Ideal Gas Equation

Solution:

The kinetic energy per molecule is given by:

Thus, the ratio of kinetic energy per molecule of Argon and Hydrogen is:

Since the temperature is the same for both gases, the ratio is 1. Hence, the correct answer is .

Step 1: Calculate pressures at the surface and depth

• Pressure at the surface
• Pressure at depth is calculated using the formula:
• Substitute

Step 2: Apply Boyle’s law

• Given the initial volume and constant temperature:
• Substitute
• Solve for :

Final Answer:

The volume of the air bubble at depth is

The force between two charges is given by Coulomb's law: where is the force, is Coulomb's constant, and are the magnitudes of the charges, and is the distance between them. If the distance is doubled, the new distance becomes . The new force is: Thus, the force decreases by a factor of 4 when the distance is doubled. Hence, the correct answer is .

Degrees of Freedom for Different Types of Gases:

Explanations:

• (A) 3 Translational degrees of freedom: This corresponds to movement along the x, y, and z axes. Monoatomic gases, consisting of single atoms, only have these three translational degrees of freedom. Thus, (A) matches with (I).
• (B) 3 Translational, 2 Rotational degrees of freedom: Diatomic molecules can translate in three directions and rotate about two axes perpendicular to the bond axis. Rigid diatomic molecules do not have vibrational degrees of freedom as the bond length is considered fixed. Thus, (B) matches with (III).
• (C) 3 Translational, 2 Rotational, and 1 Vibrational degrees of freedom: Non-rigid diatomic gases have the ability to vibrate along the bond axis, adding one vibrational degree of freedom in addition to the 3 translational and 2 rotational degrees. Thus, (C) matches with (IV).
• (D) 3 Translational, 3 Rotational, and more than 1 Vibrational degrees of freedom: Polyatomic gases, having more than two atoms, can rotate about all three axes. They also have multiple vibrational modes depending on the number of atoms and the molecular structure. Thus, (D) matches with (II).

The average kinetic energy ( ) of gas molecules is directly proportional to the absolute temperature ( ) and is given by: where is Boltzmann's constant. \bigskip
Step 1: Initial kinetic energy at . The absolute temperature at is:
Step 2: New temperature for double kinetic energy. If the kinetic energy becomes double: Since : Convert to Celsius:
Final Answer: The temperature is:

Step 1: Understanding the adiabatic process.
In an adiabatic process, there is no heat exchange between the system and its surroundings, i.e., . According to the first law of thermodynamics: Where is the change in internal energy, and is the work done by the system. Since there is no heat exchange in an adiabatic process, we have: This means that the change in the internal energy is equal to the negative of the work done by the system.
Step 2: Conclusion.
Thus, the internal energy of the gas changes during adiabatic compression. Therefore, the statement "There is no change in the internal energy" is incorrect.

Change in Internal Energy ( ):

The change in internal energy for a thermodynamic process is given by:

For an isothermal process, the temperature remains constant:

Substituting into the equation:

Conclusion:

Since the process is isothermal, there is no change in the internal energy of the system.

Final Answer:

We need to find the magnetic field produced at the nucleus of a hydrogen atom due to the electron's motion, using the Biot-Savart law.

Solution

1. Biot-Savart Law for a Circular Orbit:

The magnetic field at the center of the orbit is given by:

where:

• is the charge of the electron ( )
• is the speed of the electron ( )
• is the radius of the orbit ( )
• is the permeability of free space ( )

2. Substitute the Given Values:

3. Calculate the Magnetic Field:

Final Answer

Thus, the magnetic field produced at the nucleus of the hydrogen atom is .

The problem involves determining the dimensions of the quantity in the equation of state for a real gas given by:

First, let's analyze the dimensions of each term:

1. The term (pressure) has dimensions: .
2. The volume has dimensions: .
3. The universal gas constant has dimensions of energy per mole per Kelvin, which is generally expressed in terms of pressure, volume, and temperature: , where represents the temperature dimension.

The term should have the same dimensions as , which implies:

Thus, the dimensions of can be derived as follows:

Now, the term is a volume correction and is typically related to the volume:

So for the expression , we calculate:

It turns out that the dimensions of match those of (pressure).

Therefore, the correct answer is:

Option:

P