Problem 28 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The molecular speeds in a sample of 100 molecules are distributed as follows (Section 8.5): Number of molecules \(10 \quad 20 \quad 40 \quad 15 \quad 10 \quad 5\) Speed/ms \(^{-1}\) \(\begin{array}{llllll}60 & 80 & 100 & 120 & 140 & 160\end{array}\) (a) What is the most probable speed? (b) Calculate the mean speed of the molecules in the sample. (c) Calculate the ms speed of the molecules in the sample.

Step-by-Step Solution

Answer

(a) The most probable speed is 100 m/s. (b) The mean speed is 100 m/s. (c) The RMS speed is approximately 103 m/s.

Step 1: Identify Most Probable Speed

The most probable speed corresponds to the speed associated with the highest number of molecules. Here, 40 molecules have the speed 100 m/s, which is the maximum frequency. Therefore, the most probable speed is 100 m/s.

Step 2: Calculate Total Number of Molecules

Calculate the total number of molecules by summing the given distribution: \(10 + 20 + 40 + 15 + 10 + 5 = 100\).

Step 3: Calculate Mean Speed

To calculate the mean speed, use the formula: \(\text{Mean speed} = \frac{\sum (\text{speed} \times \text{number of molecules})}{\text{Total number of molecules}}\). Compute as follows:\[(60\times10) + (80\times20) + (100\times40) + (120\times15) + (140\times10) + (160\times5) = 10000\]Divide by the total number of molecules:\[\text{Mean speed} = \frac{10000}{100} = 100 \, \text{m/s}\].

Step 4: Calculate Mean Square Speed

Use the formula for the mean square speed: \[\text{Mean square speed} = \frac{\sum (\text{speed}^2 \times \text{number of molecules})}{\text{Total number of molecules}}\]. Compute as follows: \[((60)^2\times10) + ((80)^2\times20) + ((100)^2\times40) + ((120)^2\times15) + ((140)^2\times10) + ((160)^2\times5) = 1060000\]Divide by the total number of molecules:\[\text{Mean square speed} = \frac{1060000}{100} = 10600 \, \text{m}^2/\text{s}^2\].

Step 5: Calculate RMS Speed

The root mean square (RMS) speed is the square root of the mean square speed. Calculate as follows:\[\text{RMS speed} = \sqrt{10600} \approx 102.96 \, \text{m/s}\].

Key Concepts

Most Probable Speed

The most probable speed of a gas molecule is defined as the speed at which the greatest number of molecules are moving. It is derived from observing the distribution of molecular speeds in a given sample. In the problem at hand, a check of the provided speed distribution shows that the speed of 100 m/s is achieved by 40 molecules, the highest number in our sample set. Thus, 100 m/s is the most probable speed for this situation.

• This speed is purely statistical, indicating the peak of the speed distribution curve.
• Most probable speed might differ from mean or RMS speed, especially in non-ideal distributions.

Mean Speed Calculation

Mean speed represents the average speed at which molecules in the sample are moving. To calculate this, multiply each possible speed by the number of molecules traveling at that speed. Sum up these products and then divide by the total number of molecules in the sample.

To put this into practice: - Calculate the total momentum from each speed by multiplying the speed and its corresponding molecule count. - Sum the values: \((60 \times 10) + (80 \times 20) + (100 \times 40) + (120 \times 15) + (140 \times 10) + (160 \times 5) \)This totals to 10,000. Divide the sum by 100 (the total number of molecules) to get the mean speed, which results in 100 m/s.

• The mean speed effectively shows an average, smoothing out any outliers in the distribution.
• It provides a useful measure for understanding molecular behavior in general terms.

Mean Square Speed

The mean square speed is slightly different as it focuses on the squares of the speeds rather than the speeds themselves. This measure takes into account the energy aspect, as kinetic energy depends on speed squared.

To find it, calculate the square of each speed, multiply by the number of molecules moving at that speed, and sum these values. Finally, divide by the total molecule number:\[((60)^2 \times 10) + ((80)^2 \times 20) + ((100)^2 \times 40) + ((120)^2 \times 15) + ((140)^2 \times 10) + ((160)^2 \times 5) = 1060000\] Divide by 100 to find the mean square speed: 10600 m\(^2\)/s\(^2\).

• This speed sheds light on the variance in molecular speeds.
• Understanding mean square speed helps in context where more precise energy calculations are critical.

RMS Speed

RMS speed, or root mean square speed, is a common metric for assessing molecular motion in gases. It is derived as the square root of the mean square speed, providing a speed that accounts for both the most probable and mean speeds in the context of kinetic energy.

To calculate RMS speed: - Start with the mean square speed obtained earlier (10600 m\(^2\)/s\(^2\)). - Find the square root of this value:\(\sqrt{10600} \approx 102.96 \text{ m/s}\)

• RMS speed provides a practical understanding of an average speed reflecting energy among particles.
• It's a useful indicator in thermodynamic studies and aspects involving energy conservation.

Distribution of Molecular Speeds

The distribution of molecular speeds in a gas sample shows variation due to different energy levels among molecules. Represented most frequently with the Maxwell-Boltzmann distribution curve in statistical mechanics. This distribution provides a snapshot of how many molecules achieve each speed range.

Key considerations include:

• The shape of the distribution is typically a bell curve, known for its characteristic rise, peak, and fall pattern.
• Most probable speed is typically lower than both the mean and RMS speeds in such distributions.
• Understanding this distribution is crucial for predicting outcomes in chemical reactions and physical processes involving gaseous substances.

Real-world applications are vast, from engineering to environmental studies, wherever gas behavior under different conditions needs assessment.