One mole of an ideal gas is taken through processes AB, BC and CA as shown in the figure.The heat absorbed in process AC is

To calculate the heat absorbed in process AC for one mole of an ideal gas, we need to analyze the cyclic process represented by the paths AB, BC, and CA. Below are the steps involved in determining the heat absorbed in process AC.

1. ### Break Down the Problem

1. Identify the properties of the gas and the parameters of each process (AB, BC, and CA).
2. Determine the state variables (pressure, volume, temperature) at points A, B, and C.
3. Understand the thermodynamic processes involved (isothermal, adiabatic, isochoric, isobaric).

2. ### Relevant Concepts

1. The First Law of Thermodynamics states that: $\Delta U = Q - W$ where $\Delta U$ is the change in internal energy, $Q$ is the heat added to the system, and $W$ is the work done by the system.

2. For an ideal gas, the change in internal energy ($\Delta U$) is related only to the change in temperature: $\Delta U = nC_V\Delta T$ where $n$ is the number of moles, $C_V$ is the molar heat capacity at constant volume, and $\Delta T$ is the change in temperature.

3. The heat for an isothermal process is given by: $Q = nRT \ln \left( \frac{V_f}{V_i} \right)$

3. ### Analysis and Detail

1. If the processes AB, BC, and CA are defined in terms of pressures and volumes, we calculate $Q$ specifically for process AC.
   • If AC is an isothermal process: $Q_{AC} = nRT \ln\left(\frac{V_C}{V_A}\right)$
   • Check the coordinates of points A and C to substitute in the equation.
   • Use given or calculated values of $n$ (1 mole), $R$ (ideal gas constant), and $T$ (temperature).

4. ### Verify and Summarize

1. Confirm that the points and states involved in AC comply with assumptions for ideal gases.
2. Ensure units are consistent throughout the calculation.
3. Summarize any calculations or results in terms of the properties at point A and point C.

Final Answer

The heat absorbed in process AC can be computed using the derived equation based on the specific relationship between initial and final states, ensuring:

$Q_{AC} = nRT \ln\left(\frac{V_C}{V_A}\right)$

Insert known values to solve for $Q_{AC}$ as needed.