Problem 12 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Two identical flasks contain nitrogen gas at the same pressure. Each has an identical pin-hole. One flask is kept at \(25^{\circ} \mathrm{C}\) while the other is heated to \(125^{\circ} \mathrm{C}\). Calculate the relative rates of effusion of nitrogen from the two flasks. (Section 8.5 )

Step-by-Step Solution

Answer

The nitrogen gas effuses about 0.865 times slower from the flask at 25°C than from the flask at 125°C.

Step 1: Understand the Problem

The problem involves two identical flasks containing the same gas under the same pressure, but at different temperatures. We need to calculate the relative rates of effusion for nitrogen gas from these two flasks.

Step 2: Use Graham's Law of Effusion

Graham's Law of Effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. For gases at different temperatures, the rate of effusion is also related to the square root of temperature: \( r \propto \sqrt{T} \). Therefore, the ratio of effusion rates \( \frac{r_1}{r_2} \) for two gases at temperatures \( T_1 \) and \( T_2 \) is: \( \frac{r_1}{r_2} = \sqrt{\frac{T_1}{T_2}} \).

Step 3: Plug in the Temperatures

For the first flask at \(T_1 = 25^{\circ} \mathrm{C} = 298 \mathrm{K}\) and the second flask at \(T_2 = 125^{\circ} \mathrm{C} = 398 \mathrm{K}\), use the formula from Step 2: \( \frac{r_1}{r_2} = \sqrt{\frac{298}{398}} \).

Step 4: Calculate the Square Root of the Temperature Ratio

Calculate \( \frac{298}{398} \approx 0.749 \). Then calculate the square root: \( \sqrt{0.749} \approx 0.865 \).

Step 5: Interpret the Result

The relative rate of effusion of nitrogen gas at \(25^{\circ} \mathrm{C}\) to that at \(125^{\circ} \mathrm{C}\) is approximately 0.865. This means the nitrogen gas effuses more slowly from the flask at the lower temperature.

Key Concepts

Effusion Rate

Effusion rate refers to the speed or rate at which gas particles escape through a small hole from their container. This principle is illustrated in the study of gas behavior, such as in our problem of nitrogen gas effusing from two flasks. Understanding effusion rate relies heavily on recognizing how gas particles move. Graham's Law of Effusion is a key component in understanding effusion. It tells us that the rate of effusion is dependent on the size and mass of the gas particles. Specifically, it is inversely proportional to the square root of the gas's molar mass. This means lighter gases effuse faster than heavier ones. This is due to their higher average speeds at any given moment. For instance:

• Light gases like hydrogen will effuse quickly.
• Heavier gases like argon will do so at a slower pace.

In adding temperature effects to the equation, Graham's Law also adapts to include temperature dependence. The rate becomes tied to the square root of the temperature, which is crucial when comparing effusion rates across different thermal conditions.

Temperature Dependence

Temperature plays an essential role in the behavior of gases and their effusion rates due to its impact on kinetic energy. When you heat a gas, the kinetic energy of its molecules increases, causing them to move more quickly. This increased movement results in a higher rate of effusion.For our nitrogen gas scenario:- The flask at a higher temperature (125°C) sees nitrogen molecules with greater velocity due to more kinetic energy.- Conversely, the flask at a lower temperature (25°C) has less energetic and slower molecules.This is why we observe different effusion rates at varying temperatures. While the inherent properties of a gas contribute largely to its effusion rate, simply raising the temperature can enhance these rates significantly. Hence, when solving problems involving different temperatures, like in Step 3, it's important to convert all temperatures to Kelvin and apply the square root relationship as shown by\[ r \propto \sqrt{T} \].

Gas Laws

Gas laws encompass various principles that describe how gases behave under different conditions. Amongst these, Graham's Law of Effusion is specifically useful when dealing with the escape of gas through tiny openings. Other familiar gas laws include Boyle’s Law, Charles’s Law, and Avogadro’s Law, which collectively help explain gas behaviors in terms of pressure, volume, and temperature. In the study of effusion, the combination of these gas laws can be critical: - **Boyle's Law** discusses how pressure and volume are inversely related at a constant temperature. However, in effusion, the main variable is usually temperature, not pressure. - **Charles's Law** tells us that volume and temperature are directly proportional when pressure is held constant. This aligns well with our understanding of temperature influencing effusion because increasing the temperature can lead to expansion and increased movement of gas molecules. These laws, together with Graham's Law of Effusion, provide a comprehensive understanding of gas behavior under various scenarios, reaffirming that temperature is a vital determinant of effusion rate. Using these laws, students can predict how changes in physical conditions will affect gas particles' movement and escape efficiency through small openings.