Let's solve each part step by step: --- ## **(i) If the molar average velocity, V\*, = , what is the relation between nₐ and n_B?** **Definitions:** - \( n_A \): Mass flux of A relative to stationary coordinates (mol/(m²·s)) - \( n_B \): Mass flux of B relative to stationary coordinates - \( V_A \), \( V_B \): Velocities of A and B - \( C_A \), \( C_B \): Molar concentrations of A and B - \( C = C_A + C_B \): Total molar concentration (constant) - **Molar average velocity:** \[ V^* = \frac{C_A V_A + C_B V_B}{C} \] **Given:** \( V^* = \) \[ = \frac{C_A V_A + C_B V_B}{C} \] \[ C_A V_A + C_B V_B = \] \[ C_A V_A = - C_B V_B \] But, \[ n_A = C_A V_A \] \[ n_B = C_B V_B \] So, \[ n_A = -n_B \] ### **Final Relation:** \[ \boxed{n_A = -n_B} \] --- ## **(ii) If V\* = , and at a certain point 3C_A = C_B, what is the relation between V_A and V_B?** From part (i): \[ C_A V_A + C_B V_B = \] Given: \( 3C_A = C_B \implies C_B = 3C_A \) Substitute: \[ C_A V_A + 3C_A V_B = \] \[ V_A + 3V_B = \] \[ V_A = -3V_B \] ### **Final Relation:** \[ \boxed{V_A = -3 V_B} \] --- ## **(iii) If V\* = constant and n_B = , which of the following are independent of C_A?** We need to check for \( n_A \), \( V_A \), and \( J_B^* \). ### **Given:** - \( V^* \): constant - \( n_B = \) ### **Step 1: Write expressions** - \( n_A = C_A V_A \) - \( n_B = C_B V_B = \implies V_B = \) (since \( C_B \ne \)) - \( V^* = \frac{C_A V_A + C_B V_B}{C} \) - Since \( V_B = \): \[ V^* = \frac{C_A V_A}{C} \implies V_A = \frac{V^* C}{C_A} \] #### **Check each variable:** - **n_A:** \[ n_A = C_A V_A = C_A \left(\frac{V^* C}{C_A}\right) = V^* C \] **Independent of \( C_A \)** - **V_A:** \[ V_A = \frac{V^* C}{C_A} \] **Depends on \( C_A \)** - **J_B\^*** (Molar flux of B relative to molar average velocity): \[ J_B^* = n_B - C_B V^* = - C_B V^* = -C_B V^* \] **Depends on \( C_B \), which is \( C - C_A \), so it depends on \( C_A \)** ### **Final Answers:** \[ \boxed{ n_A \text{ is independent of } C_A; \quad V_A \text{ and } J_B^* \text{ depend on } C_A. } \] --- ## **Summary Table** | Part | Final Answer | |------|--------------| | (i) | \( n_A = -n_B \) | | (ii) | \( V_A = -3V_B \) | | (iii)| Only \( n_A \) is independent of \( C_A \) |