P$1250$ Group Work #$9$

Date: $11/25/2024$

Problem $1$

A kettle containing $1\mathrm{.}0$ kg of boiling water $(373$ K $)$ is placed on the kitchen counter. Over the course of a few hours, the water slowly cools off to room temperature $(293$ K $).$

$($a$)$ Calculate the change in entropy of the water $(S_{\text{w}\text{a}\text{t}\text{e}\text{r}})$ as it cools off. Use $4184Jk{g}^{-1}{K}^{-1}$ for the specific heat of water. $($Hint: each tiny decrease in water temperature $dT$ releases a tiny amount of heat $dQ=mcdT$ into the kitchen.$)$

$($b$)$ Calculate the change in entropy of the kitchen $(S_{\text{k}\text{i}\text{t}\text{c}\text{h}\text{e}\text{n}})$ during the same time interval. Assume that the kitchen is large enough that it remains close to room temperature $(293$ K$)$ the entire time.

$($c$)$ Has the second law of thermodynamics been obeyed? Explain your thinking.

$1$