5. A Carnot engine absorbs 440 J of energy for each cycle from a high temperature reservoir at 227oC. If it loses 330 J per cycle to its low temperature reservoir, what is
a) The temperature of the low reservoir?
b) The efficiency of this engine?
To find the temperature of the low reservoir and the efficiency of the engine, we can use the Carnot efficiency formula:
Efficiency = 1 - (T_low / T_high)
where T_low is the temperature of the low reservoir and T_high is the temperature of the high reservoir.
a) The temperature of the low reservoir:
From the given information, the engine absorbs 440 J from the high temperature reservoir and loses 330 J to the low temperature reservoir. This means that the net work done by the engine per cycle is the difference between these two values:
Net work = Energy absorbed - Energy lost = 440 J - 330 J = 110 J
The formula for the net work done by the Carnot engine is:
Net work = Efficiency * Energy absorbed
Rearranging this formula and substituting the given values, we get:
Efficiency = Net work / Energy absorbed = 110 J / 440 J = 0.25
Now, we can use the Carnot efficiency formula to find the temperature of the low reservoir:
0.25 = 1 - (T_low / 227°C)
Rearranging the equation, we have:
T_low / 227°C = 1 - 0.25 = 0.75
Multiplying both sides by 227°C and solving for T_low, we get:
T_low = 0.75 * 227°C = 170.25°C
Therefore, the temperature of the low reservoir is 170.25°C.
b) The efficiency of the engine is 0.25, or 25%.
To find the answers to this question, we can use the formulas and principles of Carnot engines.
a) The temperature of the low reservoir can be found using the equation for the efficiency of a Carnot engine:
Efficiency = 1 - (T_low / T_high)
We can rearrange the equation to solve for T_low:
T_low = T_high * (1 - Efficiency)
Given that the engine absorbs 440 J from the high temperature reservoir and loses 330 J to the low temperature reservoir, we can substitute the values into the equation:
440 J = T_high - 330 J
Simplifying the equation:
440 J + 330 J = T_high
T_high = 770 J
Now we can substitute the value of T_high into the equation for T_low:
T_low = 770 J * (1 - Efficiency)
We'll need the value of the efficiency to determine the temperature of the low reservoir.
b) The efficiency of a Carnot engine can be calculated using the equation:
Efficiency = 1 - (T_low / T_high)
Given the values provided, we can substitute them into the equation:
Efficiency = 1 - (T_low / 770 J)
To find the efficiency, we need to first calculate the temperature of the low reservoir using the equation from part a. Once we have the temperature, we can substitute it into the efficiency equation to find the answer.
So, to summarize:
a) Calculate T_low using the equation T_low = T_high * (1 - Efficiency), where T_high is 770 J.
b) Use the calculated T_low to substitute into the efficiency equation Efficiency = 1 - (T_low / T_high).