Calculate the value of `gamma=(C_P)/(C_V)` for a gaseous mixture consisting of `n_1=2` moles of `O_2` and `n_2=3` moles of `CO_2`. The gases are assumed to be ideal.
(b) Inside a container one mole of He and n moles of `H_2`. The average value of `gamma` for gaseous mixture is `(19)/(13)`. Find `n`. Also find speed of sound in gaseous mixture at temperautre `27^@C`

Explanation

(a) For `O_2`:`n_1=2`,`gamma_1=(7)/(5)`
for `CO_2`:`n_2=3`,`gamma_2=(4)/(3)`
`(n_1+n_2)/(overlinegamma-1)=(n_1)/(gamma_1-1)+(n_2)/(gamma_2-1)`
`(2+3)/(overlinegamma-1)=(2)/((7)/(5)-1)+(3)/((4)/(3)-1)`
`(5)/(overlinegamma-1)=(2)/((2)/(5))+(3)/((1)/(3))=5+9`
`overline-1=(5)/(14)impliesoverlinegamma=1+(5)/(14)=(19)/(14)`
Alternatively:
`O_2`, diatomic gas, `C_(V1)=(5R)/(2)`,`V_(P1)=(7R)/(2)`
`CO_2`, Polytomic gas `C_(V2)=3R`,`C_(P2)=4R`
`overlinegamma=(n_1C_(P1)+n_2C_(P2))/(n_1C_(V1)+n_2C_(V-2))=(2xx(7R)/(2)+3xx4R)/(2xx(5R)/(2)+3xx3R)`
`=(19)/(14)`
(b) He: `n_1=1`,`C_(V1)=(3R)/(2)`,`C_(P1)=(5R)/(2)`
`H_2`:`n_2=n`,`C_(V2)=(5R)/(2)`,`C_(P2)=(7R)/(2)`
`overlinegamma=(n_1C_(P1)+n_2C_(P2))/(n_1C_(V1)+n_2C_(V2))`
`(19)/(13)=overlinegamma=(1xx(5R)/(2)+nxx(7R)/(2))/(1xx(3R)/(2)+nxx(3R)/(2))=(5+7n)/(3+5n)`
`19(3+5n)=13(5+7n)`
`57+95n=65+91n`
`4n=8impliesn=2`
Average molecular mass
`overlineM_0=(n_1M_(He)+n_2M_(H2))/(n_1+n_2)`
`=(1xx4+2xx2)/(1+2)=(8)/(3)gm=(8)/(3)xx10^-3kg`
`T=27^@C=27+273=300K`
Speed of sound in gasneous mixture
`v_S=sqrt((overlinegammaRT)/(overlineM_0))=sqrt((((19)/(13))(8.3)(300))/(((8)/(3))xx10^-3))`
`=1168(m)/(s)`
The unit of molucular mass is gm e.g. for `O_2` molecular mass is 32 gm.