If an insulated container containing liquid is stirred with a paddle to increase the temperature, which of the following is true?

Options

A. `Delta U = W ne 0, q = 0`
B. `Delta U = W = 0, q ne 0`
C. `Delta U = 0, W = q ne 0`
D. `W = 0, Delta U = q ne 0`

Answer

To solve the question regarding the insulated container containing liquid that is stirred with a paddle to increase the temperature, we can follow these steps:

### Step 1: Understand the System
We have an insulated container (adiabatic system) with a liquid inside. Stirring the liquid with a paddle increases its temperature.

**Hint:** Remember that an insulated container does not exchange heat with its surroundings.

### Step 2: Apply the First Law of Thermodynamics
The First Law of Thermodynamics states:
\[ Q = \Delta U + W \]
Where:
- \( Q \) = heat added to the system
- \( \Delta U \) = change in internal energy
- \( W \) = work done on the system

Since the container is insulated, there is no heat exchange with the surroundings, which means:
\[ Q = 0 \]

**Hint:** In an adiabatic process, the heat transfer \( Q \) is zero.

### Step 3: Simplify the Equation
Substituting \( Q = 0 \) into the First Law equation gives:
\[ 0 = \Delta U + W \]
This can be rearranged to:
\[ \Delta U = -W \]

**Hint:** The work done on the system will affect the internal energy.

### Step 4: Analyze the Work Done
When the paddle stirs the liquid, work is done on the liquid. This work increases the internal energy of the liquid, leading to an increase in temperature. Therefore, \( W \) is positive, and consequently, \( \Delta U \) must also be positive (since \( \Delta U = -W \)).

**Hint:** If work is done on the system, the internal energy increases.

### Step 5: Relate Internal Energy to Temperature Change
The change in internal energy (\( \Delta U \)) can be related to the change in temperature (\( \Delta T \)) using the formula:
\[ \Delta U = n C_v \Delta T \]
Where \( n \) is the number of moles and \( C_v \) is the molar heat capacity at constant volume. Since the temperature is increasing (\( \Delta T > 0 \)), it follows that \( \Delta U > 0 \).

**Hint:** The internal energy change is directly related to the temperature change in the system.

### Step 6: Conclusion
From the analysis, we conclude that:
- \( \Delta U \) is not equal to zero (it is positive).
- \( Q = 0 \) (no heat exchange).
- The work done \( W \) is equal to the negative change in internal energy, which is also not zero.

Thus, the correct statement is that \( \Delta U \) is not equal to zero, and since \( Q = 0 \), the first option is true.

**Final Answer:** The correct option is **A: \( \Delta U \) is equal to \( W \) when \( Q = 0 \)**.