Thermodynamics

We are tasked with finding the ratio of the kinetic energy of a molecule of neon (Ne) to that of oxygen (Oâ‚‚) at .
Step 1: Kinetic energy of a gas molecule The average kinetic energy of a gas molecule is given by: where is the degrees of freedom, is the Boltzmann constant, and is the temperature in Kelvin.
Step 2: Degrees of freedom
For neon (monatomic gas), .
For oxygen (diatomic gas), .
Step 3: Ratio of kinetic energies
The ratio of the kinetic energy of neon to that of oxygen is: Step 4: Match with the options The ratio matches option (2).
Final Answer:

In an Isochoric process, the volume of the system remains constant. In this case, the heat supplied to the system only raises the temperature, as there is no work done due to no volume change (i.e., ). All the heat energy supplied is used to increase the internal energy and thus the temperature.
Final Answer: Isochoric process.

The problem asks for the energy spent during the expansion, which is the work done.
We know that , where is the ideal gas constant. Given and , we can find : The work done at constant pressure is given by:
We are given moles and .
Substituting the values, we get: Therefore, the amount of energy spent during the expansion is 200 Cal.

The latent heat is the heat required to melt or solidify a substance. The amount of heat required is given by: This heat is supplied by the power over a time , so: Equating both expressions: Solving for the latent heat : Thus, the latent heat of the substance is .
Final Answer: .

Step 1: Convert speed to m/s and calculate centripetal acceleration.
Speed in m/s: Centripetal acceleration : Step 2: Compare to gravitational acceleration.
Gravitational acceleration :

Step 1: Apply the gas law under constant temperature. Assuming the pressure inside each bubble is proportional to the surface tension divided by the radius and that both bubbles contain the same type of gas at the same temperature, the number of moles is proportional to the volume.
Step 2: Relate the volumes and masses of the bubbles. The volume of a sphere (bubble) is , so the ratio of the volumes (and thus the ratio of masses under constant density and temperature) is: Adjusting for the proportional relationship of the radii squared:

We are given:
Pressure ,
Density ,
Mass .
For a diatomic gas, the internal energy can be calculated using the formula for the internal energy of an ideal gas: where is the number of degrees of freedom (for a diatomic gas, ), is the number of moles, and is the universal gas constant ( ). However, we can also use the equation , where is the molar mass of the gas. Rearranging, we get: Now, substituting the given values: - , - , - . We calculate : Next, we calculate the internal energy using the formula for the internal energy of a diatomic gas: Substituting the values for , , and : Thus, the internal energy of the gas is , so the correct answer is (1).
Conclusion: The internal energy of the gas is , so the correct answer is (1).

When two ideal gases at absolute temperatures and are mixed, the total energy is conserved, and there is no loss of energy in the process. The total internal energy of the system is the sum of the internal energies of the individual gases.

The internal energy of an ideal gas is proportional to its temperature, and for each gas, we have:

where:

• and are the number of moles of gases 1 and 2,
• is the molar specific heat at constant volume (which is constant for both gases),
• and are the absolute temperatures of the two gases.

When the two gases are mixed, the total internal energy becomes:

The final temperature is such that the total energy is divided between the two gases. So, we have:

Equating the total energy before and after mixing:

Simplifying:

Thus, the final temperature is:

Therefore, the correct answer is Option (1), .

Given: Monatomic gas: 2 moles, expanding adiabatically from to . Diatomic gas: 3 moles, expanding adiabatically from to .

Step 1: Work done by a gas in an adiabatic process The work done in an adiabatic expansion or compression is given by the formula: Where:
= number of moles,
= molar heat capacity at constant volume,
= adiabatic index ( ),
= temperature change.

Step 2: Work done by the monatomic gas For a monatomic gas, . For 2 moles of the monatomic gas, the temperature change . The work done is: Substitute the values: Thus, the work done by the monatomic gas is .

Step 3: Work done by the diatomic gas For a diatomic gas, . For 3 moles of the diatomic gas, the temperature change . The work done is: Substitute the values: Thus, the work done by the diatomic gas is .

Step 4: Ratio of work done Now, we find the ratio of the work done by the diatomic gas to the work done by the monatomic gas:

Thus, the work done by the diatomic gas is times the work done by the monatomic gas.

Step 1: Use the ideal gas law. Initially, let be the number of moles of air, the volume, and the ideal gas constant: where moles (assuming air is mostly nitrogen, ), and .
Step 2: Calculate the final number of moles and the difference. From the equations, it follows that: The moles of air escaped: The mass of the air escaped:

To determine how the efficiency of a Carnot engine changes when the ratio of the absolute temperatures of the sink and source changes, we follow these steps: Given:
- Initial temperature ratio:
- New temperature ratio:

Step 1: Calculate the Initial Efficiency The efficiency of a Carnot engine is given by: For the initial ratio:

Step 2: Calculate the New Efficiency For the new ratio:

Step 3: Determine the Change in Efficiency
The change in efficiency is: However, we are interested in the absolute change relative to the initial efficiency:

Final Answer: This corresponds to option (1).

Given: - Ratio of specific heats, Step 1: Determine the fraction of heat used for work in an isobaric process For an isobaric process, the heat added to the gas is used to increase the internal energy and to do work .
The relationship is given by: For an ideal gas, the work done in an isobaric process is: The change in internal energy is: The total heat added is: The fraction of heat used for work is: Given that and , we can express as: Thus, Substituting : Step 2: Convert the fraction to a percentage Final Answer: 25 %

The cycle ABCDA consists of:
A B: Isochoric process (Volume constant, ). Pressure increases from to .
B
C: Isobaric process (Pressure constant, ). Volume increases from to . C D: Isochoric process (Volume constant, ). Pressure decreases from to .
D A: Isobaric process (Pressure constant, ). Volume decreases from to .
1. Work done in each process:
(Isochoric process)

(Isochoric process)

Net work done, .
2. Heat input in each process: For Helium (monatomic gas), and .
Process A B (Isochoric): . Using ideal gas law , . , . .
Process B C (Isobaric): . , . .
Process C D (Isochoric): (Heat rejected).
Process D A (Isobaric): (Heat rejected).
Total heat input .
3. Efficiency of the cycle: Efficiency .
Correct Answer: (4) 15.4%