Consider a cycle tyre being filled with air by a pump. Let $V$ be the volume of the tyre (fixed) and at each stroke of the pump $\Delta V$ $(< < V)$ of air is transferred to the tube adiabatically. What is the work done when the pressure in the tube is increased from $P_1$ to $P_2$ ?
Consider a cycle tyre being filled with air by a pump. Let $V$ be the volume of the tyre (fixed) and at each stroke of the pump $\Delta V$ $(< < V)$ of air is transferred to the tube adiabatically. What is the work done when the pressure in the tube is increased from $P_1$ to $P_2$ ?
The pressure is increased by $\Delta \mathrm{P}$, when volume is increased by $\Delta \mathrm{V}$ at each stroke.
$\therefore \mathrm{P}_{1} \mathrm{~V}_{1}^{\gamma}=\mathrm{P}_{2} \mathrm{~V}_{2}^{\gamma}$ (Initially)
$\therefore \mathrm{P}(\mathrm{V}+\Delta \mathrm{V})^{\gamma}=(\mathrm{P}+\Delta \mathrm{P}) \mathrm{V}^{\gamma}(\because$ volume is fixed $)$
$\mathrm{PV}^{\gamma}\left[1+\frac{\Delta \mathrm{V}}{\mathrm{V}}\right]^{\gamma}=\mathrm{P}\left[1+\frac{\Delta \mathrm{P}}{\mathrm{P}}\right] \mathrm{V}^{\gamma}$
As $\Delta \mathrm{V}<<\mathrm{V}$ so by using binomial theorem we get,
$\therefore \mathrm{PV}^{\gamma}\left(1+\gamma \frac{\Delta \mathrm{V}}{\mathrm{V}}\right)=\mathrm{PV}^{\gamma}\left(1+\frac{\Delta \mathrm{P}}{\mathrm{P}}\right)$ $\therefore \gamma \frac{\Delta \mathrm{V}}{\mathrm{V}}=\frac{\Delta \mathrm{P}}{\mathrm{P}}$ $\therefore \Delta \mathrm{V}=\frac{1}{\gamma} \cdot \frac{\mathrm{V}}{\mathrm{P}} \cdot \Delta \mathrm{P}$ but $\Delta \mathrm{V}$ and $\Delta \mathrm{P}$ are very small.
$d \mathrm{~V}=\frac{1}{\gamma} \cdot \frac{\mathrm{V}}{\mathrm{P}} \cdot d \mathrm{P}$
Hence, work done in increasing the pressure in tube from $\mathrm{P}_{1}$ to $\mathrm{P}_{2}$ is
$W=\int_{P_{1}}^{P_{2}} P d V=\int_{P_{1}} P \times \frac{1}{\gamma} \times \frac{V}{P} \cdot d P$
$=\frac{V}{\gamma} \int_{P_{1}}^{P_{2}} d P$
$=\frac{V}{\gamma}\left(P_{2}-P_{1}\right)$
$\therefore W =\frac{\left(P_{2}-P_{1}\right) V}{\gamma}$
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(image)
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$[A] (II) (iv) (R)$ $[B] (II) (iii) (P)$ $[C] (II) (iii) (S)$ $[D] (III) (iii) (P)$
($2$) Which one of the following options is the correct combination?
$[A] (III) (ii) (S)$ $[B] (II) (iv) (R)$ $[C] (II) (iv) (P)$ $[D] (IV) (ii) (S)$
($3$) Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?
$[A] (III) (iv) (R)$ $[B] (I) (ii)$ $(\mathrm{Q})$ $[C] (IV) (ii) (R)$ $[D] (I) (iv) (Q)$
An ideal gas is undergoing a cyclic thermodynamic process in different ways as shown in the corresponding $P$ $V$ diagrams in column $3$ of the table. Consider only the path from state $1$ to $2 . W$ denotes the corresponding work done on the system. The equations and plots in the table have standard notations as used in thermodynamic processes. Here $\gamma$ is the ratio of heat capacities at constant pressure and constant volume. The number of moles in the gas is $n$.
(image)
($1$) Which of the following options is the only correct representation of a process in which $\Delta U=\Delta Q-P \Delta V$ ?
$[A] (II) (iv) (R)$ $[B] (II) (iii) (P)$ $[C] (II) (iii) (S)$ $[D] (III) (iii) (P)$
($2$) Which one of the following options is the correct combination?
$[A] (III) (ii) (S)$ $[B] (II) (iv) (R)$ $[C] (II) (iv) (P)$ $[D] (IV) (ii) (S)$
($3$) Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?
$[A] (III) (iv) (R)$ $[B] (I) (ii)$ $(\mathrm{Q})$ $[C] (IV) (ii) (R)$ $[D] (I) (iv) (Q)$
- [IIT 2017]