Thermodynamics

Given: The process involves an ideal gas taken from initial state 1 to final state 4 through the paths 1 → 2 → 3 → 4. The figure shows isotherms AB, CD, and EF. We are asked to compare the most probable speed at different points of the process.

Key Concept: The most probable speed of molecules is given by the formula: where: - is the Boltzmann constant, - is the temperature, and - is the mass of a gas molecule. For an isothermal process, the temperature remains constant. Hence, the most probable speed is directly proportional to the temperature. The temperature of the gas is higher at points where the volume is smaller (since for isothermal processes, ).

Analysis: - In the figure, we see that the isotherms (AB, CD, EF) represent constant temperature curves. - For the ideal gas, the temperature is highest at the smallest volume and decreases as the volume increases. From the isotherms: - at point 3 and point 4 will be greater than at points 2 and 1 because the temperatures at points 3 and 4 are higher (due to the smaller volumes at those points). - at point 2 will be greater than at point 1 because point 2 corresponds to a higher temperature than point 1. Therefore, the most probable speed follows this order:

Final Answer: .

Given: One mole of a diatomic ideal gas undergoes a process shown in the P-V diagram. We are asked to calculate the total heat given to the gas. The value of is also provided.

Approach: The process on the P-V diagram involves a change in pressure and volume. The total heat given to the gas can be determined using the first law of thermodynamics: Where: - is the change in internal energy, - is the work done by the gas.

Internal Energy Change: For a diatomic ideal gas, the change in internal energy is given by: For one mole ( ) of a diatomic ideal gas, the specific heat at constant volume is:

Work Done: The work done by the gas during an expansion or compression process is given by: From the P-V diagram, we can calculate the work done based on the specific path shown in the diagram.

Conclusion: By calculating both and from the provided P-V diagram, we find that the total heat given to the gas is:

Final Answer: The total heat given to the gas is .

Given: The internal energy is defined as , where is a constant. The process is performed adiabatically, and we are tasked with finding the relationship between pressure and volume during this process.

Approach: For an adiabatic process, the first law of thermodynamics tells us that the change in internal energy is equal to the work done: This equation implies that: From this, we can derive a relationship between pressure and volume for an adiabatic process. Since we know the form of the internal energy , we can proceed to find the specific relationship between and under the condition that the process is adiabatic.

Solution: Rearranging terms and solving the equation for an adiabatic process, we find that the relationship between and is:

Final Answer: The correct relationship is .

We are given the polytropic equation for an ideal monatomic gas: and that the gas expands from volume to volume at a constant pressure , with molar specific heat .

Step 1: Understanding the Process
The heat absorbed during a thermodynamic process can be expressed as: where is the number of moles of the gas, is the molar specific heat at constant volume, and is the change in temperature. The polytropic process relationship gives us a way to relate the temperature change to the volume change. Since , we can write: This equation can be used to determine the temperature change, as , where is the gas constant and is the temperature.

Step 2: Deriving the Heat Absorbed
We know that the total work done in an ideal gas process is given by: The total heat absorbed is related to the work done and the change in internal energy , which for a monatomic ideal gas is . Using these relations, we can express the total heat absorbed during the polytropic expansion from to . The expression for the total heat absorbed turns out to be:

Answer:

To determine the heat absorbed along the path A → B → C, let's break down the solution step by step:

Given Data:

• Work done along the direct path A → C: W_AC = 200 J
• Heat absorbed along the direct path A → C: Q_AC = 280 J
• Work done along the path A → B → C: W_ABC = 80 J

Using the first law of thermodynamics for path A → C:

Since internal energy is a state function, ΔU is the same for both paths A → C and A → B → C.

Now, for path A → B → C:

Conclusion: The heat absorbed along the path A → B → C is 160 J.

The correct option is (C): 160 J