Chapter 3, Problem 3.1CP

Interpretation Introduction

Interpretation:

The reason for the heat capacity C_p, _mto be less than C_v, _mfor H₂O(l) which is nearly 4⁰C needs to be determined.

Concept Introduction:

If the system experiences change from one thermodynamic phase to another due to change in properties such as temperature, volume, pressure, etc., then the system is said to have a thermodynamic process. Thermal capacity or heat capacityis the physical property of the matter.

Answer to Problem 3.1CP

Because of the change in the phase of water from solid to liquid at 4⁰C.

Explanation of Solution

Generally, Cp, _m is bigger than C_v, _mand their difference is shown below:

  ${\text{Cp,}}_{\text{m}}{\text{- C}}_{\text{v}}{\text{,}}_{\text{m}}{\text{= TV}}_{\text{m}}\frac{{\beta }^{\text{2}}}{\text{k}}\text{…}\mathrm{..}\left(\text{1}\right)$

Here,

Cp, _m= at constant pressure, molar heat capacity

  $\beta$ = thermal expansion

C_v, _m = at constant volume, molar heat capacity

V_m = molar volume

T = temperature

k = isothermal compressibility

The C_p, m >C_v, m if the right side of the equation (1) is positive. The thermal compressibility and thermal expansion are positive if there is no change in the phase of the system.

But in case of H₂O (l) it is seen that C_p, m is smaller than C_v, _m. but in case of water at 4⁰C water is sometime in solid form because the melting point of ice is equal to the freezing point of water that is 0⁰C. As here is a change in phase of water from solid to liquid at 4⁰C the right side of the equation (1) becomes negative. The equation (1) can be rewritten as below:

C_p, m - C_v, m = -ve

C_p, m > C_v, m

Therefore, the heat capacity at constant volume is greater than the heat capacity at constant pressure of water at 4⁰C.