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Heat capacity, Cp, is the amount of heat required to change the heat content of 1 mole of material by exactly 1掳C.

Heat is a form of energy, often called thermal energy. Energy can be transformed from one form to another (a blender transforms electrical energy into mechanical energy), but it cannot be created nor destroyed; rather, energy is conserved.

We are given a water vapor with molar mass${\mathrm{M}}_{\mathrm{w}}=18\mathrm{g}/\mathrm{mol}$ at constant volume- The vapour molecules have threetransitional and three rotational degrees of freedom.

Solution

a .-

At constant volume means we are about to use molar heat capacity${\mathrm{C}}_{\mathrm{V}}$and before calculating the required let make adifference betWeen ${\mathrm{C}}_{\mathrm{V}}$and c- The uppercase ${\mathrm{C}}_{\mathrm{V}}$is the molar heat capacity where its unit J /mol - K and represent theheat capacity absorbed by mole range. While the lOWercase c is the speci?c heat capacity where its unit J / kg - K andrepresent the heat capacity absorbed by mass range (gram or kilogram) and it is the required to be calculated in part (a).

As given there are 6 degrees of freedom and the molar heat capaczty${\mathrm{C}}_{\mathrm{V}}=\frac{\mathrm{R}}{2}$for each degree (revxew section 18-4 In

textbook), then for the 6 degrees the total molar heat capacity would be

$$\begin{gathered} R \\ {C}_V=6脳\frac{R}{2} \\ =3R \\ =3脳8.314J/mol路K \\ =24.9J/mol路K \end{gathered}$$

Therefore The molar heat capacity is equal 24.9 J /mol - K,

calculate the specific heat capacity c by known the $\textit{molar heat capacity and molar mass Ml

$$\begin{gathered} c=\frac{C_V}{M} \\ =\frac{24.9J/mol-K}{18脳{10}^{-3}kg/mol} \\ =1380J/kg.K \end{gathered}$$

Therefore The specific heat capacity c of Water vapour at constant volume with 3 transitional and 3 rotational degrees is1380 J/kg . K

b -

Now compare the result in part (a) with the given value in part (b) we would ?nd that:

For actual vapor water the specific heat capacity c = 2000J/kgK and it is more than the

2000 - 1380 = 620 J/kgK

This increase is due to the Vibrational motion, In part (a) there is no contribution to the vibrational motion so speci?c

heat capacity is lOWer. While in part (b) the given value is for the real vapour water which has additional motion (vibrational

motion) of molecules that required more heat to vibrate and therefore higher speci?c heat capacity

Therefore there is Increase by 620 J / kg - K due to vibrational motion of molecules