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Step-by-Step Explanation
Step 1: Recognize the First Law of Thermodynamics
The first law of thermodynamics states that the change in internal energy, denoted by $ \Delta U $, of a system is equal to the heat added to the system ($q$) plus the work done on the system ($W$):
$ \Delta U = q + W $
Step 2: Analyze Each Process
Adiabatic Process (q = 0)
In an adiabatic process, there is no exchange of heat ($q = 0$), so:
$ \Delta U = W $
However, the option given states $ \Delta U = -w $, which contradicts $ \Delta U = W $. Therefore, the statement
"Adiabatic process: $ \Delta U = -w $" is incorrect.
Cyclic Process ($\Delta U = 0$)
In a cyclic process, the system returns to its initial state, so the change in internal energy is zero:
$ \Delta U = 0 $
From the first law:
$ \Delta U = q + W = 0 \quad \Longrightarrow \quad q = -W $
This matches the given statement for the cyclic process, so it is correct.
Isochoric Process ($V =$ constant)
When volume is constant, there is no expansion or compression work ($W = 0$). Hence,
$ \Delta U = q + W = q + 0 = q $
So "$\Delta U = q$" is correct for an isochoric process.
Isothermal Process ($T =$ constant)
For an ideal gas undergoing an isothermal process, the internal energy does not change because internal energy of an ideal gas depends only on temperature. Thus:
$ \Delta U = 0 $
From the first law:
$ q + W = 0 \quad \Longrightarrow \quad q = -W $
Hence, the statement "Isothermal process: $q = -W$" is correct.
Step 3: Identify the Incorrect Option
The incorrect statement is "Adiabatic process: $ \Delta U = -w$" because, in reality, if $q=0$, then $ \Delta U = W $ (not $-W$) for an adiabatic process.
