Main Answer
The temperature of the hot reservoir for an ideal heat engine with an efficiency of 16.7% and a temperature difference of 73.6°C between the hot and cold reservoirs is 499.8 K.The correct option is D.
Explanation
An ideal heat engine is a hypothetical device that converts all the heat energy it receives into useful work, with no waste or loss of energy. The efficiency of an ideal heat engine is defined as the ratio of the amount of work done by the engine to the amount of heat energy it receives from the hot reservoir.
In this problem, we are given an efficiency of 16.7% and a temperature difference between the hot and cold reservoirs of 73.6°C (equivalent to 107.1 K). We can use the formula for efficiency, which is:
Efficiency = (T_h - T_c) / T_h
where T_h is the temperature of the hot reservoir in kelvins and T_c is the temperature of the cold reservoir in kelvins. Substituting our given values into this formula, we get: 0.167 = (T_h - 273.15 - 107.1) / T_h.
We can simplify this equation by subtracting 273.15 K from both T_h and T_c, since Kelvin temperatures are measured relative to absolute zero (-273.15°C). This gives us:0.167 = (T_h - 107.1) / T_h.
We can solve for T_h by multiplying both sides by T_h:
0.167 T_h = T_h - 107.1.We can then add 107.1 to both sides to isolate T_h:T_h = 0.167 T_h + 107.1.
To solve this equation, we can use algebraic methods or a calculator to find that T_h is approximately 499.8 K, which is our answer. This means that if we have an ideal heat engine operating between two reservoirs with temperatures differing by 73.6°C, then the temperature of the hot reservoir is approximately 499.8 K in Kelvin units.The correct option is D.