Physics · Kinetic Theory · NEET
An atom in a monatomic gas only moves (translates) in 3 directions, giving 3 kinetic terms and energy (3/2)kBT. But an atom in a solid is fixed to a lattice site and can only vibrate. A vibration in one direction has BOTH kinetic energy and potential energy, so it carries 2 x (1/2)kBT = kBT. In 3 directions that is 3kBT per atom. For one mole (N_A atoms), U = 3kBT x N_A = 3RT. So C = dU/dT = 3R, which is double the (3/2)R of a monatomic gas.
Step 1: Model each atom in the solid as a 3D harmonic oscillator. Step 2: By equipartition, each vibration mode (1 dimension) has average energy kBT (half kinetic, half potential). Step 3: Each atom vibrates in 3 dimensions, so average energy per atom = 3kBT. Step 4: For one mole, U = 3kBT x N_A = 3RT, using kB x N_A = R. Step 5: A solid barely expands, so dV is about 0, meaning delta-Q = delta-U at constant pressure. Step 6: C = delta-Q/delta-T = delta-U/delta-T = 3R, which is about 3 x 8.31 = 24.9 J/mol/K.
For a gas, Cp - Cv = R because a gas expands a lot when heated and does work. A solid heated at constant pressure hardly changes volume (delta-V is nearly zero), so almost no work P delta-V is done. That means delta-Q = delta-U whether pressure or volume is kept fixed. So for solids Cp is approximately equal to Cv, and both are close to 3R. This is why we usually quote a single value 3R for a solid.
NCERT notes carbon as an exception. Its measured molar specific heat at room temperature is much less than 3R. The reason is that light, tightly-bonded atoms like carbon have very high vibration frequencies, so their vibration modes are not fully 'switched on' at ordinary temperatures. Equipartition assumes energy is shared equally among all modes, but this needs high enough temperature. Carbon reaches 3R only at much higher temperatures.
The 3R value assumes the law of equipartition holds, which is a classical result valid only when the temperature is high enough for all vibration modes to be active. At low temperature the vibrational energy levels are quantised and the higher modes 'freeze out', so they store less energy. The specific heat then drops below 3R and approaches zero as temperature goes to 0 K. This quantum behaviour is why NCERT says the agreement breaks down at low temperatures.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
The molar specific heat capacity of almost all solid elements is nearly constant and equal to 3R, about 25 J per mole per kelvin, at ordinary temperatures.
With R = 8.31 J/mol/K, 3R = 3 x 8.31 = 24.9 J/mol/K, which is close to 25 J/mol/K.
It is the molar specific heat capacity C, measured in J per mole per kelvin. To get the ordinary specific heat s in J per kg per kelvin, divide 3R by the molar mass M of the solid.
It applies well to most solid elements at ordinary temperatures. Exceptions are light, strongly-bonded solids like carbon, boron and silicon, and all solids fail the law at low temperatures where the value drops below 3R.
A vibrating atom stores energy in two forms: kinetic and potential. Equipartition gives (1/2)kBT to each, so one vibration direction carries (1/2)kBT + (1/2)kBT = kBT. Three directions give 3kBT per atom.
