Thermodynamics

The fractional change in the volume is related to the coefficient of volume expansion and the change in temperature by the equation: The fractional change in density is related to the fractional change in volume as: Substituting and , we get:
Thus, the correct answer is (d).

The first law of thermodynamics states that the change in internal energy of the gas is the sum of the heat added to the system and the work done by the system: For a cyclic process, the net change in internal energy .
Therefore, the net heat added equals the net work done: Substituting the given values:
Thus, the correct answer is (c).

The area covered by the TV tower can be calculated using the formula for the area of a circle:

where is the radius of the coverage area, which is the distance that the signal can travel from the tower. The radius can be derived using the height of the tower and the Earth's radius, based on the geometry of the situation:

where is the height of the tower (100 m) and is the radius of the Earth ( ). This relationship comes from the geometry of a line of sight between the tower and the Earth's curvature. By calculating this, we can find the radius .

The population covered is then given by multiplying the population density by the area:

After substituting the known values, we find the population covered to be approximately .

Hence, the correct answer is (c).

The specific heat of a substance depends on its molecular structure because the way molecules interact with each other determines how much heat energy is required to raise the temperature of the substance by a certain amount. Substances with more complex molecular structures may have more ways to store and transfer thermal energy, affecting the amount of energy needed for a temperature change. For example, in solids, the atomic or molecular vibrations (phonons) and bonds between molecules play a role in specific heat. In liquids and gases, the freedom of movement and interactions between molecules also influence the specific heat.

Hence, the correct answer is (d).

For an adiabatic process, the equation for pressure and temperature of an ideal gas is given by: where is the adiabatic index. Given that pressure is proportional to the cube of temperature, we have . For an ideal gas, the ratio is related to the adiabatic index as: From thermodynamics, for a monatomic ideal gas undergoing an adiabatic process, the ratio . Hence, the correct answer is (a).

When a metallic ball is heated, it expands uniformly. This causes the ball to expand and the spherical cavity at the centre to increase in size. The metal expands outward, so the cavity volume increases as well. Hence, the correct answer is (a).

An isobaric process is one in which the pressure remains constant. In such a process, the volume of the gas changes while the pressure stays the same. If the heat engine in the diagram operates under conditions of constant pressure, the process is classified as an isobaric process. During an isobaric process, the system can perform work by expanding or compressing, and the pressure does not change during these phases. The work done in an isobaric process can be calculated as the product of the constant pressure and the change in volume. Therefore, the work is done at constant pressure during the expansion and compression phases of the cycle, ensuring that the process remains isobaric.

Hence, the correct answer is (c).

When heat is added to a gas, the gas particles gain kinetic energy, which increases the temperature. If the piston is pulled out dramatically (meaning the gas expands), the gas does work against the piston, which can lead to an increase in internal energy. In an adiabatic expansion, this would generally cause the temperature to increase. Since the problem states that heat is being added (and not lost), the temperature will increase as the piston is pulled out dramatically, because work is done by the gas as it expands.

Thus, the correct answer is (b), as pulling out the piston dramatically increases the temperature due to the work done by the gas.

The efficiency of a Carnot engine is given by: where and . The efficiency is: The heat absorbed from the reservoir is , which can be converted to joules: The work done by the engine is: Hence, the correct answer is (b).

For an ideal monoatomic gas, the relationship between heat supplied, work done, and the temperature change is given by:

where is the heat supplied, is the work done, and is the change in internal energy. In a thermodynamic process, the change in internal energy is related to the temperature change. For an ideal monoatomic gas, the change in internal energy is given by , where is the number of moles of the gas, is the ideal gas constant, and is the temperature change.

At constant volume, the work done is zero because there is no change in volume. Therefore, the heat supplied equals the change in internal energy in this case, so .

For constant pressure, the heat supplied is the sum of the work done and the change in internal energy. The work done at constant pressure is , and the heat supplied is related to this by the equation . For a monoatomic ideal gas at constant pressure, the total heat supplied is , which is a well-known relationship.

Hence, the correct answer is (d).