Alen Ribic - builder, tech entrepreneur, math & physics major

Einstein Solid Heat Capacity on the HP-15C

by Alen Ribic on November 23, 2025

If you own an HP-15C, one of the greatest pocket scientific calculators ever built, you’re already holding a small numerical laboratory. In this post we’ll turn it into a statistical-mechanics engine, capable of computing:

the Einstein solid heat capacity \(C_V(T)\)
the change in internal energy \(\Delta U\)
the change in entropy \(\Delta S\)

…all from first principles, using the 15C’s built-in programmable functions and definite integral capability.

1. What Is the Einstein Solid Model?

Einstein proposed the first successful quantum model of a crystalline solid. He modeled each atom as an independent 3D harmonic oscillator with quantized energy:

\[ E_n = \left(n + \frac12\right)\hbar \omega_E. \]

This resolved the failure of classical physics, which predicted a constant heat capacity \(C_V = 3R\) for all solids.

Experimentally, solids have:

low heat capacity at low temperature (quantum regime)
saturation to \(3R\) at high temperature (Dulong–Petit limit)

The Einstein model captures this temperature dependence using a characteristic temperature.

2. The Einstein Temperature and the Ratio \(x = \Theta_E/T\)

The Einstein temperature is defined as:

\[ \Theta_E = \frac{\hbar \omega_E}{k_B}. \]

This expresses the oscillator quantum \(\hbar\omega_E\) on a temperature scale.

The key dimensionless parameter of the model is:

\[ x = \frac{\Theta_E}{T}. \]

Why? Because

\[ x = \beta \hbar\omega_E = \frac{\hbar\omega_E}{k_B T} = \frac{\Theta_E}{T}, \]

which is the ratio of:

quantum vibrational energy \(\hbar\omega_E\) to
thermal energy \(k_B T\).

Interpretation:

If \(x \gg 1\) (\(T \ll \Theta_E\)): vibrations are frozen → \(C_V \to 0\)
If \(x \ll 1\) (\(T \gg \Theta_E\)): oscillators behave classically → \(C_V \to 3R\)

The entire temperature dependence of the Einstein model is controlled by \(x\).

3. Einstein Heat Capacity Formula

The molar heat capacity is:

\[ \frac{C_V}{R} = 3\frac{x^2 e^x}{(e^x - 1)^2}, \qquad x = \frac{\Theta_E}{T}. \]

This is the expression we will program into the HP-15C.

4. Programming the HP-15C: Einstein Heat Capacity \(C_V(T)/R\)

We’ll write a program at LBL A that computes \(C_V/R\) given:

\(\Theta_E\) stored in R0
Temperature \(T\) in X

Store the Einstein temperature

Example: \(\Theta_E = 230\ \text{K}\)

230 STO 0

Program: LBL A (computes \(C_V/R\))

Enter program mode: g P/R

Then enter:

f LBL A
1/x
RCL 0
×
STO 1
RCL 1
e^x
STO 2
1
-
ENTER
×
STO 3
RCL 1
ENTER
×
RCL 2
×
3
×
RCL 3
÷
g RTN

Using it

Compute \(C_V/R\) at \(T = 210\):

210
GSB A

Result for \(\Theta_E = 230\ \text{K}\):

2.717278

5. Compute Internal Energy Change \(\Delta U\) Using the Integrator

Thermodynamics gives:

\[ \Delta U = \int_{T_1}^{T_2} C_V(T), dT. \]

Since LBL A returns \(C_V/R\), the 15C computes:

\[ I = \int_{T_1}^{T_2} \frac{C_V}{R},dT = \frac{\Delta U}{R}. \]

Multiply by \(R \approx 8.314\) J/(mol·K) to obtain \(\Delta U\).

Wrapper function for integration (LBL 0)

Append:

f LBL 0
GSB A
g RTN

Example: \(\Delta U\) between 100 K and 300 K

100 ENTER 300
f ∫xy 0
8.314 ×

Expected (for \(\Theta_E = 230\) K): \(\Delta U \approx 4.34\times 10^3\ \text{J/mol}\)

6. Compute Entropy Change \(\Delta S\)

Entropy from heat capacity:

\[ \Delta S = \int_{T_1}^{T_2} \frac{C_V(T)}{T}, dT. \]

Since LBL A returns \(C_V/R\), we need to compute

\[ \frac{1}{T}\frac{C_V}{R}. \]

Wrapper function (LBL 1)

Append:

f LBL 1
STO 9
GSB A
RCL 9
÷
g RTN

This returns \((C_V/R)/T\).

Example: \(\Delta S\) between 100 K and 300 K

100 ENTER 300
f ∫xy 1
8.314 ×

Expected value: \(\Delta S \approx 23\ \text{J/(mol·K)}\)

7. What You Just Built

Your HP-15C can now compute:

Einstein heat capacity \(C_V(T)\)
Internal energy change \(\Delta U = \int C_V,dT\)
Entropy change \(\Delta S = \int C_V/T,dT\)

All directly from statistical mechanics and all on a pocket calculator that has stood the test of time.