• Water has a high heat capacity due to strong intermolecular bonding and high-energy vibration modes.ocean image by Yulia Volodina from Fotolia.com
  
  Heat capacity relates to several concepts. Thermal energy and entropy change allow for direct calculation of heat capacity. Degrees of freedom enumerate the number of ways a molecule (or atom) moves in terms of kinetic energy as well as vibration and rotation (vib/rot) modes. For first approximations, substance heat capacity is a constant. On closer inspection, heat capacity depends on starting temperature.
  
  

Thermal Energy

• Heat capacity (H) is ratio of change in thermal energy (dQ) to change in temperature (dT). So H = dQ/dT. Joules (J) quantify thermal energy. Kinetic, vibration, and rotational motion all store thermal energy. Average kinetic energy defines overall temperature. Rotation refers to atoms rotating around chemical bonds like spheres around an axis. Vibration refers to atomic motion that regularly changes bond length and angle---as with a flexible spring.
  
  

Entropy

• Heat capacity can be expressed in terms of entropy change (dS) divided by temperature change. A temperature multiplier is present since dQ = T*dS. The complete heat capacity (H) formula is H = T*(dS/dT). Entropy is a measure of disorder. Units of energy/temperature (Joules/Kelvin) quantify entropy. Highly organized systems like a vase have only one (or relatively few) particle arrangements that maintain an unbroken vase. When that vase is shattered, many disordered configurations are possible. Broken vases have greater entropy, and are therefore more likely.
  
  

Degrees of Freedom--Kinetic

• Degrees of freedom (DoF) indicate the minimum number of variables that completely describe particle motion. Moving in a line gives one degree of freedom for changing position along a line, and another for changing momentum. Momentum incorporates how fast the particle is moving and how heavy it is. For particle linear motion, total degrees of freedom are DoF = 1+1 = 2. In a plane, DoF = 2+2 = 4 (2 position + 2 momentum values). Similarly, 3D movement gives DoF = 3+3=6.
  
  

Degrees of Freedom--vib/rot

• Rotation and vibration modes contribute to degrees of freedom. Assume a molecule is in place with one vibration mode. Another identical molecule is also not moving but has different vibration/rotation modes. Those particles are distinguishable; they may have different degrees of freedom. For multi-atom systems, number of vibration and rotation states increases. For linear molecules with N atoms, there are 3N-5 possible vibration modes. If an N-atom molecule is non-linear, 3N-6 vibration modes exist. One-atom systems such as helium gas do not have vibration modes.
  
  

Temperature Dependence

• Heat capacity depends on temperature. At higher energies, molecules can transition through more vibration/rotation modes. High kinetic energy more readily overcomes non-changing attractive forces between molecules. If intermolecular attraction is not as relevant, less energy will be absorbed before molecules' velocity increases. This is a temperature increase. Interplay between vibration/rotation modes and effect of intermolecular forces determine overall heat capacity.