Problem 28 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A gas bulb of 1 litre capacity contains \(2.0 \times 10^{21}\) molecules of nitrogen exerting a pressure of \(7.57 \times 10^{3} \mathrm{Nm}^{-2} .\) Calculate the root mean square (r.m.s) speed and the temperature of the gas molecules. If the ratio of the most probable speed to the root mean square speed is \(0.82\), calculate the most probable speed for these molecules at this temperature.

Step-by-Step Solution

Answer

RMS speed \(517.7\,\text{m/s}\), temperature \(274\,\text{K}\), most probable speed \(424.5\,\text{m/s}\).

Step 1: Identify Given Values

We need to calculate the root mean square speed \( v_{rms} \) and temperature \( T \) of the gas. The given values are volume \( V = 1 \text{ L} = 0.001 \text{ m}^3 \), pressure \( P = 7.57 \times 10^3 \text{ N/m}^2 \), and number of molecules \( n = 2.0 \times 10^{21} \). The ratio of most probable speed to RMS speed is \( 0.82 \).

Step 2: Calculate Number of Moles

To find the number of moles \( N \), we use Avogadro's number \( N_A = 6.022 \times 10^{23} \text{ mol}^{-1} \):\[ N = \frac{n}{N_A} = \frac{2.0 \times 10^{21}}{6.022 \times 10^{23}} \approx 3.32 \times 10^{-3} \text{ mol} \]

Step 3: Use Ideal Gas Law to Find Temperature

The ideal gas law states \( PV = nRT \). Rearranging for temperature, we have:\[ T = \frac{PV}{nR} \]Substitute the values \( P = 7570 \text{ Pa} \), \( V = 0.001 \text{ m}^3 \), \( n = 3.32 \times 10^{-3} \text{ mol} \), and \( R = 8.314 \text{ J/mol} \cdot \text{K} \):\[ T = \frac{7570 \times 0.001}{3.32 \times 10^{-3} \times 8.314} \approx 274 \text{ K} \]

Step 4: Calculate Root Mean Square (RMS) Speed

The formula for RMS speed is \( v_{rms} = \sqrt{\frac{3RT}{M}} \) where \( M \) is the molar mass of nitrogen \( 0.028 \text{ kg/mol} \). Substitute values:\[ v_{rms} = \sqrt{\frac{3 \times 8.314 \times 274}{0.028}} \approx 517.7 \text{ m/s} \]

Step 5: Calculate Most Probable Speed

Given the ratio, \( \frac{v_{mp}}{v_{rms}} = 0.82 \). We can rearrange to find the most probable speed:\[ v_{mp} = 0.82 \times v_{rms} \approx 0.82 \times 517.7 \approx 424.5 \text{ m/s} \]

Step 6: Conclusion: Results

The RMS speed of the nitrogen molecules is approximately 517.7 m/s, and the most probable speed is 424.5 m/s at a temperature of 274 K.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that connects various properties of a gas—such as pressure, volume, and temperature. It is expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume that the gas occupies.
• \( n \) is the number of moles of the gas.
• \( R \) is the universal gas constant, approximately \( 8.314 \text{ J/mol} \cdot \text{K} \).
• \( T \) is the temperature in Kelvin.

The beauty of the Ideal Gas Law lies in its ability to predict the behavior of a perfect gas. It assumes that the gas molecules are point particles that do not interact except via elastic collisions. When dealing with problems involving the ideal gas law, ensure you convert all measurements to SI units. For example, pressure should be in pascals, volume in cubic meters, and temperature in Kelvin. In the current exercise, using the ideal gas law helps to determine the temperature of the nitrogen molecules, given the pressure, volume, and number of moles.

Root Mean Square Speed

Root Mean Square (RMS) speed is a way to express the average speed of particles in a gas. It is particularly useful when describing molecular motion in kinetic theory. The formula is:\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]where:

• \( R \) is the universal gas constant.
• \( T \) is the temperature in Kelvin.
• \( M \) is the molar mass of the gas in kilograms per mole.

The formula is derived from kinetic theory considerations, which model the gas particles in constant, random motion. RMS speed is not the same as average speed; it gives more weight to larger velocities because of the squaring before averaging. In the exercise, we calculated the RMS speed of nitrogen to be approximately 517.7 m/s, which is indicative of how fast the nitrogen molecules in the gas are moving, on average.

Most Probable Speed

Most Probable Speed, often represented as \( v_{mp} \), is the speed at which the largest number of molecules in a gas sample are moving at a given temperature. For an ideal gas, this value can be found using the formula:\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]It is derived from the Maxwell-Boltzmann distribution, which describes the distribution of speeds of particles in a gas. The most probable speed is slightly less than the root mean square speed due to the squared nature of RMS speed calculations. This means that while RMS speed emphasizes higher speeds, the most probable speed looks at what's most common. In this scenario, given the ratio \( \frac{v_{mp}}{v_{rms}} = 0.82 \), we find the most probable speed to be roughly 424.5 m/s.

Temperature Calculation

Temperature in thermodynamics is a measure of the average kinetic energy of particles in a substance. For gases, this concept is pivotal in understanding how changes in other properties affect the state of the gas. Calculation of temperature using the ideal gas law involves rearranging the formula to solve for \( T \):\[ T = \frac{PV}{nR} \]When substituting in values, make sure to use consistent units, like pascals for pressure, cubic meters for volume, and moles for the amount of gas. In our exercise, we calculated the temperature to be 274 K, which aligns the molecular kinetic energy with the observed pressure and volume of the gas sample. Temperature, in this context, plays a crucial role in connecting macroscopic conditions with microscopic molecular behavior in gases.