Step 1: Understanding the Reaction
The given reaction is an electrophilic aromatic substitution where a benzene ring reacts with sulfuric acid \(( H_2SO_4 )\).
Step 2: Mechanism of the Reaction
- The sulfuric acid \(( H_2SO_4 )\) is acting as a sulfonating agent, introducing the sulfonic group \(( -SO_3H )\) onto the benzene ring.
- The reaction leads to the formation of benzenesulfonic acid where the sulfonic group is attached to the benzene ring.
Step 3: Conclusion
The product (X) of this reaction is benzenesulfonic acid.
A closed-loop system has the characteristic equation given by: $ s^3 + k s^2 + (k+2) s + 3 = 0 $.
For the system to be stable, the value of $ k $ is:
A digital filter with impulse response $ h[n] = 2^n u[n] $ will have a transfer function with a region of convergence.