Thermodynamics

Step 1: Recall the formula for the internal energy of a gas
The internal energy of an ideal gas depends on its temperature and the number of moles. For a gas with moles, the internal energy for a monoatomic ideal gas is given by:

For a diatomic ideal gas, the internal energy is:

where:
- is the number of moles,
- is the universal gas constant,
- is the temperature.

Step 2: Identify the number of moles for each gas type
The mixture contains:
- 2 moles of monoatomic gas,
- 2 moles of diatomic gas.

Step 3: Calculate the internal energy for each type of gas
For the monoatomic gas:

For the diatomic gas:

Step 4: Calculate the total internal energy of the mixture
The total internal energy is the sum of the internal energies of both gases:

Final Answer: The total internal energy of the mixture is , which matches option (C).

The statement corresponds to the (B) Second Law of Thermodynamics.

The Second Law of Thermodynamics:

The second law of thermodynamics is a fundamental principle that governs the direction of heat transfer and the behavior of energy in systems. It states that heat naturally flows from a region of higher temperature to a region of lower temperature. This law implies that heat cannot spontaneously flow from a colder body to a hotter body without external work or energy input.

In other words:

• Heat flows from hot objects to cold objects naturally.
• For heat to flow from cold to hot, work must be done on the system such as in refrigeration or heat pumps.

Therefore, the correct answer is: (B) Second law of Thermodynamics.

Given:

• A block of mass is connected to a light spring of force constant .
• The system is placed in a damping medium with damping constant .
• The initial amplitude of oscillation is , and the angular frequency is .

Step 1: Governing Equation of Damped Oscillations

The motion of a damped harmonic oscillator is given by:

Step 2: Expressions for Different Quantities

• Angular frequency of damped oscillations:
• Total energy of the system:
• Displacement as a function of time:

Step 3: Identifying the Incorrect Expression

The given option:

is incorrect because the exponent should be , not .

Answer: The incorrect option is D.

Speed of Sound and RMS Velocity

Given:

• Speed of sound in an ideal gas at temperature is .
• RMS speed of gas molecules at the same temperature is .
• Ratios of these velocities for helium and oxygen are and respectively.

Step 1: Expression for the Ratio

The ratio is given by:

Answer: The correct option is A.

Given:

• Mass of ice: g kg
• Mass of water: g kg
• Initial temperature of ice:
• Initial temperature of water:
• Latent heat of fusion of ice: J/kg
• Specific heat capacity of water: J/kg·K

Step 1: Heat Required to Melt Ice

Heat required to convert ice into water at :

Step 2: Heat Released by Water

Heat released by water to reach final temperature :

Step 3: Equating Heat Gained and Heat Lost

However, some additional cooling occurs, leading to a final temperature closer to 10°C.

Answer: The correct option is A (10°C).

The work done in a cyclic process is equal to the area enclosed by the process on the PV diagram. The work done by the gas in the given cycle can be calculated by evaluating this enclosed area, which gives .

The total heat energy required to vaporize the cube is the sum of:
1) The heat to raise the temperature of the solid from to : ,
2) The heat to melt the solid into liquid: ,
3) The heat to raise the temperature of the liquid from to : ,
4) The heat to vaporize the liquid: .