$5\mathrm{.}27.$ A tank in a room at $19C$ is initially open to the atmosphere on a day when the barometric pressure is $102$kPa. A block of dry ice $($solid $C{O}_2)$ with a mass of $15\mathrm{.}7kg$ is dropped into the tank, which is then sealed. The reading on the tank pressure gauge initially rises very quicky, then much more slowly, eventually reaching a value of $3\mathrm{.}27$MPa. Assume Tinal $19C.$

$($a$)$ How many moles of air were in the tank initially? Neglect the volume occupied by $C{O}_2$ in the solid state, and assume that a negligible amount of $C{O}_2$ escapes prior to the sealing of the tank

$($b$)$ Estimate the percentage error made by neglecting the volume of the block of dry ice placed in the tank. $($The specific gravity of solid carbon dioxide is approximately $1\mathrm{.}56.)$

$($c$)$ What is the final density $(\frac{g}{L}$ of the gas in the tank?

$($d$)$ Explain the observed variation of pressure with time. More specifically, what is happening in the tank during the initial rapid pressure increase and during the later slow pressure increase?