Dynamic mass balance for a CSTR: Which statement correctly expresses the unsteady-state (time-dependent) total mass balance for a reacting species in a continuous stirred tank reactor?

Introduction / Context:
CSTR modeling starts with a general balance: accumulation = in − out + generation. The question checks recognition of the unsteady (time-dependent) form for a species or conserved quantity in a well-mixed reactor.

Given Data / Assumptions:

• Well-mixed volume, variable with time in principle, though often constant.
• Inlet and outlet streams with known compositions and flow rates.
• Homogeneous reaction with a net formation rate inside the reactor.

Concept / Approach:
The correct statement must include a time derivative on the left-hand side (accumulation term). The right-hand side consists of input, output, and net reaction formation rate terms. Wording differences matter: “time rate of total amount” explicitly captures the derivative d(n)/dt.

Step-by-Step Solution:
Write general balance: d(Amount in system)/dt = In − Out + Generation.Map to CSTR: accumulation = F_inC_in − F_outC + rV.Recognize that options lacking the time derivative are steady-state forms, not unsteady.

Verification / Alternative check:
At steady state, the accumulation term goes to zero, recovering the familiar In − Out + Generation = 0, consistent with the same framework.

Why Other Options Are Wrong:
Option (b) omits time dependence; it is dimensionally inconsistent.

Option (c) is close in meaning but is ambiguously phrased and redundant; option (a) states it clearly.

“None of the above” is incorrect since (a) is correct.

Common Pitfalls:

• Forgetting the accumulation term when modeling start-up or disturbances.
• Confusing total-mass versus species balances; the structure is the same but variables differ.

Final Answer:
Time rate of total amount in the reactor = rate of addition to reactor − rate of removal + rate of formation