Question 1

Q1. Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3 Å.

Accepted Answer

The fraction of molecular volume to the actual volume occupied by oxygen gas at STP is approximately 4.24 × 10⁻⁴.

Question 2

Q2. Molar volume is the volume occupied by 1 mol of any (ideal) gas at STP. Show that it is 22.4 litres.

Accepted Answer

Using the ideal gas equation, PV = nRT, and substituting the values for 1 mole of an ideal gas at STP, the molar volume is calculated to be 22.4 litres.

Question 3

Q3. Figure 13.8 shows plot of PV/T versus P for 1.00 × 10–3 kg of oxygen gas at two different temperatures.
(a) What does the dotted plot signify?
(b) Which is true: T1 > T2 or T2 >

Accepted Answer

This question explores the behavior of real gases versus ideal gases, and the interpretation of PV/T plots. The dotted line represents ideal gas behavior, while the solid curves show deviations for real oxygen gas. The value of PV/T at the y-intercept is nR, which depends on the number of moles of the gas.

Question 4

Q4. A meter long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is

Accepted Answer

This problem involves resonance in an open-closed tube, which is a fundamental concept in wave optics and sound. The key is to understand the formation of standing waves and the relationship between tube length, wavelength, and frequency. The absence of end correction in this specific problem simplifies calculations significantly.

Question 5

Q5. A child is playing with a long rope, which is fixed at one end. The child generates a wave pulse by jerking the free end up and down. The pulse travels to the fixed end and then re

Accepted Answer

When a wave pulse travels along a rope and encounters a fixed end, it undergoes reflection. The fixed end acts as a rigid boundary, which is analogous to a denser medium for the wave. According to the principles of wave reflection, when a wave reflects from a fixed boundary (or a denser medium), it undergoes a phase change of 180 degrees (or π radians). This phase change means that the reflected pulse is inverted with respect to the incident pulse. If the incident pulse was an upward displacement, the reflected pulse will be a downward displacement, and vice-versa. This inversion occurs because the fixed end cannot move; thus, when the incident pulse tries to pull the fixed end up, the fixed end exerts an equal and opposite downward force, generating an inverted reflected pulse.

Question 6

Q6. A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted sound? Speed o

Accepted Answer

This problem demonstrates a key property of waves: frequency remains constant when a wave passes from one medium to another, while its speed and wavelength change. We will use the fundamental wave equation v = fλ to calculate the wavelengths in different media.

Question 7

Q7. A hospital uses an ultrasonic scanner to locate tumours in tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km s⁻¹? The operating frequency

Accepted Answer

This problem is a direct application of the fundamental wave equation relating speed, frequency, and wavelength. It highlights the practical use of ultrasound in medical diagnostics.

Question 8

Q8. From a certain apparatus, the diffusion rate of hydrogen is 28.7 cm³ s⁻¹. The diffusion rate of another gas under the same conditions is 7.2 cm³ s⁻¹. Identify the gas.

Accepted Answer

The gas is Oxygen (O₂).

Question 9

Q9. A gas in an open container is heated from 27 °C to 127 °C. The fraction of the original number of molecules remaining in the container will be:

Accepted Answer

The fraction of the original number of molecules remaining in the container is 0.75 or 3/4.

Question 10

Q10. An air bubble of volume 1.0 cm³ rises from the bottom of a lake 40 m deep at a temperature of 12 °C. To what volume does it grow when it reaches the surface, which is at a temperat

Accepted Answer

The air bubble grows to a volume of approximately 5.27 cm³ when it reaches the surface.

Question 11

Q11. Estimate the total number of air molecules in a room of capacity 25.0 m³ at a temperature of 27 °C and 1 atm pressure.

Accepted Answer

The total number of air molecules in the room is approximately 6.1 × 10²⁶ molecules.

Question 12

Q12. Estimate the average distance between molecules of air at S.T.P. (The number of molecules per unit volume at S.T.P. is called Loschmidt's number. Its value is 2.687 × 10²⁵ m⁻³).

Accepted Answer

The average distance between molecules of air at S.T.P. is approximately 3.3 × 10⁻⁹ m or 3.3 nm.

Question 13

Q13. Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature 17 °C. The radius of a nitrogen molecule is

Accepted Answer

The mean free path of a nitrogen molecule at 2.0 atm and 17 °C is approximately 1.11 × 10⁻⁷ m. The collision frequency is approximately 4.25 × 10⁹ s⁻¹. The time taken for a molecule to travel freely between two successive collisions is approximately 2.35 × 10⁻¹⁰ s, while the collision time is approximately 2.35 × 10⁻¹³ s. Thus, the collision time is much smaller (about 1000 times) than the time between successive collisions.

Question 14

Q14. Estimate the average speed of nitrogen molecules in an atmosphere at room temperature (27 °C). The molar mass of nitrogen molecules is 28.0 g/mol. What is the rms speed?

Accepted Answer

To estimate the average speed and calculate the root mean square (rms) speed of nitrogen molecules, we use the formulas derived from the kinetic theory of gases. The average speed (v_avg) and rms speed (v_rms) are distinct measures of molecular speeds in a gas.

**Given:**
*   Temperature, T = 27 °C
*   Molar mass of nitrogen (N₂), M = 28.0 g/mol

**Constants:**
*   Gas constant, R = 8.314 J mol⁻¹ K⁻¹
*   π ≈ 3.14159

**Step 1: Convert units**
*   Temperature T = 27 °C + 273.15 = 300.15 K (approximately 300 K for calculations).
*   Molar mass M = 28.0 g/mol = 28.0 × 10⁻³ kg/mol.

**Step 2: Calculate the rms speed (v_rms)**
The formula for the root mean square speed is:
v_rms = √(3RT/M)

v_rms = √(3 × 8.314 J mol⁻¹ K⁻¹ × 300 K / (28.0 × 10⁻³ kg/mol))
v_rms = √(7482 / 0.028) m/s
v_rms = √(267214.2857) m/s
v_rms ≈ 516.93 m/s

**Step 3: Calculate the average speed (v_avg)**
The formula for the average speed (from Maxwell-Boltzmann distribution) is:
v_avg = √(8RT/πM)

v_avg = √(8 × 8.314 J mol⁻¹ K⁻¹ × 300 K / (3.14159 × 28.0 × 10⁻³ kg/mol))
v_avg = √(19953.6 / (3.14159 × 0.028)) m/s
v_avg = √(19953.6 / 0.08796452) m/s
v_avg = √(226839.8) m/s
v_avg ≈ 476.28 m/s

**Final Answer:**
The estimated average speed of nitrogen molecules is approximately **476.28 m/s**.
The rms speed of nitrogen molecules is approximately **516.93 m/s**.

Chapter 13: Kinetic Theory