Which one of the following equations does not correctly represent the first law of thermodynamics for the given processes involving an ideal gas ? (Assume non-expansion work is zero)

Options

A. Cyclic process: `q=-w`
B. Adiabatic process: `DeltaU=-w`
C. Isochoric process: `DeltaU=q`
D. Isothermal process: `q=-w`

Answer

To determine which equation does not correctly represent the first law of thermodynamics for the given processes involving an ideal gas, we will analyze each process using the first law of thermodynamics, which states:

\[ \Delta U = Q + W \]

Where:
- \(\Delta U\) = Change in internal energy
- \(Q\) = Heat added to the system
- \(W\) = Work done on the system

We will consider the following processes: adiabatic, cyclic, isochoric, and isothermal.

### Step 1: Analyze the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. Therefore, \(Q = 0\).

Using the first law of thermodynamics:
\[ \Delta U = Q + W \implies \Delta U = 0 + W \implies \Delta U = W \]

### Step 2: Analyze the Cyclic Process
In a cyclic process, the system returns to its initial state, so the change in internal energy is zero. Therefore, \(\Delta U = 0\).

Using the first law:
\[ \Delta U = Q + W \implies 0 = Q + W \implies Q = -W \]

### Step 3: Analyze the Isochoric Process
In an isochoric process, the volume remains constant, which means no work is done (\(W = 0\)).

Using the first law:
\[ \Delta U = Q + W \implies \Delta U = Q + 0 \implies \Delta U = Q \]

### Step 4: Analyze the Isothermal Process
In an isothermal process, the temperature remains constant, which implies that the change in internal energy is zero for an ideal gas (\(\Delta U = 0\)).

Using the first law:
\[ \Delta U = Q + W \implies 0 = Q + W \implies Q = -W \]

### Summary of Results
1. **Adiabatic Process**: \(\Delta U = W\)
2. **Cyclic Process**: \(Q = -W\)
3. **Isochoric Process**: \(\Delta U = Q\)
4. **Isothermal Process**: \(Q = -W\)

### Step 5: Identify the Incorrect Equation
Now, we compare these results with the given options. The incorrect representation is found in the adiabatic process. The correct equation should be:

\[ \Delta U = W \]

If any option states \(\Delta U = -W\) for the adiabatic process, it is incorrect.

### Conclusion
The equation that does not correctly represent the first law of thermodynamics for the adiabatic process is the one that states \(\Delta U = -W\).

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