Pressure–Volume Work — Theory & Guide
In the First Law of Thermodynamics, changes in a system’s internal energy come from heat and work:
\(\mathrm{d}U=\delta q+\delta w\). For purely mechanical (pressure–volume) work with a uniform external pressure,
the infinitesimal work is \(\delta w=-P_{\text{ext}}\ \mathrm{d}V\).
The chemistry sign convention is used here: work done by the system is negative; work done on the system is positive.
Core relations
\[
\delta w \;=\; -\,P_{\text{ext}}\ \mathrm{d}V
\]
\[
w \;=\; -\!\int_{V_i}^{V_f} P_{\text{ext}}\ \mathrm{d}V
\]
\[
\text{If } P_{\text{ext}} \text{ is constant:}\qquad
w \;=\; -\,P_{\text{ext}}\,(V_f - V_i) \;=\; -\,P_{\text{ext}}\,\Delta V
\]
Ideal-gas shortcut using \(\Delta n_{\text{gas}}\)
For (i) ideal gases, (ii) isothermal processes, and (iii) processes where the system pressure closely
matches the (slowly changing) external pressure, one often uses \(P\,\Delta V \approx \Delta n_{\text{gas}}\,R\,T\).
\[
P\,\Delta V \;\approx\; \Delta n_{\text{gas}}\,R\,T
\]
\[
w \;\approx\; -\,\Delta n_{\text{gas}}\,R\,T
\]
Here \(\Delta n_{\text{gas}} = \sum \nu_k\) over gaseous species (products − reactants), \(R=8.314\ \mathrm{J\cdot mol^{-1}\cdot K^{-1}}\),
and \(T\) is absolute temperature in K. This approximation is not used for liquids/solids.
Units & handy conversions
- \(1\ \mathrm{L\cdot atm} = 101.325\ \mathrm{J}\)
- \(1\ \mathrm{bar} = 10^{5}\ \mathrm{Pa}\), \(\;1\ \mathrm{atm} = 101\,325\ \mathrm{Pa}\)
- \(1\ \mathrm{m^{3}} = 10^{3}\ \mathrm{L}\)
Sign interpretation
- Expansion \((\Delta V > 0)\): \(w = -P_{\text{ext}}\Delta V < 0\) → work done by the system.
- Compression \((\Delta V < 0)\): \(w > 0\) → work done on the system.
- Free expansion into vacuum \((P_{\text{ext}}=0)\): \(w=0\).
- Constant volume \((\Delta V=0)\): \(w=0\).
Context with the First Law
\[
\Delta U \;=\; q + w
\]
For an ideal gas at constant temperature, \(\Delta U \approx 0\), so \(q \approx -\,w\).
At constant external pressure, calorimetry often gives \(q_p=\Delta H\), and then \(\Delta U=\Delta H + w\).
The calculator here reports only the mechanical \(pV\) work term \(w\).
When to use each method in the calculator
- \(w = -P_{\text{ext}}\Delta V\): use when you know (or control) \(P_{\text{ext}}\) and the volume change,
especially in irreversible steps at constant external pressure (e.g., piston with fixed weight).
- \(w \approx -\Delta n_{\text{gas}}RT\): use for gas-phase, near-isothermal processes when
\(\Delta n_{\text{gas}}\) and \(T\) are known and the ideal-gas approximation is reasonable.
Assumptions & limitations
- Pressure is spatially uniform and \(P_{\text{ext}}\) is the mechanical load opposing expansion.
- The \(\Delta n_{\text{gas}}\) shortcut presumes ideal-gas behavior and small pressure mismatch
between system and surroundings (quasi-mechanical work).
- Non-\(pV\) work (electrical, surface, shaft, etc.) is not included.
- If the process is reversible and isothermal for an ideal gas, the exact integral (not used by this tool) is:
\[
w_{\text{rev}} \;=\; -\,n\,R\,T \,\ln\!\left(\frac{V_f}{V_i}\right)
\]
For finite-step, constant-load processes, the constant-\(P_{\text{ext}}\) formula above is appropriate.
How the tool computes
- Mode selection. You choose either \(\Delta V\) or \(\Delta n_{\text{gas}}\) mode.
- Resolve the change.
- If \(\Delta V\) is blank, it uses \(V_f - V_i\) with your units.
- If \(\Delta n_{\text{gas}}\) is blank, it uses \(n_f - n_i\).
- Convert units. Internally, \(P\) is converted to Pa and \(V\) to \(\mathrm{m^3}\) so \(w\) is in J.
- Compute \(w\).
\[
w \;=\; -\,P_{\text{ext}}\,\Delta V
\]
\[
\text{or}\qquad w \;\approx\; -\,\Delta n_{\text{gas}}\,R\,T
\]
- Report. The result is shown in J and kJ with your chosen significant figures,
plus a short sign interpretation (expansion/compression).
Common pitfalls
- Using atm with \(\mathrm{m^3}\) directly. Convert consistently or use the L·atm → J factor.
- Forgetting absolute temperature. In the \(\Delta n_{\text{gas}}\) shortcut, always use K.
- Including non-gaseous species in \(\Delta n_{\text{gas}}\). Only gas counts there.
- Sign mistakes. Remember: expansion → \(w<0\); compression → \(w>0\).
