A heat engine operating between 100°C and 700°C has efficiency equal to 40% of the maximum theoretical efficiency .how much energy does this engine extract from the hot reservoir in order to do 5000J of mechanical work?
100 + 273 = 373 deg K
700 + 273 = 973 deg K
ideal Carnot eff = 1 - Tl/Th = 1 - 373/973 = 0.6166
real efficiency = 0.40* .6166 = 0.2467
so 5000 J / 0.2467 = 20,268 Joules
To determine the energy extracted from the hot reservoir by the heat engine, we need to calculate the maximum theoretical efficiency first.
The maximum efficiency of a heat engine can be determined using the Carnot efficiency formula:
Efficiency = (Th - Tc) / Th
Where:
Efficiency = Maximum theoretical efficiency
Th = Temperature of the hot reservoir (in Kelvin)
Tc = Temperature of the cold reservoir (in Kelvin)
Given:
Th = 700°C = 700 + 273 = 973 K
Tc = 100°C = 100 + 273 = 373 K
Using the formula, we can calculate the maximum theoretical efficiency:
Efficiency = (973 - 373) / 973
Efficiency ≈ 0.6179
Now, we need to calculate the efficiency of the heat engine compared to the maximum theoretical efficiency:
Actual Efficiency = 0.4 * Max Efficiency = 0.4 * 0.6179
Actual Efficiency ≈ 0.2472
To calculate the energy extracted from the hot reservoir, we will use the formula for efficiency of a heat engine:
Actual Efficiency = Energy Extracted / Energy Input
We rearrange the formula to solve for Energy Extracted:
Energy Extracted = Actual Efficiency * Energy Input
Given:
Mechanical work output = 5000 J
Substituting the values into the formula:
Energy Extracted = 0.2472 * 5000
Energy Extracted ≈ 1236 J
Therefore, the heat engine extracts approximately 1236 J of energy from the hot reservoir to do 5000 J of mechanical work.
To determine how much energy the engine extracts from the hot reservoir, we need to use the efficiency of the engine and the amount of mechanical work done.
First, let's calculate the maximum theoretical efficiency of the heat engine using the Carnot efficiency formula:
Carnot efficiency = (T1 - T2) / T1
where T1 and T2 are the temperatures of the hot and cold reservoirs respectively.
Given that the temperatures of the hot and cold reservoirs are 700°C and 100°C respectively, we can convert them to Kelvin by adding 273:
T1 = 700°C + 273 = 973 K
T2 = 100°C + 273 = 373 K
Now we can calculate the maximum theoretical efficiency:
Carnot efficiency = (973 - 373) / 973 = 0.617
Next, we need to find the actual efficiency of the engine. The problem states that the efficiency is 40% of the maximum theoretical efficiency, so:
Actual efficiency = 0.4 * 0.617 = 0.247
Now let's use the efficiency formula to calculate the heat extracted from the hot reservoir:
Heat extracted = (output work) / (efficiency) = 5000 J / 0.247
Heat extracted = 20161.93 J (rounded to five decimal places)
Therefore, the engine extracts approximately 20161.93 Joules of energy from the hot reservoir in order to do 5000 Joules of mechanical work.