Specific Heat and Internal Energy Change

Physics Thermodynamics • First Law of Thermodynamics

Written by STEM Calculators Team Published January 2, 2026 Updated February 24, 2026

3. Specific Heat and Internal Energy Change

Compute ideal-gas energy changes using \(\Delta U = nC_V\Delta T\) and (constant-pressure) \(Q_p = nC_P\Delta T\), with \(C_P=C_V+R\). Choose a model (monatomic/diatomic/polyatomic), or enter \(\gamma=C_P/C_V\) directly.

Inputs support: pi, e, sqrt( ), sin, cos, exp, log. Use * for multiplication. Temperatures are in K.

Model & inputs

n=—, ΔT=—

Compute: Amount \(n\) (mol): Temperature change \(\Delta T\) (K): How to set \(C_V, C_P\): Preset: Degrees of freedom \(f\): \(\gamma\): \(C_V\) (J/mol·K):

Equipartition model (optional): \(C_V=\tfrac{f}{2}R\), so \(C_P=C_V+R\) and \(\gamma=\dfrac{f+2}{f}\). (At higher temperatures, additional modes can increase the effective \(f\).)

Graph controls (interactive)

Progress \(\tau\in[0,1]\): Play speed: Show \(\Delta U\) line: Show \(Q_p\) line:

Drag to pan, wheel to zoom, double-click or “Reset view” to restore. The marker moves along the input \(\Delta T\) as \(\tau\) changes.

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Steps

Enter values and click Solve.

Graph

\(\Delta U(\Delta T)\) and \(Q_p(\Delta T)\) with marker at \(\tau\)

Hover to probe.

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Theory — Specific Heat and Internal Energy Change

For an ideal gas, the internal energy depends only on temperature. This calculator uses constant (effective) heat capacities to compute energy changes for a specified temperature change \(\Delta T\).

1) Heat capacities and \(\gamma\)

Heat capacities per mole:

\[ C_P = C_V + R,\qquad \gamma=\frac{C_P}{C_V} \]

If \(\gamma\) is known, then:

\[ C_V=\frac{R}{\gamma-1},\qquad C_P=\frac{\gamma R}{\gamma-1} \]

2) Internal energy change \(\Delta U\)

For an ideal gas:

\[ \Delta U = nC_V\Delta T \]

This holds regardless of the path (isochoric/isobaric/isothermal, etc.) as long as the gas is ideal and \(C_V\) is treated as constant over the temperature range.

3) Heat added at constant pressure

For a constant-pressure temperature change:

\[ Q_p = nC_P\Delta T \]

Compare to constant volume where \(W=0\) and \(Q_v=\Delta U=nC_V\Delta T\).

4) Degrees of freedom (equipartition preview)

A common model links heat capacity to the effective number of degrees of freedom \(f\):

\[ C_V=\frac{f}{2}R,\qquad C_P=C_V+R,\qquad \gamma=\frac{f+2}{f} \]

Monatomic (typical): \(f=3\Rightarrow C_V=\tfrac{3}{2}R,\ \gamma=\tfrac{5}{3}\)
Diatomic (room temp typical): \(f=5\Rightarrow C_V=\tfrac{5}{2}R,\ \gamma=\tfrac{7}{5}\)
Polyatomic (often): \(f\approx6\Rightarrow C_V=3R,\ \gamma=\tfrac{4}{3}\)

University note: as temperature increases, vibrational modes may become active, increasing the effective \(f\), so \(C_V\) and \(C_P\) can vary with temperature.

5) Worked example (detailed)

Example: \(n=2\) mol diatomic (typical \(f=5\Rightarrow C_V=\tfrac{5}{2}R\)), \(\Delta T=100\) K.

\[ \Delta U=nC_V\Delta T =2\left(\frac{5}{2}R\right)(100) =5R(100) \approx 4.16\times10^3\ \text{J} \]

If additional modes are included (larger effective \(f\)), the value increases accordingly.

Steps

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