Problem 156 - A Complete Resource Book in Chemistry for JEE Main

Question

For gaseous state if most probable speed is denoted \(\mathrm{C}^{*}\), average speed by \(\overline{\mathrm{C}}\) and and mean square speed by C, then for a large number of molecules the ratios of these speeds are: (a) \(C^{*}: \overline{\mathrm{C}}: \mathrm{C}=1: 128: 1.225\) (b) \(\mathrm{C}^{*}: \overline{\mathrm{C}}=1: 1.225: 1.128\) (c) \(\mathrm{C}^{*}: \overline{\mathrm{C}}: \mathrm{C}=1.225: 1.128: 1\) (d) \(\mathrm{C}^{*}: \overline{\mathrm{C}}: \mathrm{C}=1.128: 1.225: 1\)

Step-by-Step Solution

Answer

The correct option is (d): \(C^* : \overline{C} : C = 1 : 1.128 : 1.225\).

Step 1: Understanding Speed Formulas

In the gaseous state, the speeds of gas molecules can be categorized into three types: most probable speed, average speed, and root mean square speed. They are given by different formulas: Most probable speed \(C^* = \sqrt{\frac{2RT}{M}}\), Average speed \(\overline{C} = \sqrt{\frac{8RT}{\pi M}}\), and Root mean square speed \(C = \sqrt{\frac{3RT}{M}}\).

Step 2: Calculate Ratios

To find the ratios of these speeds, we compute the fractions between each pair of speeds using their formulas. To start, calculate the ratio \( \frac{\overline{C}}{C^*} = \frac{\sqrt{\frac{8RT}{\pi M}}}{\sqrt{\frac{2RT}{M}}} = \sqrt{\frac{8}{2 \pi}} = \sqrt{\frac{4}{\pi}} \approx 1.128.\)

Step 3: Compute the Next Ratio

Now compute the ratio \( \frac{C}{C^*} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{2RT}{M}}} = \sqrt{\frac{3}{2}} \approx 1.225.\)

Step 4: Arrange the Ratios

Using the computed ratios, the arrangement \(C^* : \overline{C} : C\) becomes \(1 : 1.128 : 1.225.\)

Step 5: Select Correct Answer

Compare the arranged ratios to the options provided: option (d) matches the calculated result: \(C^* : \overline{C} : C = 1 : 1.128 : 1.225\).

Key Concepts

Most Probable Speed

In the study of gases, the concept of most probable speed refers to the speed at which the largest number of particles in a gas are moving. It's important to note that this is not the average speed, but the mode of the speed distribution in a gas. For molecules in a gas, this speed can be calculated using a mathematical formula:

• The formula is given by \(C^* = \sqrt{\frac{2RT}{M}}\).
• Here, \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas.

The most probable speed is an important concept as it helps us understand the kinetic behavior of gas molecules at a given temperature. The formula indicates that the most probable speed increases with temperature and decreases with molecules of higher molar mass. Essentially, lighter and hotter gas molecules move more quickly than heavier, cooler ones. By understanding this behavior, we can predict how gases will interact in different conditions.

Average Speed

Average speed in the context of gases is exactly what it sounds like: the mean speed of all the molecules in a gas sample. It's a measure of the typical speed that gas molecules are moving at a given temperature. To compute this, one uses the formula:

• \(\overline{C} = \sqrt{\frac{8RT}{\pi M}}\)
• Again, \(R\) is the gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass.

It's interesting to observe that the average speed relies on similar factors as the most probable speed. Temperature and molar mass play key roles. With an increase in temperature, the average speed goes up, and it decreases for gases with a higher molar mass.
This concept is meaningful in real-world applications as it helps in predicting the behavior of gases under different circumstances, such as in chemical reactions or in gas diffusion.

Root Mean Square Speed

Root mean square speed (often abbreviated as RMS speed) is a particularly useful statistical measure in understanding the kinetic energies of gases. It is essentially the square root of the average of the squares of the individual speeds of gas molecules, which gives insight into their energetic behavior. The formula is:

• \(C = \sqrt{\frac{3RT}{M}}\)
• Where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas.

The RMS speed is greater than both the most probable speed and the average speed. This reflects that the squaring of speeds (particularly the higher speeds) increases the average and thus gives a more comprehensive idea about the energy of the system as a whole.
It gives us an idea about the kinetic energy possessed by the gas particles, which is critical in many thermodynamic calculations and is fundamental to the derivation of important gas laws. By comparing RMS speed with other speed measurements, we can gain a deeper understanding of gas molecule behavior in various conditions.