An external presseure P is applied on a cube at 273K hence it compresses equally from all sides `alpha` is coefficient of linear expansion & K is bulk modulus of material. To bring cube to its original size by heating the temperature rise must be

Options

A. P/3`alpha`K
B. P/`alpha`K
C. P/2`alpha`K
D. P/4`alpha`K

Answer

To solve the problem, we need to determine the temperature rise required to return a cube to its original size after it has been compressed by an external pressure \( P \). We will use the concepts of bulk modulus and linear expansion.

### Step-by-Step Solution:

1. **Understanding Bulk Modulus**:
The bulk modulus \( K \) is defined as:
\[ K = -\frac{\Delta P}{\frac{\Delta V}{V}} \]
where \( \Delta P \) is the change in pressure, \( \Delta V \) is the change in volume, and \( V \) is the original volume.

2. **Relating Pressure and Volume Change**:
Rearranging the formula gives:
\[ \Delta V = -\frac{\Delta P \cdot V}{K} \]
Here, \( \Delta P = P \) (the applied pressure).

3. **Volume Change in Terms of Linear Expansion**:
The change in volume \( \Delta V \) can also be expressed in terms of the coefficient of linear expansion \( \alpha \) and the change in temperature \( \Delta \theta \):
\[ \Delta V = \gamma \cdot V \cdot \Delta \theta \]
where \( \gamma = 3\alpha \) for a cube (since volume is proportional to the cube of the linear dimensions).

4. **Setting the Equations Equal**:
From the two expressions for \( \Delta V \):
\[ -\frac{P \cdot V}{K} = 3\alpha \cdot V \cdot \Delta \theta \]
We can cancel \( V \) from both sides (assuming \( V \neq 0 \)):
\[ -\frac{P}{K} = 3\alpha \Delta \theta \]

5. **Solving for Temperature Change**:
Rearranging the equation to solve for \( \Delta \theta \):
\[ \Delta \theta = -\frac{P}{3\alpha K} \]
Since we are interested in the magnitude of the temperature rise, we can drop the negative sign:
\[ \Delta \theta = \frac{P}{3\alpha K} \]

### Final Answer:
The temperature rise required to bring the cube back to its original size is:
\[ \Delta \theta = \frac{P}{3\alpha K} \]