Question

At \( 4^\circ C \) for water, \( C_P = C_V \) because:

• A. \( \left( \frac{\partial P}{\partial V} \right)_T = 0 \)
• B. \( \left( \frac{\partial V}{\partial T} \right)_P = 0 \)
• C. \( \left( \frac{\partial P}{\partial T} \right)_V = 0 \)
• D. None of the three

Hint:

Water is most dense at \( 4^\circ C \), which results in no change in volume with temperature at this point.

Answer 1

The Correct Option is B

At \( 4^\circ C \), water exhibits anomalous expansion, meaning that its volume does not change with temperature under constant pressure: \[ \left( \frac{\partial V}{\partial T} \right)_P = 0 \] Since the heat capacities are related by:
\[ C_P - C_V = - T V \left( \frac{\partial P}{\partial T} \right)_V \left( \frac{\partial V}{\partial T} \right)_P \] when \( \left( \frac{\partial V}{\partial T} \right)_P = 0 \), it follows that \( C_P = C_V \).