For a first-order reaction, the rate law is given by: \( r = k[A] \), where \( k \) is the rate constant and \( [A] \) is the concentration of reactant A.

Solution 1:

Volume = 100 mL, Concentration = 10 M

\( r_1 = k \times 10 \)

Solution 2:

Volume = 200 mL, Concentration = 10 M

\( r_2 = k \times 10 \)

Solution 3:

Initial volume of A = 100 mL, Initial concentration = 10 M

Final volume = 100 mL (A) + 100 mL (water) = 200 mL

Using the dilution formula, \( M_1V_1 = M_2V_2 \):

\( 10 \times 100 = M_2 \times 200 \)

\( M_2 = \frac{10 \times 100}{200} = 5 \text{ M} \)

\( r_3 = k \times 5 \)

Comparing the rates:

\( r_1 = 10k \), \( r_2 = 10k \), \( r_3 = 5k \)

Therefore, \( r_1 = r_2 > r_3 \)