Final answer:
The rate of entropy change for the two reservoirs is calculated by dividing the heat transferred by the absolute temperature of each reservoir in Kelvins. For a reversible air conditioner, the total rate of entropy change will sum to zero, as the entropy decrease in the cool space is equal to the entropy increase in the warm space.
Step-by-step explanation:
The question involves calculating the rate at which the entropies of the two reservoirs change during the operation of a completely reversible air conditioner. The air conditioner provides 35,000 btu/h of cooling at an inside temperature of 70°F while rejecting heat to the outside at 110°F. To find the rate of entropy change, we can use the formula ΔS = Q/T, where ΔS is the change in entropy, Q is the heat transferred, and T is the absolute temperature in Kelvins.
The rate of entropy decrease for the inside, which is being cooled, is found by taking the heat removed from the cool space (35,000 btu/h) and dividing by the absolute temperature of the inside space. Similarly, the rate of entropy increase for the outside space is found by taking the heat delivered to the warm space (35,000 btu/h, since it's reversible and no work is wasted) and dividing by the absolute temperature of the outside space. Note that you must convert the temperatures from Fahrenheit to Kelvin by using the conversion formula K = (°F + 459.67) × 5/9.
Converting the temperatures:
- Inside: 70°F = 294.261 K
- Outside: 110°F = 316.483 K
Now, we will convert 35,000 btu/h into Joules per second (Watts) by using the conversion 1 btu/h = 0.29307107 W. Therefore, 35,000 btu/h = 35,000 × 0.29307107 W = 10,257.48745 W.
Calculating the entropy change:
- Rate of entropy decrease for inside: ΔS_in = -10,257.48745 W / 294.261 K
- Rate of entropy increase for outside: ΔS_out = 10,257.48745 W / 316.483 K
After calculating these, the total rate of entropy change can be found by adding the rate of entropy decrease of the cool space and the rate of entropy increase of the warm space, which should sum to zero for a reversible process.