How Is Entropy Related to the Minimum Work Needed to Cool Water?

[metalboyxp]

This question buzz my head Plz help
A mass of water (mass m=5kg, specific heat capacity at constant pressure: Cp=4184j/kg*C) initially in thermal equilibrium with the atmosphere at 20 degree celcius, is cooled at constant pressure to 4 degree celcius by means of heat pumps operating between water & atmosphere. What is the minimum work required?
The hint for this question is that using the change in entropy, but how?

 

[GCT]

what is the expression for constant pressure entropy change?

 

[quicknote]

The change in entropy is a measure of the disorder or randomness in a system. In this scenario, the water is initially in thermal equilibrium with the atmosphere, meaning that the molecules are evenly distributed and there is no net movement of heat. As the water is cooled, the molecules become more ordered and the entropy decreases.

To determine the minimum work required, we can use the equation for change in entropy (ΔS) = Q/T, where Q is the heat transferred and T is the temperature. In this case, the heat transferred is the amount of energy required to cool the water from 20°C to 4°C, and the temperature is the final temperature of the water (4°C).

We can also use the equation for work (W) = -ΔH, where ΔH is the change in enthalpy. Enthalpy is a measure of the total energy of a system, including both the internal energy and the energy required to do work. In this scenario, the change in enthalpy is equal to the heat transferred (Q) since the process is at constant pressure.

Therefore, the minimum work required can be calculated as W = -ΔH = -Q = -mCpΔT, where m is the mass of water, Cp is the specific heat capacity, and ΔT is the change in temperature (20°C - 4°C = 16°C).

Substituting the given values, we get W = -(5kg)(4184j/kg*C)(16°C) = -334720j or -334.72kJ.

In conclusion, the minimum work required to cool the water from 20°C to 4°C is -334.72kJ. This can be achieved by using heat pumps that operate between the water and the atmosphere, transferring heat from the water to the atmosphere and decreasing the entropy of the system.