Calculate root mean square speed most probable speed and average speed of
(a) `SO_(2)` at `STP`
(b) Ethane at `27^(@)C` and 760 mm of hg
(c ) `O_(2)` if its density is `0.0081 g mL^(-1)` at 1 atm
(d) `O_(2)` if 6.431g of it occupies 5 litre at 750 mm
(e) `O_(3)` at `92 cm` `20^(@)C` .

Explanation

Given `T =273K`
`u_(rms)of So_(2) =sqrt((3RT)/(M)) =sqrt((3 xx8.314xx10^(7)xx273)/(64)`
`= 3.26 xx 10^(4) cm sec^(-1)`
Now `u_(MP) =u_(rms) xx 0.816 = 2.66 xx 10^(4) cm sec^(-1)`
`u_(AV) = u_(rms) xx 0.9213 = 3.00 xx 10^(4) cm sec^(-1)`
(b) `u_(rms) of C_(2)H_(6) = sqrt((3RT)/(M))`
`=sqrt(((3xx8.314 xx10^(7)xx300))/(30)`
`=4.99 xx10^(4) cm sec^(-1)`
` ( :' Given T = 273 + 27 =300K)`
Note if `T` is given, always use `u_(rms) = sqrt(3RT//M)`
Calculate `u_(MP)` and `u_(AV)` as above
`u_(MP)` for `C_(2)H_(6) =4.07 xx 10^(4) cm sec^(-1)`
`u_(AV) for C_(2)H_(6) = 4.60 xx 10^(4) cm sec^(-1)`
(c) Given density (d) of `O_(2) = 0.0081 g mL^(-1) , P=1 atm`
`u_(rms) = sqrt((3P)/(d)) = sqrt((3xx1xx76xx13.6xx981)/(0.0081))`
` =1.94 xx 10^(4) cm sec^(-1)`
Calculate `u_(MP)` and `u_(AV)` as above
`u_(MP) = 1.58 xx 10^(4) cm sec^(-1)`
`u_(AV) = 1.78 xx 10^(4) cm sec^(-1)`
(d) Given for `O_(2): w = 6.431 g`
`V =5 litre, P =750 mm`
Using `PV =(w //m) RT`
`(750)/(760) xx 5 = (6.431)/(32) xx 0.0821 xx T`
`:. T =299.05K`
`u_(rms) = sqrt((3xx8.314 xx10^(7) xx 299.5)/(32))`
` =4.83 xx 10^(4) cm sec^(-1)`
`:. u_(MP) = 3.94 xx 10^(4) cm sec^(-1)`
`u_(AV) = 4.45 xx 10^(4) cm sec^(-1)`
(e) Given for `O_(3) : T =293 K,P =(92//76)atm`
`u_(rms) =sqrt((3RT)/M)=sqrt((3 xx 8.314 xx 10^(7)xx293)/(48)`
`=3.9 xx 10^(4) cm sec^(-1)`
`u_(MP) = 3.18 xx 10^(4) cm sec^(-1)`
`u_(AV) = 3.59 xx 10^(4) cm sec^(-1)` .