Question

Inversion of sucrose $\left( C _{12} H _{22} O _{11}\right)$ is a first order reaction and is studied by measuring angle of rotation at different interval of time
$\underset{\text{Sucrose d-}}{C _{12} H _{22} O _{11}}+ H _{2} O \xrightarrow{ H ^{+}} \underset{\text{Glucose d- }}{C _{6} H _{12} O _{6}}+ \underset{\text{Fructose l-}}{C _{6} H _{12} O _{6}}$
$r _{0}=$ angle of rotation at the start, $r _{ t }=$ angle of rotation at time $t , r _{\infty}=$ angle of rotation at the complete reaction. There is $50 \%$ inversion, when

• A. $r_{0}=r_{t}-2 r_{\infty}$
• B. $r_{0}=r_{t}-r_{\infty}$
• C. $r_{0}=2 r_{t}-r_{\infty}$
• D. $r_{0}=r_{t}+r_{\infty}$

Answer 1

Answer: (C)

$r _{0}=$ rotation at the start, $\,\,\,\,\,\,\,r _{ t }=$ rotation at time $t$

$r _{\infty}=$ rotation at the end

then $\left(r_{\infty}-r_{0}\right)=a$ and $\left(r_{\infty}-r_{t}\right)=(a-x)$

When $50 \%$ inversion has taken place

$a-x=\frac{a}{2} $

$ \therefore 2(a-x)=a$

$2\left(r_{\infty}-r_{t}\right)=r_{\infty}-r_{0}$ or $2 r_{\infty}-2 r_{t}=r_{\infty}-r_{0}$

$\therefore r_{0}=\left(2 r_{t}-r_{\infty}\right)$