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The gas phase decomposition of dimethyl ...

The gas phase decomposition of dimethyl ether follows the first order kinetics as `CH_3 - O - CH_(3) (g) to CH_(4) (g) + H_(2) (g) + CO(g)` This reaction is carried out in a constant volume container at temperature of 500°C and has a half-life of 14.5 min. Initially, only dimethyl ether is present at a pressure of 0.40 atm. After 12 min, find the total pressure of the system. Assume ideal gas behaviour.

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`k = (0.693)/( t_(1//2) ) = (0.693)/( 14.5" min") = 4.779 xx 10^(-2)" min"^(-1)`
Let decrease in pressure of `CH_3 - O - CH_3` in 12 min = `p` atm
`a= 0.40" atm", (a- x) = (0.40 - p)" atm"`
`k = (2.303)/( t) log"" (a)/( (a-x) )`
`4.779 xx 10^(-2) = (2.303)/( 12) log"" (0.40)/( 0.40 - p)`
`log"" (0.40)/( 0.40 - p) = (4.779 xx 10^(-2) xx 12)/( 2.303) = 0.2490`
`= (0.40)/(0.40-p) =" antilog" (0.2490) = 1.774`
`rArr p = 0.40- (0.40)/( 1.774)" atm " p = 0.175" atm"`
When pressure of `CH_3- O - CH_3` decreases by p atm, the pressure of each of `CH_(4), H_(2) and CO` increases by `p` atm. Hence, the total pressure after 12 min
`= (0.4- p) + p + p+ p`
`= 0.4+ 2p = 0.4 + 2 xx 0.175`
`=0.4 + 0.350 = 0.750" atm"`
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