What is an adiabatic process ? Derive an expression for work done in an adiabatic process.
What is an adiabatic process ? Derive an expression for work done in an adiabatic process.
The process during which there is no exchange of heat takes place between a system and its environment is known as an adiabatic process.
From first law of thermodynamics,
$\Delta \mathrm{Q}=\Delta \mathrm{U}+\Delta \mathrm{W}, \Delta \mathrm{Q}=0$ for adiabatic process
$\therefore \Delta \mathrm{U}=-\Delta \mathrm{W}$ or $-\Delta \mathrm{U}=\Delta \mathrm{W}$
Mean work done by gas decreases the internal energy and work done on the gas increases the internal energy.
For an ideal gas in adiabatic process,
$\mathrm{PV}^{\gamma}=$ constant
Where $\gamma$ is the ratio of specific heats at constant pressure and at constant volume.
$\therefore \gamma=\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}$
The work done in an adiabatic change of an ideal gas from the state $\left[\mathrm{P}_{1}, \mathrm{~V}_{1}, \mathrm{~T}_{1}\right]$ to the state $\left[\mathrm{P}_{2}, \mathrm{~V}_{2}, \mathrm{~T}_{2}\right]$
$\Delta \mathrm{W}=\mathrm{P} \Delta \mathrm{V}$
The total work done $\mathrm{W}$ by total change,
$\mathrm{W}=\int_{\mathrm{V}_{1}}^{\mathrm{V}_{2}} \mathrm{P} d \mathrm{~V}$
but $PVY$ = constant
$\therefore \mathrm{P}=\frac{\text { constant }}{\mathrm{V}^{\gamma}} \ldots(2)$
$\therefore$ Putting the value of equ. $(2)$ in equ. $(1)$
$\mathrm{W}=$ constant $\times \int_{\mathrm{V}_{1}}^{\mathrm{v}_{2}} \frac{d \mathrm{~V}}{\mathrm{~V}^{\gamma}}$
$=$ constant $\times\left[\frac{\mathrm{V}^{\gamma+1}}{\mathrm{~V}_{2}^{\nearrow}-\gamma+1}\right]_{\mathrm{V}_{1}}^{\mathrm{V}_{2}}$
$=\frac{\text { constant }}{1-\gamma} \times\left[\mathrm{V}_{2}^{-\gamma+1}-\mathrm{V}_{1}^{\gamma+1}\right]$
$=\frac{1}{1-\gamma}\left[\frac{\text { constant }}{\mathrm{V}_{2}^{\gamma-1}}-\frac{\text { constant }}{\mathrm{V}_{1}^{\gamma-1}}\right]$
Similar Questions
A mixture of ideal gas containing $5$ moles of monatomic gas and $1$ mole of rigid diatomic gas is initially at pressure $P _0$, volume $V _0$ and temperature $T _0$. If the gas mixture is adiabatically compressed to a volume $V _0 / 4$, then the correct statement(s) is/are,
(Give $2^{1.2}=2.3 ; 2^{3.2}=9.2 ; R$ is gas constant)
$(1)$ The final pressure of the gas mixture after compression is in between $9 P _0$ and $10 P _0$
$(2)$ The average kinetic energy of the gas mixture after compression is in between $18 RT _0$ and $19 RT _0$
$(3)$ The work $| W |$ done during the process is $13 RT _0$
$(4)$ Adiabatic constant of the gas mixture is $1.6$
A mixture of ideal gas containing $5$ moles of monatomic gas and $1$ mole of rigid diatomic gas is initially at pressure $P _0$, volume $V _0$ and temperature $T _0$. If the gas mixture is adiabatically compressed to a volume $V _0 / 4$, then the correct statement(s) is/are,
(Give $2^{1.2}=2.3 ; 2^{3.2}=9.2 ; R$ is gas constant)
$(1)$ The final pressure of the gas mixture after compression is in between $9 P _0$ and $10 P _0$
$(2)$ The average kinetic energy of the gas mixture after compression is in between $18 RT _0$ and $19 RT _0$
$(3)$ The work $| W |$ done during the process is $13 RT _0$
$(4)$ Adiabatic constant of the gas mixture is $1.6$
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$1.$ The change in internal energy $\Delta U = 0$ in a cyclic process.
$2.$ In an adiabatic process temperature remains constant.
$3.$ The internal energy of a system during isothermal process decreases.
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$1.$ The change in internal energy $\Delta U = 0$ in a cyclic process.
$2.$ In an adiabatic process temperature remains constant.
$3.$ The internal energy of a system during isothermal process decreases.