A certain refrigerator is rated as being 39% as efficient as a Carnot refrigerator. To remove 102 J of heat from the interior at 0 °C and eject it to the outside at 23 °C, how much work must the refrigerator motor do?
To determine how much work the refrigerator motor must do, we can use the efficiency formula for a Carnot refrigerator:
Efficiency = (Work Output / Heat Input)
Given that the refrigerator is 39% as efficient as a Carnot refrigerator, we can substitute this value into the formula:
0.39 = (Work Output / Heat Input)
Next, we need to calculate the heat input of the refrigerator. The heat input is the heat extracted from the interior (102 J) plus the work done by the refrigerator motor. So we have:
Heat Input = Heat Extracted + Work Output
Since the heat is extracted from the interior at 0 °C and ejected outside at 23 °C, we can calculate the heat extracted using the formula:
Heat Extracted = Mass * Specific Heat Capacity * Temperature Difference
Since the temperature difference is given by 23 °C - 0 °C, we get:
Heat Extracted = Mass * Specific Heat Capacity * (23 °C - 0 °C)
Since the specific heat capacity of water is approximately 4.18 J/g°C, we can use this value:
Heat Extracted = Mass * 4.18 J/g°C * (23 °C - 0 °C)
Now we substitute the heat extracted into the heat input equation:
Heat Input = Mass * 4.18 J/g°C * (23 °C - 0 °C) + Work Output
Finally, we can substitute the heat input and efficiency into the efficiency formula:
0.39 = (Work Output / (Mass * 4.18 J/g°C * (23 °C - 0 °C) + Work Output))
Now, we can solve this equation to find the value of Work Output, which represents the work that the refrigerator motor must do.