Process identification by p*v behavior: A gas is heated or expanded such that the product p * v remains constant throughout the process. What is this process called?

Introduction / Context:
Many thermodynamic processes are classified by simple functional relations among p, v, and T. Recognizing the label associated with p * v = constant helps with quick diagramming and integration of work terms.

Given Data / Assumptions:

• Throughout the process, p * v = constant for the working gas.
• Ideal-gas assumption may or may not be invoked; the definition of the process is algebraic.
• No additional constraints (like heat transfer conditions) are specified.

Concept / Approach:
The general polytropic relation is p * v^n = constant. When n = 1, the p–v curve is a rectangular hyperbola, and the process is called a hyperbolic process. For an ideal gas, an isothermal process also satisfies p * v = m * R * T = constant if temperature is constant; thus, for ideal gases, a hyperbolic process with n = 1 coincides with isothermal. However, the broader, name-by-equation classification is “hyperbolic.”

Step-by-Step Solution:

Verification / Alternative check:
On a p–v plot, p = C/v gives a rectangular hyperbola; the area under the curve ∫p dv = C * ln(v2/v1) is the characteristic work expression for n = 1.

Why Other Options Are Wrong:

• Isothermal: True for ideal gases with constant T, but the strict name tied to p * v = constant is “hyperbolic.”
• Adiabatic: Follows p * v^γ = constant for an ideal gas; n = γ ≠ 1.
• Polytropic: Too general; n can be any real number, the specific case n = 1 is hyperbolic.

Common Pitfalls:
Assuming p * v = constant automatically implies no heat transfer; that is only guaranteed for ideal-gas isothermal processes, not for real gases or non-ideal conditions.

Final Answer:
hyperbolic process