Problem 85 - Chemistry The Central Science

Question

(a) Place the following gases in order of increasing average molecular speed at \(25^{\circ} \mathrm{C}: \mathrm{Ne}, \mathrm{HBr}, \mathrm{SO}_{2}, \mathrm{NF}_{3}, \mathrm{CO}\) (b) Calculate the rms speed of \(\mathrm{NF}_{3}\) molecules at \(25^{\circ} \mathrm{C} .\) (c) Calculate the most probable speed of an ozone molecule in the stratosphere, where the temperature is \(270 \mathrm{~K}\).

Step-by-Step Solution

Answer

(a) The order of gases based on increasing average molecular speed at 25°C is: HBr < NF3 < SO2 < CO < Ne. (b) The rms speed of NF3 molecules at 25°C is 460.71 m/s. (c) The most probable speed of an ozone molecule in the stratosphere, at 270 K, is 352.72 m/s.

Step 1: Identify the equation for average molecular speed

The equation to find the average molecular speed (u) is given by: \(u = \sqrt{(\frac{8RT}{\pi M})}\) Where: u = average molecular speed R = gas constant = 8.314 J/mol-K T = temperature in Kelvin M = molar mass of the gas in kg/mol. We will use this equation to order the gases based on their average molecular speed at 25°C.

Step 2: Order the gases based on average molecular speed

To order the gases, we will need to convert the given temperature, 25°C, to Kelvin. T = 25 + 273.15 = 298.15 K Now, we will use the given equation to calculate the molecular speed for each gas. As inversely proportional, higher molar mass gas molecules will have lower average molecular speeds at the same temperature. The molar masses for the given gases are: Ne: 20.18 g/mol = 0.02018 kg/mol HBr: 80.91 g/mol = 0.08091 kg/mol SO2: 64.07 g/mol = 0.06407 kg/mol NF3: 71.00 g/mol = 0.07100 kg/mol CO: 28.01 g/mol = 0.02801 kg/mol By comparing molar mass, we order the gases (lowest to highest molecular speed): HBr < NF3 < SO2 < CO < Ne

Step 3: Calculate the rms speed of NF3 molecules

We will use the following equation for the rms speed (vrms): \(v_{rms} = \sqrt{(\frac{3RT}{M})}\) We will use temperature T = 298.15 K and M = 0.07100 kg/mol for NF3. \(v_{rms} = \sqrt{(\frac{3 \times 8.314 \times 298.15}{0.07100})}\) \(v_{rms} = 460.71 \, m/s\) The rms speed of NF3 molecules at 25°C is 460.71 m/s.

Step 4: Calculate the most probable speed of an ozone molecule

For the most probable speed (vmp), we will use the following equation: \(v_{mp} = \sqrt{(\frac{2RT}{M})}\) Given, the temperature in the stratosphere is 270 K, we need the molar mass of the ozone (O3) molecule: O3: 48.00 g/mol = 0.04800 kg/mol \(v_{mp} = \sqrt{(\frac{2 \times 8.314 \times 270}{0.04800})}\) \(v_{mp} = 352.72 \, m/s\) The most probable speed of an ozone molecule in the stratosphere, at 270 K, is 352.72 m/s.

Key Concepts

Average Molecular Speed

When we talk about the average molecular speed in chemistry, we are referring to the mean velocity that molecules in a gas possess at a given temperature. It's essential to understand that as the temperature increases, the kinetic energy of the molecules increases, which in turn increases their speed.

This average speed is estimated using an equation that combines the ideal gas constant (\( R \)), the temperature (\( T \)), in Kelvin, and the molecular mass (\( M \)) of the gas. The formula is given by: \( u = \sqrt{(\frac{8RT}{\pi M})} \) where \( u \) represents the average molecular speed. For students comparing different gases at the same temperature, as in the case of the exercise, it's crucial to recognize the inverse relationship between the molecular speed and the molar mass: lighter molecules move faster. The speed ordering challenge is central to mastering how to predict molecular behavior under various conditions.

Root Mean Square Speed

The root mean square (rms) speed is a term often encountered when studying gases and their molecular motions. It represents the square root of the average of the squares of the individual speeds of the gas molecules. This value is particularly significant because it correlates directly to the kinetic energy of the gas.

The equation for rms speed is: \( v_{rms} = \sqrt{(\frac{3RT}{M})} \) where \( v_{rms} \) stands for the root mean square speed of the molecules, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass expressed in kg/mol. For students tackling problems like the calculation of rms speed of \(NF_3\) molecules, understanding this formula helps to comprehend the kinetic theory of gases. It's the rms speed that's directly related to the temperature and intrinsic energy within a gas sample.

Most Probable Speed

The most probable speed in a gas is the speed at which the maximum number of molecules are moving. It is slightly different from the average or rms speed because it focuses on the peak of the speed distribution curve for a gas, known as the Maxwell-Boltzmann distribution.

To find the most probable speed (\( v_{mp} \) ), we use this equation: \( v_{mp} = \sqrt{(\frac{2RT}{M})} \) where \( T \) is the absolute temperature, \( R \) is the ideal gas constant, and \( M \) is the molar mass. In the exercise involving calculating the most probable speed for ozone in the stratosphere, it demonstrates the direct application of this concept. Understanding the most probable speed is vital for grasping how gases behave and distribute their molecular speeds at specific temperatures.