Name: Macroscopic and Microscopic Variables
Uploaded: 2019-07-19
Duration: 5 min 3 s
Description: This video enables students to understand the definition and examples of the macroscopic and microscopic variables of the thermodynamic system.

Question

Derive the work done in an adiabatic process.

Accepted Answer

Adiabatic process is a process in which no heat flows into and out of the system (Q=0) but gas can be compressed through some external work. Hence, pressure, volume and temperature may change in this process.
So, first law of thermodynamics become, $∆ U = - W$
The equation of state for an adiabatic process is given by ${PV}^{γ} =$ Constant
Consider $μ$ moles of an ideal gas enclosed in a cylinder having perfectly non-conducting walls and base. A frictionless and insulating piston of cross-sectional area A is fitted in the cylinder.

Let W be the work done when the system goes from the initial state $(P_{i}, V_{i}, T_{i})$ to the final state $(P_{f}, V_{f}, T_{f})$ adiabatically.
$W = \int_{Vi}^{Vf} PDV . . . . . . . . . . . . . . . . . . . . . (1)$
Adiabatic process occurs quasi-statically, at every stage the ideal gas law is valid. Under this condition
${PV}^{γ} = constant P = \frac{constant}{V^{γ}}$
So, from equation (1)

$W_{adia} = \int_{Vi}^{Vf} \frac{constant}{V^{γ}} dV W_{adia} = \int_{Vi}^{Vf} {V^{-}}^{γ} dV W_{adia} = (\int_{vi}^{vf} \frac{V^{- γ + 1}}{γ + 1}) Vf / Vi = \frac{1}{1 - γ} [\frac{constant}{{V_{f}}^{γ - 1}} - \frac{constant}{{V_{i}}^{γ - 1}}] But P_{i} {V_{i}}^{γ} = P_{f} {V_{f}}^{γ} = constant = \frac{1}{1 - γ} [\frac{P_{f} {V_{f}}^{γ}}{{V_{f}}^{γ - 1}} - \frac{P_{i} {V_{i}}^{γ}}{{V_{i}}^{γ - 1}}] = \frac{1}{1 - γ} [P_{f} V_{f} - P_{i} V_{i}] . . . . . . . . . . . . . . . . . . . . (2)$

From ideal gas law,
$P_{f} V_{f} = {μRT}_{f}$ and $P_{i} V_{i} = μRT,$
Substituting in equation (2), we get
$W_{adia} = \frac{μR}{1 - γ} [T_{i} - P_{f}]$
When $Ti > Tf$ , the gas cools during adiabatic expansion and work is done by a gas. So, W_adiabatic is positive.
When $Ti < Tf$ , temperature of gas decreases during adiabatic compression, work is done on gas. So, W_adiabatic is negative.

Derive the work done in an adiabatic process.

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