In this question, we are considering an isothermal atmosphere, where the temperature remains constant with elevation. Our task is to understand how the pressure changes with elevation in such conditions.

Let's break it down using the barometric formula, which describes how atmospheric pressure decreases as altitude increases under the assumption of a constant temperature (isothermal conditions).

Concept Explanation:

In an isothermal atmosphere, the equation for pressure as a function of altitude can be given by the barometric formula:

P(h) = P_0 \cdot e^{-\frac{Mgh}{RT}}

Where:

• P(h) is the pressure at height h .
• P_0 is the pressure at sea level.
• M is the molar mass of air.
• g is the acceleration due to gravity.
• R is the universal gas constant.
• T is the absolute temperature.

Analysis:

Under constant temperature, the pressure decreases exponentially with an increase in elevation. Now, let's evaluate the given options:

1. Decreases linearly with elevation: This is incorrect as pressure decreases exponentially, not linearly.
2. Remains constant: Incorrect, as pressure decreases with elevation.
3. Varies in the same way as the density: Correct, because in an isothermal process, both pressure and density decrease exponentially with the same factor e^{-\frac{Mgh}{RT}} .
4. Increases exponentially with elevation: Incorrect, pressure decreases, not increases, with elevation.

Conclusion:

The correct answer is that in an isothermal atmosphere, the pressure varies in the same way as the density.