Question:
The pressure applied on a cube from all sides is \(P\). In order to keep its volume constant, the temperature of the cube is to be raised by (The bulk modulus and the coefficient of volume expansion of the material of the cube are \(\beta\) and \(\alpha\) respectively)
The pressure applied on a cube from all sides is \(P\). In order to keep its volume constant, the temperature of the cube is to be raised by (The bulk modulus and the coefficient of volume expansion of the material of the cube are \(\beta\) and \(\alpha\) respectively)
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Remember:
\[
\frac{\Delta V}{V}
=
-\frac{P}{\beta}
\]
for compression under pressure, and
\[
\frac{\Delta V}{V}
=
\alpha\Delta T
\]
for thermal expansion.
For constant volume, set the algebraic sum of the two volume strains equal to zero.
Updated On: Jul 9, 2026
- \(\dfrac{P}{\alpha\beta}\)
- \(\dfrac{P\alpha}{\beta}\)
- \(\dfrac{P\beta}{\alpha}\)
\(\dfrac{\alpha\beta}{P}\)
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The Correct Option is A
Solution and Explanation
Concept:
The change in volume due to pressure is given by
\[
\frac{\Delta V}{V}
=
-\frac{P}{\beta},
\]
where \(\beta\) is the bulk modulus.
The change in volume due to temperature rise is
\[
\frac{\Delta V}{V}
=
\alpha\Delta T,
\]
where \(\alpha\) is the coefficient of volume expansion.
For the volume to remain constant, these two effects must exactly cancel each other.
Step 1: Write the total volumetric strain. \[ \left(\frac{\Delta V}{V}\right)_{\text{total}} = \alpha\Delta T-\frac{P}{\beta}. \]
Step 2: Use the condition of constant volume. Since the volume remains unchanged, \[ \left(\frac{\Delta V}{V}\right)_{\text{total}} = 0. \] Therefore, \[ \alpha\Delta T-\frac{P}{\beta}=0. \]
Step 3: Calculate the required rise in temperature. \[ \alpha\Delta T = \frac{P}{\beta}. \] \[ \Delta T = \frac{P}{\alpha\beta}. \]
Step 4: Write the final answer. \[ \boxed{\Delta T=\frac{P}{\alpha\beta}} \] \[ \boxed{\text{Answer = (A)}} \]
Step 1: Write the total volumetric strain. \[ \left(\frac{\Delta V}{V}\right)_{\text{total}} = \alpha\Delta T-\frac{P}{\beta}. \]
Step 2: Use the condition of constant volume. Since the volume remains unchanged, \[ \left(\frac{\Delta V}{V}\right)_{\text{total}} = 0. \] Therefore, \[ \alpha\Delta T-\frac{P}{\beta}=0. \]
Step 3: Calculate the required rise in temperature. \[ \alpha\Delta T = \frac{P}{\beta}. \] \[ \Delta T = \frac{P}{\alpha\beta}. \]
Step 4: Write the final answer. \[ \boxed{\Delta T=\frac{P}{\alpha\beta}} \] \[ \boxed{\text{Answer = (A)}} \]
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