Problem 84 - Chemistry

Question

At a certain temperature the speeds of six gaseous molecules in a container are \(2.0 \mathrm{~m} / \mathrm{s}, 2.2 \mathrm{~m} / \mathrm{s}, 2.6 \mathrm{~m} / \mathrm{s}\) \(2.7 \mathrm{~m} / \mathrm{s}, 3.3 \mathrm{~m} / \mathrm{s},\) and \(3.5 \mathrm{~m} / \mathrm{s} .\) Calculate the root- mean-square speed and the average speed of the molecules. These two average values are close to each other, but the root-mean-square value is always the larger of the two. Why?

Step-by-Step Solution

Answer

The average speed of the gas molecules is 2.72 m/s and the root mean square speed is 2.23 m/s. The root mean square speed is larger than the average speed because when calculating the root mean square, the speeds are squared before averaging, which gives more weight to higher speeds.

Step 1: Calculate Average Speed

Find the sum of all the speeds: \(2.0m/s + 2.2m/s + 2.6m/s + 2.7m/s + 3.3m/s + 3.5m/s = 16.3m/s\). Divide the sum by the total number of molecules, which is 6, to find the average speed: \(16.3m/s ÷ 6 = 2.72m/s\).

Step 2: Calculate Root Mean Square Speed

Find the square of each of the speeds, then find their sum: \(2.0m/s^2 + 2.2m/s^2 + 2.6m/s^2 + 2.7m/s^2 + 3.3m/s^2 + 3.5m/s^2 = 29.8(m/s)^2. Divide this sum by the total number of molecules, which is 6, to find the mean square speed: \(29.8(m/s)^2 ÷ 6 = 4.97(m/s)^2. Finally, find the root of this number to get the rms speed: √4.97(m/s)^2 = 2.23m/s.

Step 3: Reason for Difference in Values

RMS value is larger than the average value because when calculating rms, the speeds are squared before averaging. This gives more weight to higher speeds compared to lower speeds. The weights are then equalized by taking the square root, but the result is still higher compared to the simple average.

Key Concepts

Root Mean Square Speed

The Root Mean Square (RMS) speed is a statistical measure used extensively in the kinetic theory of gases to describe the speed of particles in a gas. To calculate the RMS speed, you first square the speeds of each individual gas molecule. For example, with speeds of 2.0 m/s, 2.2 m/s, and so on, you square each one. - Then, you find the mean of these squared speeds by adding them all together and dividing by the number of particles. - Finally, you take the square root of this mean value to obtain the RMS speed. This method emphasizes the higher speeds because squaring larger numbers results in even larger numbers, making the RMS speed always slightly higher than the average speed. This characteristic is particularly useful in physics as it aligns closely with our concepts of energy and force. The calculated RMS speed in our scenario was 2.23 m/s, highlighting that it accounts for variations in molecular speed by giving more weight to the faster moving molecules.

Average Speed

Average speed is a fundamental concept that is easier to calculate compared to the Root Mean Square speed. To find the average speed, you sum up all the individual speeds of molecules, then divide this total by the number of molecules. This gives you the mean, or average, speed. In our example, with speeds given as 2.0 m/s to 3.5 m/s, the sum of these speeds is calculated to be 16.3 m/s, and dividing by the 6 molecules provided gives an average speed of 2.72 m/s. It’s important to note that the average speed simply considers all speeds equally without taking into account the variation or distribution of those speeds. This can sometimes make it a limited descriptor, especially in the realm of kinetic theory, where energy distribution and variations in speed matter greatly.

Kinetic Theory of Gases

The Kinetic Theory of Gases offers a foundational framework for understanding how gases behave in terms of motion and energy. According to this theory: - Gas molecules are in constant, random motion. - The particles collide with each other and the walls of their container without losing energy. - Temperature is directly proportional to the average kinetic energy of the gas molecules, which relates to speed. This theory helps us understand why different measures like RMS speed and average speed are significant. The higher the temperature, for instance, the higher the average kinetic energy, leading to higher speeds overall. The kinetic theory not only supports the differences in how RMS and average speeds are calculated but also emphasizes why RMS is useful as it reflects energy distribution more accurately. This theory allows us to make connections between macroscopic properties like pressure and temperature with microscopic dynamics of molecular movement.