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?
To calculate how much ice the Carnot engine can melt from its exhaust, we need to determine the heat transfer and the amount of energy required to melt ice.
First, let's calculate the heat transfer in the Carnot engine. The Carnot engine is an idealized engine that operates between two temperatures: the hot reservoir temperature (Th) and the cold reservoir temperature (Tc).
Given:
Th = 170°C (which is 170 + 273 = 443 K)
Tc = 23°C (which is 23 + 273 = 296 K)
The efficiency of a Carnot engine is given by:
Efficiency = 1 - (Tc / Th)
Substituting the values:
Efficiency = 1 - (296 K / 443 K) = 1 - 0.667 = 0.333
Now, we can calculate the heat transfer from the hot reservoir. The heat transfer in a Carnot engine is given by:
Qh = W / Efficiency
Given:
W = 4.5 * 10^4 J
Efficiency = 0.333
Substituting the values:
Qh = (4.5 * 10^4 J) / 0.333 ≈ 1.35 * 10^5 J
The heat transfer from the hot reservoir (Qh) is 1.35 * 10^5 J.
Next, we need to determine the energy required to melt ice. The latent heat of fusion of ice is the energy required to convert 1 gram of ice at 0°C into 1 gram of water at 0°C. The latent heat of fusion of ice is approximately 334 J/g.
To calculate the amount of ice that can be melted, we need to divide the heat transfer from the hot reservoir by the latent heat of fusion of ice. Let's assume the mass of ice melted is m grams.
Qh = m * (334 J/g)
Solving for m:
m = Qh / (334 J/g)
Substituting the value of Qh:
m = (1.35 * 10^5 J) / (334 J/g) ≈ 404.19 g
Therefore, the Carnot engine can melt approximately 404 grams of ice from its exhaust after it has done 4.5 * 10^4 J of work.