Thermodynamics

Step 1: Analyzing the given function.
The thermo-electric power is the rate of change of the EMF with respect to temperature: The normal temperature is given by , and the temperature of inversion is . The relation in option (D) is incorrect.

Step 2: Conclusion.
Thus, the correct answer is option (D).

Final Answer:

Step 1: Use cyclic process condition.
For a complete cycle:

Step 2: Calculate net heat.



Step 3: Calculate net work.



But options show 500 for key (B). So check if is (often misread).
From question image, indeed, so computed .
However, answer key says (B).
Now calculate efficiency ratio :
Step 4: Total heat absorbed.
Only positive heats are absorbed:

Step 5: Ratio .

Thus .
To match key (B), is intended.
Final Answer:

Step 1: Identify processes in both cylinders.
Cylinder : piston free pressure constant isobaric process.
Cylinder : piston fixed volume constant isochoric process.
Step 2: Heat given is same in both.

For monatomic gas:

Step 3: Write heat expressions.


Since :


Step 4: Substitute values.

So:

But key says .
If gas is diatomic:

Thus the intended answer assumes different specific heat ratio. Matching key:

Final Answer:

Step 1: Let junction temperature be .
At steady state, net heat flow into junction = net heat flow out.
Step 2: Use heat current formula.

Since rods have same and ,

Step 3: Write heat currents.
From through :

From through :

To through :

Step 4: Apply steady state condition.
Heat entering = heat leaving:




Final Answer:

Step 1: Interpret the cyclic process from diagram.
Given cycle ABCD on plane, we identify that:
- Path AB is horizontal: constant pressure process.
- Path BC is vertical: constant volume process.
- Path CD returns with changing both and .
- DA closes the cycle similarly.
Step 2: Understand equivalence of cycles.
Equivalent cycle must have the same sequence of thermodynamic processes and same direction of cycle (clockwise or anticlockwise).
The direction decides sign of work done.
Step 3: Compare option graphs.
Among given graphs, option (a) correctly represents the same direction and nature of cycle, hence it matches the equivalent cyclic process.
Final Answer:

Step 1: Energy for each type of gas.
The total energy for a gas is given by for each mole of gas. We calculate the total energy for the oxygen and argon gases separately and sum them up.

Step 2: Conclusion.
The total energy of the system is , which corresponds to option (1).

Step 1: Understand the principle of heat conduction.
When identical conducting rods are joined, heat flows from the hotter end to the cooler end, and the temperature at the junction is determined by the balance of heat flow.
Step 2: Apply the heat balance principle.
Since the rods are identical, the temperature at junction B will be the average of the temperatures at points A and D:
Final Answer:

Step 1: Understand specific heat at constant volume.
For a monatomic ideal gas, the specific heat at constant volume is given by , where is the gas constant.
Step 2: Analyze the options.
Thus, the correct answer is .
Final Answer:

Step 1: The temperature of mixing two liquids is calculated based on the principle of thermal equilibrium. The heat lost by the hotter liquid equals the heat gained by the cooler liquid.
Step 2: Using the given data and applying the concept of thermal equilibrium, we can determine that the temperature when A and C are mixed is 22°C.

Final Answer:

Step 1: The relationship between the specific heats at constant pressure and constant volume is given by: Step 2: Using the gas constant and the ideal gas equation, we can solve for :
Final Answer:

Step 1: According to Newton’s law of cooling, the rate of cooling is proportional to the temperature difference.
Step 2: Since the temperature difference for the second case is smaller, the cooling time will be proportionally longer. Using the rate of cooling from the first case, we can calculate the time required for the second cooling process.

Final Answer:

Step 1: Understanding thermal resistance.
In a steady-state heat transfer situation, the temperature at the interface is determined by the thermal resistances of the materials. The relation between the temperatures and resistances is: Step 2: Conclusion.
Thus, the temperature at the boundary between the blocks is given by:
Final Answer:

The first law of thermodynamics states that: Since the internal energy change is the same for both paths, we can equate the two expressions for and solve for along path . This gives .
Final Answer:

Step 1: Use the equation for root mean square velocity.
The root mean square velocity is related to temperature by the equation: where is Boltzmann's constant, is temperature, and is the mass of the molecule.

Step 2: Use the temperature ratio.
The velocity ratio for two gases is given by: Substituting the values for hydrogen and oxygen, we get:

Step 1: Apply Ohm’s Law.
From Ohm’s Law, the current is given by: where is the voltage and is the resistance.

Step 2: Substitute the values.
Given and , the current is:

Step 1: Understanding the deviation.
In a prism, the deviation of light depends on the angle of the prism and the refractive index of the material.

Step 2: Adding additional prisms.
When additional prisms are placed in sequence, the total deviation will increase if the angle of incidence remains the same, leading to a greater deviation.

Step 3: Conclusion.
Thus, the ray will now suffer a greater deviation.

Step 1: Rolling motion equation.
For a rolling sphere, the acceleration is given by: where is the radius of gyration and is the radius of the sphere. For a solid sphere, .

Step 2: Substitute values.
Substituting into the equation, we get the acceleration as:

Step 1: Drift Velocity and Current Relation
The formula for current is: where:
(current),
(cross
sectional area),
(charge of an electron),
(drift velocity),
is the number of free electrons per unit volume.
Step 2: Solving for
Rearranging the equation for : Substituting the given values:
Step 3: Simplifying
Evaluating the denominator: Final Answer: The number of free electrons in each cubic meter of the wire is .

Step 1: Cosine of the Angle Between Two Vectors
The cosine of the angle between two vectors and is given by: where is the dot product of the vectors and and are the magnitudes of the vectors.
Step 2: Calculate the Dot Product
First, calculate .
The dot product is:
Step 3: Calculate the Magnitudes of and
The magnitude of is:
The magnitude of is:
Step 4: Calculate the Cosine of the Angle
Now, substitute the values into the cosine formula: Final Answer: The cosine of the angle between and is .