For the first-order decomposition reaction of ${\mathrm{N}}_2{\mathrm{O}}_5$ , it is found that

$2{\mathrm{N}}_2{\mathrm{O}}_{\mathrm{g}}\left(\mathrm{g}\right)\to 4{\mathrm{NO}}_2\left(\mathrm{g}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right)$                                   $-\frac{\mathrm{d}\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]}{\mathrm{d}t}=k\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]$

${\mathrm{N}}_2{\mathrm{O}}_5\left(\mathrm{g}\right)\to 2{\mathrm{NO}}_2\left(\mathrm{g}\right)+\frac{1}{2}{\mathrm{O}}_2\left(\mathrm{g}\right)$                                   $-\frac{\mathrm{d}\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]}{\mathrm{dt}}{k}^{\mathrm{\prime }}\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]$     

Which of the following is true?

For the reaction    $2{\mathrm{N}}_2{\mathrm{O}}_5\left(\mathrm{g}\right)\to 4{\mathrm{NO}}_2\left(\mathrm{g}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right)$ the rate expression is

$-\frac{1}{2}\frac{\mathrm{d}\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]}{\mathrm{d}t}=k^{\prime \prime }\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]\text{ i.e. }-\frac{\mathrm{d}\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]}{\mathrm{d}t}=2k^{\prime \prime }\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]$$=k\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]$

and for the reaction     ${\mathrm{N}}_2{\mathrm{O}}_5\left(\mathrm{g}\right)\to 2{\mathrm{NO}}_2\left(\mathrm{g}\right)+\frac{1}{2}{\mathrm{O}}_2\left(\mathrm{g}\right)$  the rate expression is $-\frac{\mathrm{d}\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]}{\mathrm{d}t}=k^{\mathrm{\prime }}\left[{\mathrm{N}}_2{\mathrm{O}}_5\right]$

Since$-\mathrm{d}[{\mathrm{N}}_2{\mathrm{O}}_5 ]/\mathrm{dt}$ does not depend upon the way the chemical reaction is formulated, we will have $k=k^{\mathrm{\prime }}$