Chemistry · Thermodynamics · NEET
Work done on the gas is w = -p_ext x (V_f - V_i), which is the area under the pressure-volume curve. In a single step you apply ONE large external pressure (equal to the final high pressure) for the whole volume change, so the rectangle of area is tall and wide, giving a big area. In a stepwise compression you first use a smaller external pressure, compress a little, then a bit more pressure, and so on. Each small step covers a smaller rectangle, and adding them gives a total area that is SMALLER than the single tall rectangle. Smaller area means less work is done on the gas.
During compression the surroundings push the piston in, so work is done ON the gas. In the chemistry (new IUPAC) sign convention this makes w positive. During expansion the gas pushes out and does work on the surroundings, so w is negative. This concept page is about compression, so we compare how much work the surroundings must do on the gas by different paths.
More steps means you compress the gas slowly, increasing the external pressure by only a small amount at each stage instead of all at once. If you keep increasing the number of steps until each step is infinitely small, the external pressure is always just a tiny bit larger than the gas pressure. This infinite-step limit is the REVERSIBLE compression, and it requires the LEAST work to be done on the gas.
The reversible (infinitely many tiny steps) compression needs the MINIMUM work on the gas. A single large step needs the MAXIMUM work on the gas. Careful: for EXPANSION it flips - the reversible path gives the MAXIMUM work done BY the gas. NEET asks both, so read whether the process is compression or expansion.
Draw pressure on the y-axis and volume on the x-axis. The work is the area under the process line between the initial and final volume. A single-step compression is a tall flat line at the high external pressure, giving a large rectangle. A multi-step compression looks like a staircase that hugs the smooth gas curve, so its total area is smaller. The reversible curve is the smooth hyperbola with the smallest area for compression.
The work is always calculated using the EXTERNAL pressure that acts on the piston, because that is the force the surroundings actually apply. The gas's own internal pressure only equals the external pressure in the special reversible case, where p_ext is kept just infinitesimally different from p_gas. That is exactly why reversible work is calculated with an integral of nRT dV / V instead of a simple p_ext x change in V.
Which of the following p-V curves represents the maximum work done?
Under isothermal condition, a gas at 300 K expands from 0.1 L to 0.25 L against a constant external pressure of 2 bar. The work done by the gas is (1 L bar = 100 J)
An ideal gas expands isothermally from 10^-3 m^3 to 10^-2 m^3 at 300 K against a constant pressure of 10^5 N m^-2. The work done on the gas is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Yes, for the same start and end states a single large-pressure step always does more work on the gas than any multi-step path, because the single tall rectangle has more area than the staircase. The reversible (infinite-step) path needs the least.
Not exactly. Stepwise means a finite number of steps. Reversible is the limiting case of INFINITELY many infinitely small steps, where p_ext is always just a tiny bit larger than the gas pressure. Reversible gives the true minimum work on the gas.
For expansion, reversible gives the MAXIMUM work done by the gas; a single step gives less. For compression, a single step gives the MAXIMUM work done on the gas; reversible gives the minimum. Always check the direction before choosing your answer.
Use w = -p_ext x (V_f - V_i) with the constant external pressure. For a reversible isothermal process use w = -2.303 nRT log(V_f / V_i).
