Compressed air energy storage $($CAES$)$ is a method of storing excess electrical energy in the form of compressed air. The air can be used to produce electrical power when the demand requires it$.$ CAES plants usually store air underground in excavated salt caverns produced by solution mining, or in underground porous and permeable rock structures. When the demand is high, this compressed air is used instead of compressing it$,$ thereby eliminating the work of compression in the high demand periods. Moreover, the compression is usually done by electricity derived from wind or solar energy, whereas the compressed air is utilized for use with fossil$-$fuel production. Of course, here, we will solve a toy problem. Consider air of mass $100$ kg at $300$ K and $1$ bar pressure. How much work is needed to compress air quasistatically to a pressure of $8$ bar by a polytropic process with k $=1\mathrm{.}2.$ You can assume air to be a diatomic gas with cv $=2\mathrm{.}5$R$.$ What is the final temperature of the compressed air? What is the exergy of compressed air? One can define the $($exergy$)$ efficiency of the storage scheme as caes $=$ Ex$($stored gas$)$W$0$stored state $,(1)$ where $$W$0$ stored state is the work done in compressing the gas from the dead state to stored state, and Ex$($stored gas$)$ is the exergy of the gas in the stored state. What is its value?