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 bring a cube back to its original size after it has been compressed by an external pressure \( P \). The relevant concepts include the bulk modulus \( K \) and the coefficient of linear expansion \( \alpha \).

### Step-by-Step Solution:

1. **Understanding Bulk Modulus**:
The bulk modulus \( K \) is defined as the ratio of the change in pressure to the fractional change in volume:
\[ K = -\frac{P}{\frac{\Delta V}{V}} \]
From this, we can express the volumetric strain (change in volume per unit volume) as:
\[ \frac{\Delta V}{V} = -\frac{P}{K} \]

2. **Relating Volumetric Strain to Linear Strain**:
For a cube, the volumetric strain is related to the linear strain by the formula:
\[ \text{Volumetric Strain} = 3 \times \text{Linear Strain} \]
Therefore, we can write:
\[ \frac{\Delta V}{V} = 3 \times \text{Linear Strain} \]
Substituting the expression for volumetric strain from step 1, we have:
\[ -\frac{P}{K} = 3 \times \text{Linear Strain} \]
Thus, the linear strain can be expressed as:
\[ \text{Linear Strain} = -\frac{P}{3K} \]

3. **Using Coefficient of Linear Expansion**:
The linear strain due to temperature change is given by:
\[ \text{Linear Strain} = \alpha \Delta T \]
where \( \Delta T \) is the change in temperature. Setting the two expressions for linear strain equal gives us:
\[ -\frac{P}{3K} = \alpha \Delta T \]

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

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