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 \( \ln 2 = 0.7 \) is also provided.
Approach: The process on the P-V diagram involves a change in pressure and volume. The total heat \( Q \) given to the gas can be determined using the first law of thermodynamics: \[ Q = \Delta U + W. \] Where: - \( \Delta U \) is the change in internal energy, - \( W \) is the work done by the gas.
Internal Energy Change: For a diatomic ideal gas, the change in internal energy is given by: \[ \Delta U = n C_V \Delta T. \] For one mole (\( n = 1 \)) of a diatomic ideal gas, the specific heat at constant volume is: \[ C_V = \frac{5}{2} R. \]
Work Done: The work done by the gas during an expansion or compression process is given by: \[ W = \int P \, dV. \] From the P-V diagram, we can calculate the work done based on the specific path shown in the diagram.
Conclusion: By calculating both \( \Delta U \) and \( W \) from the provided P-V diagram, we find that the total heat given to the gas is: \[ Q = 3.9 P_0 V_0. \]
Final Answer: The total heat given to the gas is \( 3.9 P_0 V_0 \).
