Temperature of nitrogen in a vessel of volume $$2{m^3}$$ is $$288$$ $$K.$$ $$A$$ $$U$$-tube manometer connected to the vessel shows a reading of $$70$$ $$cm$$ of mercury (level higher in the end open to atmosphere). The universal gas constant is $$8314$$ $$J/kmol$$-$$K,$$ atmospheric pressure is $$1.01325$$ bar, acceleration due to gravity is $$9.81$$ $$m/{s^2}$$ and density of mercury is $$13600$$ $$kg/{m^3}.$$ The mass of nitrogen (in $$kg$$) in the vessel is _________

A certain amount of an ideal gas is initially at a pressure $${p_1}$$ and temperature $${T_1}$$. First, it undergoes a constant pressure process $$1$$-$$2$$ such that $$T{}_2 = 3{T_1}/4.$$ Then, it undergoes a constant volume process $$2$$-$$3$$ such that $${T_3} = {T_1}/2.$$ The ratio of the final volume to the initial volume of the ideal gas is

Match items from groups, $${\rm I},$$ $${\rm II},$$ $${\rm III},$$ $${\rm IV}$$ and $$V$$

A football was inflated to a gauge pressure of $$1$$ bar when the ambient temperature was $${15^ \circ }C$$. When the game started next day, the air temperature at the stadium was $${5^ \circ }C$$.
Assume that the volume of the football remains constant at $$2500c{m^3}$$

The amount of heat lost by the air in the football and the gauge pressure of air in the football at the stadium respectively equal