Gas storage modeling in process control: For an ideal gas contained in a rigid vessel, the (molar) capacitance of the vessel is defined as C = (∂n/∂P) at constant temperature. Using P = pressure, V = vessel volume, n = moles of gas, R = universal gas constant, and T = absolute temperature, which expression correctly represents the gas capacitance?

Introduction / Context:
Dynamic modeling of tanks and process vessels often uses the electrical analogy, where a vessel acts like a “capacitor” storing mass or moles. For gases, this storage capacity depends on how sensitive the contained moles are to changes in pressure at a fixed temperature. The question asks for the ideal-gas expression for this capacitance in a rigid vessel.

Given Data / Assumptions:

• Ideal-gas behavior: P * V = n * R * T.
• Vessel is rigid: V is constant.
• Capacitance definition: C = ∂n/∂P at constant T and V.
• R is the universal gas constant; T is absolute temperature.

Concept / Approach:
The molar capacitance for an ideal gas is obtained by differentiating the ideal-gas equation with respect to pressure while holding temperature and volume constant. This directly provides how many moles “fit” per unit pressure rise, i.e., the storage property that governs pressure dynamics when inflow and outflow are unbalanced.

Step-by-Step Solution:

Verification / Alternative check:
If T increases, V/(R*T) decreases, meaning fewer moles per unit pressure rise (gas is “softer”), which matches physical intuition. Larger volumes proportionally increase storage, aligning with C ∝ V.

Why Other Options Are Wrong:

• R * T / V: This is the inverse of the correct result; it would imply smaller vessels hold more per unit pressure, which is nonphysical.
• n * R / P or P * V / (n * R): These arise from rearrangements but do not equal ∂n/∂P at constant T and V.
• R / (P * V): Has incorrect units and dependence.

Common Pitfalls:
Confusing capacitance with compressibility, or forgetting the derivative must be taken at constant volume and temperature; using total derivatives that include T or V changes leads to wrong expressions.

Final Answer:
V / (R * T)