<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/55822734-cfa1-4e4e-b3dc-c5f6325ab74e/Heat_capacity.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/55822734-cfa1-4e4e-b3dc-c5f6325ab74e/Heat_capacity.png" width="40px" /> Heat capacity: how much temperature $\Delta T\to0$ raised after absorbing $\Delta Q$ heat

$$ C=\frac{\text d Q}{\text dT} $$

it can take units $\rm J \, K^{-1} \, kg^{-1}$ or $\rm J \, K^{-1} \, Mol^{-1}$

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🗒️ Notes:

• Heat capacity with $V$ constant $\sim$ to with $P$ constant ie $C_V\approx C_P$ to a few $\%$ for solids

• Dulong and petits Law: $50\%$ of solid monoatomic elements have $C\sim 25 \,\rm JK^{-1}mol^{-1}\approx 3R$

  Universal gas constant $R=k_B N_A\approx 8.31 \rm J K^{-1} mol^{-1}$, Boltzmann constant $\times$ Avogadro $\rm Nb$

• Temperature dependence of heat capacity is very similar for different metals

Classical model

💼 Case: Consider $3$ atoms in a $1D$ chain, there is a restoring force returning it to its equilibrium here we’re imagining springs

🧽 Assume: only nearest neighbour interactions and the spring constant $K$ is the restoring force in all $xyz$ directions

• In $3\rm D$, the total energy $u$ for an atom of mass $m$ is

  $$ u=\frac 12 mv^2 +\frac 12 K|\vec r -\vec r_0 |^2 $$

  where $\vec r_0$ is the equilibrium position.

<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/54b6bc6c-6bf9-408f-b94f-ee0207ba84a9/Equipartition_principle.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/54b6bc6c-6bf9-408f-b94f-ee0207ba84a9/Equipartition_principle.png" width="40px" /> Equipartition principle: The average energy in each accessible degree of freedom of a system in thermal equilibrium is $k_B T/2$

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• The total internal energy of one mole with $p$ atoms per the number of atoms per unit basis is

  $$ U=U_0+pN_A\times 6\times \frac{k_BT}{2} =U_0 +3 pRT $$

  where $U_0$ is the cohesive energy at $T=0$

🗒️ Notes:

• $C=\text dU/\text dT=3R$
• $6$ degrees of freedom: $(p_x,p_y,p_z)$ kinetic energy and $(x,y,z)$ vibrational potential
• Here $C_V$ is independent of temperature which is not true at low temp, thus we need QM

Quantum Model: the Einstein Model

<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/fb173472-056a-49f0-9d7e-1750909b8e12/Einstein_model.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/fb173472-056a-49f0-9d7e-1750909b8e12/Einstein_model.png" width="40px" /> Einstein model: The quantum model assumes the atoms are vibrating independently as a QM SHO in each direction $x,y,z$

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• The allowed energies of vibration are given by

  $$ E=\left ( n_x+n_y+n_z+\frac 32 \right )\hbar \omega \qquad \omega =\sqrt{\frac Km} \qquad n_x,n_y,n_z\in \N $$

  We expect a reduction in accessibility of quantum states when $k_B T<\hbar \omega$ when:

  $$ T<\Theta_E=\frac{\hbar }{k_B}\sqrt{\frac Km} $$

  where $\Theta_E$ is the Einstein temperature and is the limit where the first excited state is occupied

🗒️ Note: we use a statistical approach valid for large numbers to explain macroscopic properties