To determine the rate of the reaction following the given rate law, we will use the provided values for the rate constant ([tex]\( k \)[/tex]), concentrations ([tex]\([ A ] \)[/tex] and [tex]\([ B ] \)[/tex]), and the orders of the reaction with respect to each reactant ([tex]\( m \)[/tex] and [tex]\( n \)[/tex]). The rate law is given by:
[tex]\[ \text{rate} = k [A]^m [B]^n \][/tex]
Let's input the given values step by step:
1. The rate constant [tex]\( k \)[/tex] is 0.2 [tex]\(( \text{mol} / \text{L} ) / \text{s} \)[/tex].
2. The concentration of [tex]\( A \)[/tex] is 3 M.
3. The concentration of [tex]\( B \)[/tex] is 3 M.
4. The reaction order with respect to [tex]\( A \)[/tex] ([tex]\( m \)[/tex]) is 1.
5. The reaction order with respect to [tex]\( B \)[/tex] ([tex]\( n \)[/tex]) is 2.
Plugging these values into the rate law, we get:
[tex]\[ \text{rate} = 0.2 \left( 3 \text{ M} \right)^1 \left( 3 \text{ M} \right)^2 \][/tex]
Evaluate the exponents first:
[tex]\[ 3^1 = 3 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
Now substitute these values back into the equation:
[tex]\[ \text{rate} = 0.2 \times 3 \times 9 \][/tex]
Next, carry out the multiplications:
[tex]\[ \text{rate} = 0.2 \times 27 \][/tex]
[tex]\[ \text{rate} = 5.4 \][/tex]
Therefore, the rate of the reaction is [tex]\( 5.4 \, (\text{mol} / \text{L}) / \text{s} \)[/tex]. So the correct answer is:
B. [tex]\( 5.4 \, (\text{mol} / \text{L}) / \text{s} \)[/tex]
