The Kinetic Theory Of Gases

Step 1: Understanding the overtone frequencies.
The frequency of the first overtone for an open pipe is , where is the fundamental frequency, and for a closed pipe, the first overtone is the third harmonic, .
Step 2: Frequency and length relationship.
The frequency of an organ pipe is inversely proportional to its length. So, for the same overtone frequencies, the ratio of the lengths of the pipes is given by:
Step 3: Conclusion.
The ratio of the lengths of the open and closed pipes is , so the correct answer is (D).

Step 1: Write expression for magnetic moment of a revolving charge.
Magnetic moment is given by:
where is current and is area of the circular orbit.

Step 2: Calculate current due to revolving electron.

Step 3: Write area of orbit.

Step 4: Combine expressions.

Step 5: Substitute given values.

Step 6: Conclusion.
The magnetic moment of the revolving electron is .

Step 1: Expression for centripetal force.
For circular motion of electron:

Step 2: Use Bohr’s postulates.
Radius of orbit:
Velocity of electron in orbit:

Step 3: Substitute and in force equation.

Step 4: Simplify.

Step 5: Conclusion.
The required centripetal force is .

Step 1: Using the formula for r.m.s velocity.
The r.m.s. velocity of a gas is given by: where is Boltzmann's constant, is temperature, and is the molecular mass. For hydrogen and nitrogen, we know the ratio of r.m.s velocities: Substituting the given values and solving for , we get: Step 2: Final Answer.
Thus, the temperature is 1050 K.

Step 1: Ideal Gas Law.
The ideal gas law is given by: where is the pressure, is the volume, is the number of moles, is the gas constant, and is the temperature. The number of moles is related to the density by: where is the molar mass of the gas. Substituting this into the ideal gas law, we get: Thus, the ratio of the densities of gases A and B is related to their molecular weights by: Given that , we find: Thus, the ratio of the molecular weights of A and B is .
Step 2: Final Answer.
Thus, the ratio of the molecular weights of A and B is .

Step 1: Relationship between pressure and temperature.
For an ideal gas, the pressure is directly proportional to the temperature. The average force on the walls of the container is related to the pressure of the gas, which depends on the temperature as . The force is proportional to the pressure, so the force depends on .
Step 2: Conclusion.
Thus, the value of is 1, corresponding to option (A).

Step 1: Understanding the problem.
The average kinetic energy of a gas molecule is given by the equation: where is Boltzmann's constant and is the temperature in Kelvin. The average kinetic energy is the same for all ideal gases at the same temperature, regardless of the type of molecule (whether , , or others). Therefore, the average kinetic energy for will be the same as for .
Step 2: Conclusion.
Thus, the average kinetic energy of molecule at 300K is , corresponding to option (B).

Step 1: Understanding the fringe spacing.
The fringe spacing in a double-slit experiment is given by the formula: where is the wavelength of the light, is the distance between the slits and the screen, and is the separation between the slits.

Step 2: Analyzing the options.
(A) Reducing the separation between the slits: This would increase the fringe spacing, not decrease it.
(B) Using blue light instead of green light: Blue light has a shorter wavelength, which would result in closer fringes.
(C) Using red light instead of green light: Red light has a longer wavelength, which results in closer fringes.
(D) Moving the screen away from the slits: This would increase the fringe spacing.

Step 3: Conclusion.
The correct answer is (C) Using red light instead of green light, as it will produce a longer wavelength and cause the fringes to be more closely spaced.

Step 1: Understanding the specific heat ratio.
For a rigid diatomic molecule, the number of degrees of freedom is , and for a monatomic molecule, . The ratio of specific heats is given by: Step 2: Finding for rigid diatomic and monatomic molecules.
For a rigid diatomic molecule: For a monatomic molecule: Step 3: Finding the ratio of to .
The ratio is: Step 4: Conclusion.
The correct answer is (C), .

Step 1: Formula for .
The ratio of specific heats for a gas is related to the number of degrees of freedom by: Step 2: Calculate for monoatomic gas.
For a monoatomic gas, the number of degrees of freedom , since it has 3 translational degrees of freedom. Therefore: Step 3: Calculate for diatomic gas.
For a rigid diatomic gas, the number of degrees of freedom (3 translational + 2 rotational). Therefore: Step 4: Calculate the ratio of values.
The ratio of for the monoatomic gas to the diatomic gas is: Step 5: Conclusion.
Thus, the ratio of for the monoatomic gas to the diatomic gas is , which corresponds to option (A).

Step 1: Understanding the equation.
The equation relates pressure , volume , temperature , and the Boltzmann constant . It is a part of the ideal gas law in terms of molecules. The ideal gas law in molecular form is: Where is the number of molecules. Therefore, gives the number of molecules of gas in the given volume. Step 2: Conclusion.
Thus, represents the number of molecules of gas in one mole. The correct answer is (D) number of molecules of gas in one mole.

Step 1: Formula for RMS velocity.


Step 2: Equate RMS velocities.


Step 3: Substitute given values.

Step 1: Understanding the relation between and .
The ratio , where is the specific heat at constant pressure and is the specific heat at constant volume. This ratio is related to the degrees of freedom 'n' of the gas particles. For ideal gases, the ratio is dependent on the number of translational, rotational, and vibrational degrees of freedom.
Step 2: Formula application.
The general expression for in terms of the degrees of freedom 'n' is derived from the kinetic theory of gases. The correct formula is , which accounts for the translational and rotational degrees of freedom.
Step 3: Conclusion.
The correct answer is , as it matches the known formula for the ratio of specific heats in terms of degrees of freedom.

Step 1: Write the formula for r.m.s. velocity.
The r.m.s. velocity of gas molecules is given by where is absolute temperature.
Step 2: Convert temperatures into Kelvin scale.

Step 3: Take the ratio of velocities.

Step 4: Conclusion.
The required ratio is .

Step 1: Understanding the r.m.s. velocity and isothermal compression.
The root mean square (r.m.s.) velocity of gas molecules is given by the equation: where is the Boltzmann constant, is the temperature, and is the molecular mass.
Step 2: Analyzing the effect of isothermal compression.
In an isothermal process, the temperature remains constant. Since the r.m.s. velocity depends only on the temperature (and not the pressure or volume directly in an isothermal process), the r.m.s. velocity does not change.
Step 3: Conclusion.
Therefore, the correct answer is (A) because the r.m.s. velocity of gas molecules remains constant during isothermal compression.

Step 1: Formula
Maximum safe speed .
Since are constant, .
Step 2: Proportionality
.
Given , so .
Step 3: Increase in Radius
Increase .
Final Answer: (C)

Concept:
In vertical circular motion:

• Maximum tension occurs at lowest point
• Minimum tension occurs at highest point

Step 1: Given ratio.
Step 2: Apply energy conservation. Between top and bottom:
Step 3: Substitute in tension ratio. Cancel :
Step 4: Solve equation. Let
Step 5: Conclusion.