26. Imagine the gas that forms the thin atmosphere of a small exoplanet at a constant
temperature. The speed of gas molecules at equilibrium are described by the Maxwell
Boltzmann distribution. Molecules whose speed exceeds the escape velocity will
occasionally be lost to space. Once the system re-equilibriates at the same temperature,
which of the following statements is correct about the distribution of speeds:
a.It will be identical to the original distribution, but truncated beyond the
escape velocity.
b. It will be identical to the original distribution.
c. It will be a new Maxwell-Boltzmann distribution with a lower average
velocity than before.
d. It will be a new Maxwell-Boltzmann distribution with a higher average
velocity than before.
The correct answer is d. It will be a new Maxwell-Boltzmann distribution with a higher average velocity than before.
In a thin exoplanet atmosphere at constant temperature, the Maxwell-Boltzmann speed distribution governs molecular velocities. Molecules exceeding escape velocity escape to space, preferentially removing the high-speed tail. Upon re-equilibration at fixed temperature through collisions, the system reforms a Maxwell-Boltzmann shape (determined solely by temperature), but with reduced total molecules and higher average speed due to selective loss of fastest particles.
Option Analysis
-
a. Identical to original but truncated beyond escape velocity: Incorrect. Truncation occurs only in rapid, non-equilibrium evaporation where slower molecules cannot gain speed fast enough; thin atmospheres re-equilibrate via collisions, restoring full Maxwell-Boltzmann shape.
-
b. Identical to original distribution: Incorrect. Loss of high-speed molecules changes the population distribution; average kinetic energy per remaining molecule increases despite constant temperature.
-
c. New Maxwell-Boltzmann with lower average velocity: Incorrect. Opposite effect occurs—fastest molecules escape, leaving slower ones behind initially, but collisions redistribute energy, yielding higher mean speed for the depleted population at same T.
-
d. New Maxwell-Boltzmann with higher average velocity: Correct. Jeans escape removes high-velocity tail; re-equilibration maintains MB form at fixed T (same most probable speed), but fewer total molecules shift mean speed upward as ∫v f(v)dv/∫f(v)dv increases.
Maxwell Boltzmann distribution exoplanet atmosphere escape velocity questions test atmospheric retention physics in CSIR NET exams, distinguishing Jeans escape effects from equilibrium assumptions. Understanding selective fast-molecule loss explains planetary atmosphere evolution, with hydrogen depletion on terrestrial worlds versus helium retention on gas giants.
