To determine the order of the reaction for the given rate equation, we need to sum the exponents of the concentrations in the rate law expression.
The rate equation is given as:
[tex]\[ \text{rate} = k[A]^{1/2}[B]^{1/3}[C]^{1/4} \][/tex]
Here are the exponents for each reactant:
- For [tex]\([A]\)[/tex], the exponent is [tex]\(\frac{1}{2}\)[/tex].
- For [tex]\([B]\)[/tex], the exponent is [tex]\(\frac{1}{3}\)[/tex].
- For [tex]\([C]\)[/tex], the exponent is [tex]\(\frac{1}{4}\)[/tex].
The order of the reaction is determined by adding these exponents together:
[tex]\[ \text{Order of reaction} = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \][/tex]
To add these fractions, we need to find a common denominator. The least common multiple (LCM) of the denominators 2, 3, and 4 is 12. We convert each fraction to have the common denominator of 12:
[tex]\[ \frac{1}{2} = \frac{6}{12} \][/tex]
[tex]\[ \frac{1}{3} = \frac{4}{12} \][/tex]
[tex]\[ \frac{1}{4} = \frac{3}{12} \][/tex]
Now, we can sum these fractions:
[tex]\[ \frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12} \][/tex]
Therefore, the order of the reaction is [tex]\(\frac{13}{12}\)[/tex]. Among the given choices, the correct answer is:
(c) [tex]\(\frac{13}{12}\)[/tex]
