Question

For the reaction $2NO\left(g\right)+2H_2\left(g\right)\to N_2\left(g\right)+2H_{2}O\left(g\right)$
The rate expression can be written in the following ways
$\frac{d\left[N_2\right]}{dt}=k_1\left[NO\right]\left[H_2\right]\frac{d\left[H_{2}O\right]}{dt}=k\left[NO\right]\left[H_2\right]-\frac{d\left[NO\right]}{dt}=k_1^{\prime }\left[NO\right]\left[H_2\right]-\frac{d\left[H_2\right]}{dt}=k_1^{\prime \prime }\left[NO\right]\left[H_2\right]$
The relationship betyween $k,k_1,k_1^{\prime }$ and $k_1^{\prime \prime }$

• A. $k=k_1=k_1^{\prime }=k_1^{\prime \prime }$
• B. $k=2k_1=k_1^{\prime }=k_1^{\prime \prime }$
• C. $k=k_1=2k_1^{\prime }=k_1^{\prime \prime }$
• D. $k=k_1=k_1^{\prime }=2k_1^{\prime \prime }$

Answer 1

The correct option is B $k=2k_1=k_1^{\prime }=k_1^{\prime \prime }$

For the reaction $2NO\left(g\right)+2H_2\left(g\right)\to N_2\left(g\right)+2H_{2}O\left(g\right)$
The rate expression can be written in the following ways
$\frac{d\left[N_2\right]}{dt}=k_1\left[NO\right]\left[H_2\right]\frac{d\left[H_{2}O\right]}{dt}=k\left[NO\right]\left[H_2\right]-\frac{d\left[NO\right]}{dt}=k_1^{\prime }\left[NO\right]\left[H_2\right]-\frac{d\left[H_2\right]}{dt}=k_1^{\prime \prime }\left[NO\right]\left[H_2\right]$

The rate of reaction is
$r=\frac{-1}{2}\frac{d\left[NO\right]}{dt}=\frac{-1}{2}\frac{d\left[H_2\right]}{dt}=\frac{d\left[N_2\right]}{dt}=\frac{1}{2}\frac{d\left[H_{2}O\right]}{dt}\Rightarrow \frac{1}{2}{k}_1^{\prime }\left[NO\right]\left[H_2\right]=\frac{1}{2}{k}_1^{\prime \prime }\left[NO\right]\left[H_2\right]=k_1\left[NO\right]\left[H_2\right]=\frac{1}{2}k\left[NO\right]\left[H_2\right]$
Comparing the above equations, we get
$k_1^{\prime }=2k_1{k}_1^{\prime \prime }=2k_{1}k=2k_1$

$\therefore k=2k_1=k_1^{\prime }=k_1^{\prime \prime }$