Nitrogen gas (molecular weight $$28$$) is enclosed in a cylinder by a piston, at the initial condition of $$2$$ bar, $$298$$ $$K$$ and $$1\,\,{m^3}$$. In a particular process, the gas slowly expands under isothermal condition, until the volume becomes $$2{m^3}.$$ Heat exchange occurs with the atmosphere at $$298$$ $$K$$ during this process.

The work interaction for the Nitrogen gas is

Considering the relationship $$TdS=dU+PdV$$ between the entropy $$(S),$$ internal energy $$(U),$$ pressure $$(P),$$temperature $$(T)$$ and volume $$(V),$$
Which of the following statements is correct?

An industrial heat pump operates between the temperatures of $${27^ \circ }C$$ and $$ - {13^ \circ }C.$$ The rates of heat addition and heat rejection are $$750$$ $$W$$ and $$1000$$ $$W,$$ respectively. The $$COP$$ for the heat pump is

A $$2$$ $$kW,$$ $$40$$ liter water heater is switched on for $$20$$ minutes. The heat capacity $${C_p}$$ for water is $$4.2$$ $$kJ/kg$$ $$K.$$ Assuming all the electrical energy has gone into heating the water, increase of the water temperature in degree centigrade is