Question

It is possible to make crystalline solids that are only one layer of atoms thick. Such “two-dimensional” crystals can be created by depositing atoms on a very flat surface. (a) If the atoms in such a two-dimensional crystal can move only within the plane of the crystal, what will be its molar heat capacity near room temperature? Give your answer as a multiple of R and in (b) At very low temperatures, will the molar heat capacity of a two-dimensional crystal be greater than, less than, or equal to the result you found in part (a)? Explain why.

Short answer

(a) Molar heat capacity of two dimensional crystal move within the plane of the crystal is

(b)The molar heat capacity of a two-dimensional crystal be less than result found in part (a)

Step-by-step solution

Determine Molar heat capacity of solid

The molar heat capacity of achemical substance is the amount ofenergy that must be added, in the form ofheat, to onemole of the substance in order to cause an increase of one unit in itstemperature.

The molar heat capacity of the three dimensional solid near the room temperature is

where R is the gas constant

Determine molar heat capacity of two dimensional crystal move within the plane of the crystal

Solve for molar heat capacity in two dimension

Therefore ,molar heat capacity of two dimensional crystal move within the plane of the crystal is .

Determine molar heat capacity of two dimensional crystal at very low temperature 

The molar heat capacity of a two-dimensional crystal be less than result found in part (a)

Because at low temperature heat capacities decreases and for the same reason the vibrational degree of the molecule are frozen out at the low temperature

Therefore ,The molar heat capacity of a two-dimensional crystal be less than result found in part (a)