a) The specific heat of ice at constant volume is 833 J/kg K at –180 °C, 1640 J/

Question

a) The specific heat of ice at constant volume is 833 J/kg K at –180 °C, 1640 J/kg K at –60 °C, and 2060 J/kg K at –5.0 °C. Calculate CV, the molar heat capacity at constant volume for ice, at each of these temperatures. The molar mass of H2O is 18.0 g/mol.

b) Why do you suppose the value of CV increases with increasing temperature?

c) Do the values that you calculate approach the value of 3R (given in the rule of Dulong and Petit) as the temperature increases? Speculate about why this should be so.

Explanation / Answer

a) the molar heat capacity at constant volume at -180 degrees = 833 * 18/1000

                                                                                                       = 14.994 J/mol .K

     the molar heat capacity at constant volume at -60 degrees = 1640 * 18/1000

                                                                                                       = 29.52 J/mol .K

    the molar heat capacity at constant volume at -5 degrees = 2060 * 18/1000

                                                                                                       = 37.08 J/mol.K

b)   Three things that contribute to a molecule's average energy are: translation, rotation, and vibration.

     When a substance is cold, collisions with neighboring molecules does not provide enough energy to get out of the ground state for rotation or vibration, so the average energy only increases from translation .

As the substance heats up, the average temperature of the molecules increases, so when they collide, they are more likely to impart enough energy to allow rotation and vibration to occur as the energy jumps to a higher state.

So, the value of CV increases with increasing temperature .

c)    Yes,   values that calculated above approach the value of 3R as the temperature increases .

      This is because , Dulong–Petit law offers fairly good prediction for the specific heat capacity of many elementary solids with relatively simple crystal structure at high temperatures.

   The heat capacity of solids approaches a maximum of 3R per mole of atoms, due to the fact that full vibrational-mode degrees of freedom amount to 3 degrees of freedom per atom each corresponding to a quadratic kinetic energy term and a quadratic potential energy term .

The average of each quadratic term is 12kT, or 12RT per mole . Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity.

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