Question

A certain amount of gas at \(25^{\circ} \mathrm{C}\) and at a pressure of 0.800 atm is contained in a glass vessel. Suppose that the vessel can withstand a pressure of \(2.00 \mathrm{~atm} .\) How high can you raise the temperature of the gas without bursting the vessel?

Short answer

The final temperature (\(T_2\)) that can be reached without bursting the vessel is approximately 745 K

Step-by-step solution

Convert Temperatures to Kelvin

Before proceeding the calculation, convert the initial temperature from Celsius to Kelvin. The Celsius to Kelvin conversion is done by adding 273 to the Celsius temperature. Thus, the initial temperature is \( 25^{\circ} \mathrm{C} + 273 = 298 \mathrm{K} \) .

Apply the Ideal Gas Law

The ideal gas law equation is \( P \propto T \), implying that \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). Here, \(P_1\) and \(T_1\) are the initial pressure and temperature, and \(P_2\) and \(T_2\) are the final pressure and temperature. Plug in the known values: \( \frac{0.800 \mathrm{atm}}{298 \mathrm{K}} = \frac{2.00 \mathrm{atm}}{T_2} \)

Solve for Final Temperature

Rearrange the equation from step 2 to solve for \(T_2\). Then calculate \( T_2 = \frac{2.00 \mathrm{atm} \times 298 \mathrm{K}}{0.800 \mathrm{atm}} \)

Key concepts

Pressure-Temperature Relationship

In the world of gases, the pressure-temperature relationship is a fundamental principle described by Gay-Lussac's Law. This law states that, for a given mass of gas at a constant volume, the pressure is directly proportional to its absolute temperature. This means if you increase the temperature, the pressure will also increase, provided the gas does not expand or contract in volume.

This relationship is crucial in scenarios involving sealed containers. If the container's ability to hold pressure is surpassed due to excessive heating, it could burst. For example, in our exercise, a gas initially at a pressure of 0.800 atm and 25°C (or 298K) is in a vessel that can withstand up to 2.00 atm of pressure. This law helps us figure out the maximum temperature the gas can reach before the container fails.

The mathematically expressed form of Gay-Lussac's Law is:

• Direct Proportionality: \( P \propto T \)
• Equation Form: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \)

Understanding this relationship allows us to solve for the unknown variable, in this case, the temperature \( T_2 \), using the given pressure constraints.

Temperature Conversion

One might ask why we need to convert temperature from Celsius to Kelvin when dealing with gas laws. The Kelvin scale is used because it is an absolute temperature scale, starting at absolute zero, where theoretically, all molecular motion ceases. This provides a direct relationship with energy.

For calculations involving gases, such as employing the ideal gas law, temperature must be in Kelvins to maintain proportionality between pressure and temperature. Converting Celsius to Kelvin is straightforward: simply add 273 to the Celsius temperature. For example, the initial temperature of 25°C in the exercise becomes 298K when converted.

These conversions are essential because they ensure our calculations accurately align with the physical behavior of gases, maintaining the correct relationship between variables such as pressure, volume, and temperature.

Gas Behavior

Understanding gas behavior is crucial for predicting how gases will react under various conditions. Gases are made up of particles in constant, rapid motion, and they follow the principles encompassed in the ideal gas law, which combines several individual gas laws.

In our exercise, the behavior of the gas helps us determine how temperature affects pressure in a confined space. Since gases will expand when heated, this behavior could potentially lead to increased pressure in a rigid container, highlighting why temperature increases must be calculated and controlled.

Key principles in gas behavior include:

• Particles move independently and rapidly.
• There are negligible forces between particles unless collisions occur.
• The average kinetic energy of the gas particles is directly proportional to temperature in Kelvin.

These foundational ideas are critical when making predictions or calculations about real-world situations involving gases, like calculating how hot we can get the gas before risking vessel failure. Understanding these behaviors allows us to safely manage and manipulate gases in everyday applications.