Step 1: Understanding the Concept:
The force exerted by a gas on the walls of its container is the result of the constant macroscopic pressure of the gas acting over the area of the walls.
We need to relate this force to the absolute temperature of the ideal gas.
Step 2: Key Formula or Approach:
1. Pressure and Force: Pressure $P$ is defined as force $F$ per unit area $A$, so $F = P \times A$.
2. Ideal Gas Law: The state of an ideal gas is governed by the equation $PV = nRT$, where $P$ is pressure, $V$ is volume, $n$ is number of moles, $R$ is the universal gas constant, and $T$ is absolute temperature.
Step 3: Detailed Explanation:
From the Ideal Gas Law, we can express pressure as:
\[ P = \frac{nRT}{V} \]
Substitute this expression for pressure into the force equation:
\[ F = \left( \frac{nRT}{V} \right) \times A \]
Since the gas is in a "closed container", its volume $V$, the surface area $A$ of the container walls, and the amount of gas $n$ are all constants. The gas constant $R$ is inherently constant.
Let $k = \frac{nRA}{V}$, which represents a combined constant value.
Then the equation becomes:
\[ F = k \cdot T \]
This shows that the average force $F$ is directly proportional to the temperature $T$ to the first power.
\[ F \propto T^1 \]
Comparing this with the given relationship $F \propto T^x$, we find:
\[ x = 1 \]
Step 4: Final Answer:
The value of $x$ is 1.