A Carnot engine operates between 170°C and 23°C. How much ice can the engine melt from its exhaust after it has done 4.5 104 J of work? (kg)
T1=273+170=443 K
T2=273+23=296 K
efficiency =(Q1-Q2)/Q1 =W/Q1
efficiency =(T1-T2)/T1
W/Q1 = (T1-T2)/T1
Q1=W•T1/(T1-T2)
Q1= r•m
r= 333•10⁵ J/kg
W•T1/(T1-T2) =r•m
r= W•T1/(T1-T2)•m=...
m = W•T1/(T1-T2)• r =...
To determine how much ice the Carnot engine can melt, we need to calculate the heat absorbed by the engine. The Carnot engine is working as a heat engine, so the heat absorbed is related to the work done and the efficiency of the engine.
The efficiency of a Carnot engine is given by the equation:
Efficiency = 1 - (Tc/Th)
Where Tc is the temperature of the cold reservoir (23°C) and Th is the temperature of the hot reservoir (170°C).
First, let's convert the temperatures to Kelvin:
Tc = 23°C + 273.15 = 296.15 K
Th = 170°C + 273.15 = 443.15 K
Now, let's calculate the efficiency:
Efficiency = 1 - (296.15 K / 443.15 K) = 1 - 0.6684 = 0.3316
The efficiency of the Carnot engine is 0.3316, which means that 33.16% of the heat absorbed is converted into work.
We can now calculate the heat absorbed by the engine using the equation:
Heat absorbed = Work done / Efficiency
Heat absorbed = 4.5 104 J / 0.3316 = 1.36 105 J
Next, we need to calculate the amount of heat required to melt ice. The heat required to melt ice can be calculated using the equation:
Heat required = Mass of ice Specific heat of ice + Mass of ice Latent heat of fusion
The specific heat of ice is 2.09 J/(g·°C) and the latent heat of fusion of ice is 334 J/g.
We want to find the mass of ice, so we rearrange the equation to solve for the mass:
Mass of ice = Heat absorbed / (Specific heat of ice + Latent heat of fusion)
Mass of ice = 1.36 105 J / (2.09 J/(g·°C) + 334 J/g)
Let's calculate the mass now:
Mass of ice = 1.36 105 J / (2.09 J/(g·°C) + 334 J/g)
= 1.36 105 J / (336.4 J/g)
= 404.994 g
Therefore, the Carnot engine can melt approximately 404.994 g (or 0.404994 kg) of ice from its exhaust after doing 4.5 104 J of work.
To solve this problem, we need to make a few assumptions and use some concepts from thermodynamics.
First, let's assume that the carnot engine is operating in an idealized, reversible cycle. This means that the engine operates at maximum efficiency and has no energy loss.
Next, we can use the concept of the Carnot efficiency (η) to determine the amount of heat (Qh) absorbed by the engine from the high-temperature reservoir, and the amount of heat (Qc) rejected by the engine to the low-temperature reservoir.
The Carnot efficiency (η) is given by the equation:
η = 1 - (Tc/Th)
Where Tc is the temperature of the low-temperature reservoir and Th is the temperature of the high-temperature reservoir.
Given that the engine operates between 170°C and 23°C, we can calculate the Carnot efficiency:
η = 1 - (23°C / 170°C) = 0.8647
Now, we can use the work done by the engine (W) to calculate the amount of heat absorbed by the engine from the high-temperature reservoir (Qh):
W = Qh - Qc
We are given that the engine has done 4.5 x 10^4 J of work, so we can rearrange the equation to solve for Qh:
Qh = W + Qc
Qh = 4.5 x 10^4 J
Given that Qh = ηQc, we can substitute this into the equation to solve for Qc:
Qc = (1/η) * Qh = (1/0.8647) * (4.5 x 10^4 J) = 5.20 x 10^4 J
Finally, we can use the heat of fusion (Lf) of ice to determine the amount of ice melted from the exhaust. The heat of fusion of ice is the amount of heat required to melt a unit mass of ice:
Lf = 3.34 x 10^5 J/kg
The amount of ice melted (m) can be calculated using the equation:
m = Qc / Lf
m = (5.20 x 10^4 J) / (3.34 x 10^5 J/kg) = 0.155 kg
Therefore, the Carnot engine can melt approximately 0.155 kg of ice from its exhaust after doing 4.5 x 10^4 J of work.