Question

An outdoor storage vessel for hydrogen gas with a volume of \(300 . \mathrm{m}^{3}\) is at \(1.5\) atm and \(10 .{ }^{\circ} \mathrm{C}\) at 2:00 AM. By 2:00 PM, the temperature has risen to \(30 .{ }^{\circ} \mathrm{C}\). What is the new pressure of the hydrogen in the vessel?

Short answer

The new pressure of the hydrogen in the vessel at 2:00 PM is approximately 1.6 atm.

Step-by-step solution

Convert temperatures to Kelvin

Convert the initial and final temperatures from degrees Celsius to Kelvin by adding 273.15 to each temperature. This is necessary because the gas laws require absolute temperatures. Initial temperature (T1): 10°C + 273.15 = 283.15 K Final temperature (T2): 30°C + 273.15 = 303.15 K

Apply Charles’s Law

Since the volume of the gas remains constant, use Charles’s Law, which states that the pressure of a given mass of an ideal gas is directly proportional to its Kelvin temperature at constant volume. The formula is: \( \frac{P1}{T1} = \frac{P2}{T2} \).Solving for the new pressure (P2), we get: \( P2 = \frac{P1 \cdot T2}{T1} \)

Insert the known values

Insert the initial pressure (P1), initial temperature (T1), and final temperature (T2) into the equation from Step 2. We have \( P1 = 1.5 \) atm, \( T1 = 283.15 \) K, and \( T2 = 303.15 \) K. So, \( P2 = \frac{1.5 \text{ atm} \cdot 303.15 \text{ K}}{283.15 \text{ K}} \)

Calculate the final pressure

Perform the calculation from Step 3 to find the final pressure: \( P2 = \frac{1.5 \times 303.15}{283.15} \) atm. Calculate this to get the value of P2 in atm.

Key concepts

Understanding Gas Laws

Gas laws are foundational principles in chemistry that describe how gases behave under different conditions of temperature, pressure, and volume. These laws are crucial for predicting the behavior of gases in various scenarios, such as in understanding weather patterns, designing engines, and even in our example of a hydrogen gas storage vessel.

One fundamental aspect of these laws is the direct and inverse relationships shared between the gas variables. As seen in our exercise, Charles's Law focuses specifically on the direct relationship between temperature and pressure of a gas when the volume is held constant. This means if the temperature of a gas increases, the pressure will also increase if the volume is not changed.

Utilizing the appropriate gas law as in our exercise, allows us to reliably calculate changes in the gas's behavior when subjected to different temperatures. Simplifying the process of these calculations and understanding the underlying principles can empower students to solve complex problems effortlessly.

The Role of Absolute Temperature in Gas Laws

Absolute temperature is measured in Kelvin and is essential in the study of gas laws because it provides a standard starting point, unlike Celsius or Fahrenheit scales that have arbitrary zero points. Absolute zero in the Kelvin scale represents the theoretical point at which particles have minimal thermal motion, indicating no kinetic energy.

To apply gas laws correctly, as in our exercise, temperatures must be converted to Kelvin. This avoids negative values and ensures the mathematical relationships in the equations remain consistent. Using Kelvin allows us to appreciate the importance of absolute temperature in evaluating how a gas will respond to thermal changes.

By converting Celsius temperatures to Kelvin, we acknowledge the absolute nature of thermal energy in gas behaviors and provide the precision required for accurate calculations, which is necessary for deriving the correct new pressure in a constant volume scenario like in the provided exercise.

Ideal Gas Behavior and Real-world Applications

The concept of ideal gas behavior simplifies the complex interactions between gas particles by assuming ideal conditions: gases consist of point particles that do not attract or repel each other, collisions between gas particles and with the walls of the container are perfectly elastic, and there are no volume effects of the particles themselves.

In reality, no gas perfectly fits these criteria, but many behave closely enough to the ideal model under certain conditions that the model is extremely useful. In our exercise, assuming the hydrogen gas behaves as an ideal gas allows us to apply Charles’s Law for a reliable estimation of how temperature changes influence pressure.

Understanding this model helps explain and predict gas behavior in a variety of real-world applications, from the workings of household refrigerators to the engineering of high-speed jet engines. Always remember the assumptions behind the ideal gas law when making calculations or predictions about real gases, and account for deviations from ideality under extreme conditions of temperature and pressure.