186(6)] m/∆t = 5.97 x 10^4 kg/sec

Ch 22 train

A 20.0% efficient real engine is used to speed up a train from rest to 5.00 m/s. It is known that an ideal (Carnot) engine having the same cold and hot reservoirs would accelerate the same train from rest to a speed of 6.50 m/s using the same amount of fuel. Assuming that the engines use air at 300 K as a cold reservoir, find the temperature of the steam serving as a hot reservoir.

Work = D½mv^2 Work = D½m5^2 Work = 12.5m e = Weng / Qh 20% = 12.5m / Qh Qh = 62.5m

ecarnot = 1 – Tc/Th ecarnot = Workmax / Qh ecarnot = D½m6.5^2 / Qh 1 – Tc/Th = 21.125m / Qh 1 – 300/Th = 21.125m / 62.5m Th = 453.2 K