Suppose you seal a rigid container that has been open to air at sea level when the temperature is \(0^{\circ} \mathrm{C}(273 \mathrm{K}) .\) The pressure inside the sealed container is now exactly equal to the outside air pressure: \(10^{5} \mathrm{N} / \mathrm{m}^{2}\) a. What would be the pressure inside the container if it were left sitting in the desert shade where the surrounding air temperature was \(50^{\circ} \mathrm{C}(323 \mathrm{K}) ?\) b. What would be the pressure inside the container if it were left sitting out in an Antarctic night where the surrounding air temperature was \(-70^{\circ} \mathrm{C}(203 \mathrm{K}) ?\) c. What would you observe in each case if the walls of the container were not rigid?

In the realm of gas behavior, understanding how pressure and temperature are related is key. The Ideal Gas Law, expressed as \( PV = nRT \), provides the foundation. Here, P stands for pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is temperature in Kelvin.
When certain conditions, such as the volume and the number of moles, are constant, the relationship simplifies to \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). This shows that the pressure of a gas is directly proportional to its temperature. Simply put, if the temperature increases, the pressure increases, provided the volume stays unchanged.
In our exercise, we used this direct relationship to determine new pressures at different temperatures. First, we calculated the pressure at 50°C (323 K), and found that it increased to approximately 1.18 \( \times\ \( 10^5 \) N/m\^2\). Conversely, for -70°C (203 K), the pressure decreased to about 0.74 \( \times\ \( 10^5 \) N/m\^2\). This exercise highlights the practical importance of the pressure-temperature relationship in understanding gas behavior.

A sealed container is one where no gas can enter or exit. This means the amount of gas inside remains constant. In our scenario, we initially sealed the container at sea level and set the temperature at 0°C (273 K). The pressure inside matched the outside pressure of \( 10^5 \) N/m^2.
Because the container is sealed, we could use the Ideal Gas Law's simplified form, \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), to find pressures at different temperatures. The sealed nature ensures that the variables involved--like volume and moles of gas--remain constant, simplifying our calculations.
The rigidity of the container also plays a part. For rigid walls, the volume doesn't change, but if the walls were not rigid, the volume would adjust with temperature changes. This alternation would alter gas behavior further, demonstrating another aspect of the Ideal Gas Law.

Gas behavior under varying conditions is central to thermodynamics and everyday phenomena. By using the Ideal Gas Law, we explore how a gas's pressure changes in different temperature scenarios within a sealed container.
In normal conditions, gas molecules move randomly and collide with the container walls, creating pressure. An increase in temperature injects more energy into the system, causing molecules to move faster and exert more force, thus increasing pressure.
Conversely, a decrease in temperature slows molecule movement, decreasing the pressure. Observing the container's behavior in different environments, such as a hot desert or cold Antarctic night, allows for practical application of these principles. The rigid container ensures volume remains steady, thus making pressure the variable of interest when temperatures change.
If the container were non-rigid, the situation becomes more complex. The container would expand or contract, influencing gas behavior in noticeable ways. This makes understanding rigid versus non-rigid scenarios valuable for both theoretical and practical applications.