question_answer

What is meant by molar specific heat of a gas? The molar specific heat of hydrogen \[{{\mathbf{H}}_{\mathbf{2}}}\] is about \[\frac{\mathbf{5}}{\mathbf{2}}\mathbf{R}\]in the temperature range of about \[\mathbf{250}\text{ }\mathbf{K}\] to\[\mathbf{750}\text{ }\mathbf{K}\]. At lower temperatures, molar specific heat of hydrogen decreases to the value typical of monoatomic gases: \[\frac{\mathbf{3}}{\mathbf{2}}\mathbf{R}.\] At higher temperatures, it tends to the value \[\frac{7}{2}\] \[R\]. What do you think is happening?      

Answer:

                The molar specific heat of a gas is defined as the amount of heat required to raise the temperature of 1 mole of a gas through\[\text{1}{}^\circ \text{C}\]. The molar specific heat of a gas at constant volume is given by\[{{C}_{V}}=\frac{f}{2}R,\], where \[f\] is degrees of freedom of the gas. In the range of about 250 to\[\text{75}0\text{ K,}\], a diatomic gas such as \[{{H}_{2}}\] gas possesses 5 degrees of freedom (3 corresponding to translational and 2 corresponding to rotational modes of motion Of the gas). Hence \[{{C}_{V}}=\frac{5}{2}R\]            (between\[\text{25}0\text{ to 75}0\text{ K}\]) But at lower temperatures, the rotational motion is not excited and hydrogen gas molecule possesses only 3 degrees of freedom corresponding to translational motion. Hence at lower temperatures,                                                                                                                                         \[{{C}_{V}}=\frac{3}{2}R\] At higher temperatures, hydrogen gas   molecule has 2 additional degrees of freedom corresponding to vibrational mode of the motiod, so that total degrees of freedom become \[f=5+2=7.\] Hence at higher temperatures, \[{{C}_{V}}=\frac{7}{2}R.\]