Q.Graphically show the total work done in an expansion when the state of an ideal gas is changed reversibly and isothermally from (pi, Vi) to (pf, Vf). With the help of a pV plot compare the work done in the above case with that carried out against a constant external pressure pf.

The key idea is the pressure-volume work in a thermodynamic process, a form of energy transfer central to the First Law of Thermodynamics. The magnitude of work done depends on the specific path taken between the initial and final states.

Consider an ideal gas expanding from an initial state $(p_i, V_i)$ to a final state $(p_f, V_f)$.

1. Reversible Isothermal Expansion: For a reversible process, the external pressure $p_{ext}$ is always infinitesimally close to the gas pressure $p_{gas}$. For an ideal gas undergoing isothermal expansion, $p_{gas} = nRT/V$. The work done by the gas is $W_{rev} = -\int_{V_i}^{V_f} \frac{nRT}{V} dV = -nRT \ln\left(\frac{V_f}{V_i}\right)$. Graphically, this is the area under the smooth $pV$ isotherm curve.

2. Expansion against Constant External Pressure $p_f$: For an irreversible expansion against a constant external pressure $p_f$, the work done by the gas is $W_{const} = -p_f(V_f - V_i)$. Graphically, this is the area of a rectangle with height $p_f$ and width $(V_f - V_i)$.

3. Comparison on a $pV$ Plot: …

In thermodynamics, the work done by or on a system during a change in its state is a crucial concept, especially when dealing with gases. The First Law of Thermodynamics states that the change in internal energy ($\Delta U$) of a system is equal to the heat added to the system ($Q$) minus the work done by the system ($W$):

Work done by a gas during expansion is typically negative, as the system expends energy to push against its surroundings. Conversely, work done on a gas during compression is positive. For an infinitesimal change in volume $dV$ against an external pressure $P_{ext}$, the work done $dW$ is given by:

The total work done for a finite change in volume from $V_i$ to $V_f$ is the integral of this expression:

Let's analyze the two scenarios:

1. Reversible Isothermal Expansion

A reversible process is one that can be reversed by an infinitesimal change in conditions, meaning the system is always in equilibrium with its surroundings. For a gas, this implies that the internal pressure of the gas ($P_{gas}$) is always infinitesimally close to the external pressure ($P_{ext}$), so $P_{ext} \approx P_{gas}$. An isothermal process occurs at a constant temperature ($T$).

For an ideal gas, the ideal gas law states $PV = nRT$. Since $T$ is constant, $PV = \text{constant}$. This means $P = \frac{nRT}{V}$.

Substituting $P_{ext} = P_{gas} = \frac{nRT}{V}$ into the work formula:

1. Graphical Representation: On a pV plot, the path of a reversible isothermal expansion is a hyperbola ($PV = \text{constant}$). The initial state is $(P_i, V_i)$ and the final state is $(P_f, V_f)$. The curve smoothly connects these two points. The work done is the entire area under this hyperbolic curve, bounded by the curve, the initial volume $V_i$, the final volume $V_f$, and the volume axis.
   • Since $V_f > V_i$ for expansion, $\ln(V_f/V_i)$ is positive, making $W$ negative, indicating work done by the system.

2. Expansion Against a Constant External Pressure $P_f$

This is an irreversible process. The external pressure $P_{ext}$ is constant and equal to the final pressure of the gas, $P_f$. The expansion occurs rapidly, and the system is not in equilibrium with its surroundings throughout the process. The gas expands from an initial volume $V_i$ to a final volume $V_f$ against this constant external pressure $P_f$.

1. Work Done Calculation:

Since $P_{ext}$ is constant ($P_{ext} = P_f$):

Again, for expansion, $V_f > V_i$, so $W$ is negative.

2. Graphical Representation: On the same pV plot, the initial state is $(P_i, V_i)$. For the expansion to occur, $P_i$ must be greater than $P_f$. The process path is represented by:

* A vertical drop from $(P_i, V_i)$ to $(P_f, V_i)$ (this represents an instantaneous drop in internal pressure to match the external pressure, or the external pressure being suddenly lowered to $P_f$). No work is done during this vertical drop as volume is constant.

* A horizontal line from $(P_f, V_i)$ to $(P_f, V_f)$. This represents the expansion at constant external pressure $P_f$. …