Problem 68 - Chemistry The Science in Context

Question

A tank containing 5.00 L of nitrogen at 75.0 atm pressure and \(28^{\circ} \mathrm{C}\) is left standing in sunlight. Its temperature rises to \(50.0^{\circ} \mathrm{C} .\) What is the gas pressure at this higher temperature?

Step-by-Step Solution

Answer

Answer: The new gas pressure in the tank at 50.0°C is approximately 80.4 atm.

Step 1: Recall the Ideal Gas Law and Convert Temperatures to Kelvin

The Ideal Gas Law can be written as: PV = nRT where P is the pressure, V is the volume, n is the amount of gas (in moles), R is the ideal gas constant, and T is the temperature (in Kelvin). First, we need to convert the given Celsius temperatures to Kelvin. \(T_1\) = \(28^{\circ}\mathrm{C}\) + 273.15 = 301.15 K \(T_2\) = \(50^{\circ}\mathrm{C}\) + 273.15 = 323.15 K

Step 2: Use the relationship between initial and final states

Since the volume and the amount of gas are constant, we can rearrange the ideal gas law and use the relationship: \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\) where \(P_1\) and \(T_1\) are the initial pressure and temperature, and \(P_2\) and \(T_2\) are the final pressure and temperature.

Step 3: Solve for the final pressure

Now we can solve for \(P_2\): \(P_2 = \frac{P_1T_2}{T_1}\) Plug in the given values for \(P_1\) and the temperatures in Kelvin: \(P_2 = \frac{75.0 \,\text{atm} \times 323.15 \,\text{K}}{301.15\, \text{K}}\) Calculate the new pressure: \(P_2 \approx 80.4\, \text{atm}\)

Step 4: State the final answer

The gas pressure in the tank at 50.0°C is approximately 80.4 atm.

Key Concepts

Gas Pressure

Gas pressure is the force exerted by a gas when it collides with the walls of its container. The pressure of a gas is a crucial concept in understanding how gases behave under different conditions. It is measured in units such as atmospheres (atm), Pascals (Pa), and pounds per square inch (psi), among others. For instance, in our exercise, we talk about nitrogen gas in a tank exerting pressure initially at 75.0 atm.

One of the key things to remember is that the pressure of a gas is directly proportional to its temperature, assuming the volume of the gas and the number of moles remain constant. This means that as the temperature increases, as it did when the tank was left in sunlight, the gas particles move more rapidly, colliding with the walls of their container more frequently and with greater force, resulting in an increase in pressure.

Understanding the concept of gas pressure is essential because it influences many real-world applications, including tire pressure in vehicles, weather patterns, and even the working principle of engines and refrigerators. It's one of the central aspects of the Ideal Gas Law used to predict the behavior of gases under various conditions.

Temperature Conversion

Temperature conversion is a necessary step in gas behavior calculations since temperatures must often be in a standard unit, and for gas equations, that unit is Kelvin. To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. This is exactly what we do in our solution process when converting the initial temperature of 28°C and the final temperature of 50°C to Kelvin.

To convert Celsius to Kelvin:

• Add 273.15 to the Celsius temperature.

To convert Kelvin to Celsius:

• Subtract 273.15 from the Kelvin temperature.

This step is crucial because the Kelvin scale is an absolute temperature scale, meaning it starts at absolute zero, the point where particles theoretically stop moving. In contrast, the Celsius scale is based on the properties of water, with 0°C being the freezing point and 100°C being the boiling point at standard atmospheric pressure.

Kelvin Scale

The Kelvin scale is a thermodynamic temperature scale that's vital to the sciences, especially physics and chemistry. It starts at absolute zero, the theoretical point where particles have minimum thermal motion. The size of one Kelvin is the same as one degree Celsius, but there is no negative temperature on the Kelvin scale since it starts from the absolute zero point, where the temperature would be 0 K.

The Kelvin scale is crucial for scientific calculations because it allows for the use of zero in equations, such as the Ideal Gas Law, eliminating the issue of negative temperatures that can complicate calculations. For example, in our exercise, we saw that temperatures need to be in Kelvin to properly utilize the Ideal Gas Law formula. This practice simplifies the calculations and provides a clear base for understanding thermal energy in a system.

Using the Kelvin scale allows students and scientists to calculate the behavior of gases more accurately, especially at extremely high or low temperatures, making it the preferred scale for many applications, including astrophysics, aerodynamics, and material science.