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 find out how much ice the Carnot engine can melt, we need to use the formula for the efficiency of a Carnot engine:
Efficiency = 1 - (T_cold / T_hot)
Where T_cold is the temperature of the cold reservoir (in this case, 23°C) and T_hot is the temperature of the hot reservoir (in this case, 170°C).
Let's convert the temperatures from Celsius to Kelvin:
T_cold = 273 + 23 = 296 K
T_hot = 273 + 170 = 443 K
Now we can calculate the efficiency:
Efficiency = 1 - (296 K / 443 K) = 1 - 0.668 = 0.332
The efficiency of the Carnot engine is 0.332, meaning that only 33.2% of the heat energy it takes in can be converted into work.
To find out the amount of ice melted, we need to calculate the heat energy taken in by the engine and then multiply it by the efficiency. The formula for heat energy is:
Q = W / Efficiency
Where Q is the heat energy, W is the work done (4.5 * 10^4 J in this case), and Efficiency is the efficiency of the engine (0.332).
Let's calculate the heat energy taken in by the engine:
Q = (4.5 * 10^4 J) / 0.332 = 135542.16 J
Now we can calculate the amount of ice melted by using the latent heat of fusion of ice. The latent heat of fusion of ice is the amount of heat energy required to convert 1 gram of ice into water at its melting point, which is 334 J/g.
Amount of ice melted = Q / latent heat of fusion = 135542.16 J / 334 J/g = 405.64 g
Therefore, the Carnot engine can melt approximately 405.64 grams of ice from its exhaust after it has done 4.5 * 10^4 J of work.
To determine how much ice the Carnot engine can melt from its exhaust, we need to calculate the efficiency of the engine and then use the energy conservation principle to find the amount of ice melted.
1. Calculate the efficiency of the Carnot engine:
The efficiency (η) of a Carnot engine is given by the formula:
η = 1 - (Tc/Th)
where Tc is the lower temperature (in Kelvin) and Th is the higher temperature (in Kelvin).
To convert the temperatures from Celsius to Kelvin, we use the formula:
T(K) = T(°C) + 273.15
Given:
Tc = 23°C
Th = 170°C
Converting to Kelvin:
Tc(K) = 23 + 273.15 = 296.15 K
Th(K) = 170 + 273.15 = 443.15 K
Calculating the efficiency:
η = 1 - (296.15/443.15) = 1 - 0.6687 = 0.3313 or 33.13%
2. Calculate the energy absorbed by the engine:
The work done by the Carnot engine is given as 4.5 * 10^4 J (joules). This is the energy absorbed from the high-temperature reservoir.
3. Calculate the energy given by the engine as waste:
Since the efficiency of the Carnot engine is 33.13%, the remaining energy is given as waste:
Waste energy = (1 - efficiency) * Energy absorbed
Waste energy = (1 - 0.3313) * 4.5 * 10^4 J = 0.6687 * 4.5 * 10^4 J = 3.00315 * 10^4 J
4. Convert the waste energy to heat energy:
According to the principle of energy conservation, the waste energy given by the engine will be converted to heat energy, which can be used to melt the ice.
Now, we need to use the heat of fusion of water to determine how much ice can be melted:
The heat of fusion of water is approximately 333.55 kJ/kg.
To find the amount of ice melted, divide the waste energy by the heat of fusion:
Amount of ice melted = Waste energy / Heat of fusion
Converting the heat of fusion to joules:
Heat of fusion = 333.55 kJ/kg = 333.55 * 10^3 J/kg
Substituting the values:
Amount of ice melted = 3.00315 * 10^4 J / (333.55 * 10^3 J/kg)
Calculating the amount of ice melted will depend on the quantity of ice that can be melted by a joule of energy released.
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