Question

The response of a first-order instrument to a step input is:

• A. Instantaneous
• B. Linear with time
• C. Exponential with time
• D. Sinusoidal

Hint:

Logic Tip: Imagine pulling a cold drink out of the fridge. It warms up really fast at first, but takes hours to finally reach exact room temperature. That classic fast-then-slow curve is an exponential response.

Answer 1

The Correct Option is C

Concept:
In process dynamics, a "first-order" instrument (like a standard thermometer) has a specific mathematical way it reacts when the environment around it suddenly changes (a "step input").

Step 1: A mercury thermometer (first-order system) sitting in a $20^{\circ}\text{C}$ room is suddenly plunged into boiling water ($100^{\circ}\text{C}$). The sudden change in the environment is the "step input."

Step 2: The thermometer does not instantly read $100^{\circ}\text{C}$ (Ruling out option A) because it takes time for the glass and mercury to heat up (thermal lag).

Step 3: Initially, the temperature difference between the boiling water and the cold thermometer is huge, so heat transfers very quickly, causing the reading to shoot up rapidly.

Step 4: As the thermometer gets hotter (e.g., reaches $90^{\circ}\text{C}$), the temperature difference shrinks. Heat transfer slows down. The reading crawls very slowly toward the final $100^{\circ}\text{C}$ mark.

Step 5: A curve that starts by changing very fast and then gradually slows down as it approaches a final value is defined mathematically by an exponential equation (specifically, $1 - e^{-t/\tau}$). Therefore, the response is exponential with time.