Optical rotation of sucrose in 1 N HCl at various times was found as shown below :
`{:("Time (sec)",0,7.18,18.0,27.05,oo),("Rotation (deg)",+24.09,+21.7,+17.7,+15.0,-10.74):}`
Show that the inversion of sucrose is a first order reaction

Explanation

Initial concentration of sucrose = `r_(0)-r_(infty) = +24.09 -(-10.74)=+34.83`
`k = 2.303/t loga/(a-x) = (2.303/t) log ((r_(0)-r_(infty))/(r_(t)-r_(infty)))`
i) At t=7.18s, `k=(2.303)/(7.18s) log(34.83)/(21.4-(-10.74))= (2.303)/(7.18s) log(34.83)/(32.14)`
`=(2.303)/(7.18s) (log 34.83-log 32.14) = 2.303/7.18s(1.542-1.507)`
`(2.303 xx 0.035)/(7.18)s = 1.23 xx 10^(-3)s^(-1)`
At t=18s, `k=(2.303)/18s log(34.83)/(15-(-10.74)) = (2.303)/(18s)log(34.83)/(28.44)`
`=(2.303 xx 0.087)/(18s) = 1.13 xx 10^(-3)s^(-1)`
(iii) At t=27.05 s, `k=(2.303)/(27.05)s log(34.83)/(25.74)`
`=(2.303)/(27.05s) log34.83 - log 25.74 = 2.303/27.05s (1.542-1.411)`
`(2.303 xx 0.131)/(27.05s) = 1.15 xx 10^(-3)s^(-1)`
Since the value of k comes out to be nearly constant, the inversion of cane sugar is the reaction of first order,