A gas can be taken from A to B via two different paths ACB and ADB.
When path ACB is used, 60J of heat flows into the system and 30J of work is done by the system. If path ADB is used, work done by the system is 10J. The heat flow into the system in path ADB is}
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This type of problem is a direct application of the fact that internal energy (U) is a state function. The key is to calculate \(\Delta\)U using the path for which complete information is given, and then use that value of \(\Delta\)U to find the missing variable for the other path.
Step 1: Understanding the First Law of Thermodynamics and State Functions.
The first law of thermodynamics states that the change in internal energy (\(\Delta\)U) of a system is equal to the heat (q) added to the system minus the work (w) done by the system.
\[ \Delta U = q - w \]
(Note: The sign convention w = work done *by* the system is used here).
Internal energy (U) is a state function. This means that the change in internal energy (\(\Delta\)U) between two states (A and B) is independent of the path taken. Therefore, \(\Delta U_{ACB} = \Delta U_{ADB}\). Step 2: Calculate the change in internal energy for path ACB.
For path ACB:
Heat flow into the system, q\(_{ACB}\) = +60 J
Work done by the system, w\(_{ACB}\) = +30 J
Using the first law:
\[ \Delta U_{ACB} = q_{ACB} - w_{ACB} = 60 \text{ J} - 30 \text{ J} = 30 \text{ J} \]
Step 3: Use the property of state functions to find the heat flow for path ADB.
Since internal energy is a state function, the change in internal energy from A to B is the same for both paths.
\[ \Delta U_{ADB} = \Delta U_{ACB} = 30 \text{ J} \]
For path ADB:
Work done by the system, w\(_{ADB}\) = +10 J
Heat flow, q\(_{ADB}\) = ?
Using the first law for path ADB:
\[ \Delta U_{ADB} = q_{ADB} - w_{ADB} \]
\[ 30 \text{ J} = q_{ADB} - 10 \text{ J} \]
\[ q_{ADB} = 30 \text{ J} + 10 \text{ J} = 40 \text{ J} \]
Step 4: Final Answer.
The heat flow into the system in path ADB is 40 J.
