Answer the following by appropriately matching the lists based on the information given in the paragraph.
In a thermodynamics process on an ideal monatomic gas,the infinitesimal heat absorbed by the gas is given by $T \Delta X$,where $T$ is the temperature of the system and $\Delta X$ is the infinitesimal change in a thermodynamic quantity $X$ of the system. For a mole of monatomic ideal gas,$X = \frac{3}{2} R \ln \left(\frac{T}{T_A}\right) + R \ln \left(\frac{V}{V_A}\right)$. Here,$R$ is the gas constant,$V$ is the volume of the gas,$T_A$ and $V_A$ are constants.
The $List-I$ below gives some quantities involved in a process and $List-II$ gives some possible values of these quantities.
List-$I$ List-$II$ $(I)$ Work done by the system in process $1 \rightarrow 2 \rightarrow 3$ $(P)$ $\frac{1}{3} R T_0 \ln 2$ $(II)$ Change in internal energy in process $1 \rightarrow 2 \rightarrow 3$ $(Q)$ $\frac{1}{3} R T_0$ $(III)$ Heat absorbed by the system in process $1 \rightarrow 2 \rightarrow 3$ $(R)$ $R T_0$ $(IV)$ Heat absorbed by the system in process $1 \rightarrow 2$ $(S)$ $\frac{4}{3} R T_0$ $(T)$ $\frac{1}{3} R T_0 (3 + \ln 2)$ $(U)$ $\frac{5}{6} R T_0$
If the process carried out on one mole of monatomic ideal gas is as shown in the figure in the $PV$-diagram with $P_0 V_0 = \frac{1}{3} R T_0$,the correct match is:
$(1)$ $I \rightarrow Q, II \rightarrow R, III \rightarrow P, IV \rightarrow U$
$(2)$ $I \rightarrow S, II \rightarrow R, III \rightarrow Q, IV \rightarrow T$
$(3)$ $I \rightarrow Q, II \rightarrow R, III \rightarrow S, IV \rightarrow U$
$(4)$ $I \rightarrow Q, II \rightarrow S, III \rightarrow R, IV \rightarrow U$
If the process on one mole of monatomic ideal gas is as shown in the $TV$-diagram with $P_0 V_0 = \frac{1}{3} R T_0$,the correct match is:
$(1)$ $I \rightarrow S, II \rightarrow T, III \rightarrow Q, IV \rightarrow U$
$(2)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow S$
$(3)$ $I \rightarrow P, II \rightarrow R, III \rightarrow Q, IV \rightarrow T$
$(4)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow P$
Give the answer for question $(1)$ and $(2)$.
In a thermodynamics process on an ideal monatomic gas,the infinitesimal heat absorbed by the gas is given by $T \Delta X$,where $T$ is the temperature of the system and $\Delta X$ is the infinitesimal change in a thermodynamic quantity $X$ of the system. For a mole of monatomic ideal gas,$X = \frac{3}{2} R \ln \left(\frac{T}{T_A}\right) + R \ln \left(\frac{V}{V_A}\right)$. Here,$R$ is the gas constant,$V$ is the volume of the gas,$T_A$ and $V_A$ are constants.
The $List-I$ below gives some quantities involved in a process and $List-II$ gives some possible values of these quantities.
| List-$I$ | List-$II$ |
| $(I)$ Work done by the system in process $1 \rightarrow 2 \rightarrow 3$ | $(P)$ $\frac{1}{3} R T_0 \ln 2$ |
| $(II)$ Change in internal energy in process $1 \rightarrow 2 \rightarrow 3$ | $(Q)$ $\frac{1}{3} R T_0$ |
| $(III)$ Heat absorbed by the system in process $1 \rightarrow 2 \rightarrow 3$ | $(R)$ $R T_0$ |
| $(IV)$ Heat absorbed by the system in process $1 \rightarrow 2$ | $(S)$ $\frac{4}{3} R T_0$ |
| $(T)$ $\frac{1}{3} R T_0 (3 + \ln 2)$ | |
| $(U)$ $\frac{5}{6} R T_0$ |
If the process carried out on one mole of monatomic ideal gas is as shown in the figure in the $PV$-diagram with $P_0 V_0 = \frac{1}{3} R T_0$,the correct match is:
$(1)$ $I \rightarrow Q, II \rightarrow R, III \rightarrow P, IV \rightarrow U$
$(2)$ $I \rightarrow S, II \rightarrow R, III \rightarrow Q, IV \rightarrow T$
$(3)$ $I \rightarrow Q, II \rightarrow R, III \rightarrow S, IV \rightarrow U$
$(4)$ $I \rightarrow Q, II \rightarrow S, III \rightarrow R, IV \rightarrow U$
If the process on one mole of monatomic ideal gas is as shown in the $TV$-diagram with $P_0 V_0 = \frac{1}{3} R T_0$,the correct match is:
$(1)$ $I \rightarrow S, II \rightarrow T, III \rightarrow Q, IV \rightarrow U$
$(2)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow S$
$(3)$ $I \rightarrow P, II \rightarrow R, III \rightarrow Q, IV \rightarrow T$
$(4)$ $I \rightarrow P, II \rightarrow R, III \rightarrow T, IV \rightarrow P$
Give the answer for question $(1)$ and $(2)$.
- A
- B
- C
- D
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$A$ student records $\Delta Q, \Delta U,$ and $\Delta W$ for a thermodynamic cycle $A \rightarrow B \rightarrow C \rightarrow A$. Certain entries are missing in the table below. Find the correct entry from the following options.
$AB$ $BC$ $CA$ $\Delta W$ $40 \, J$ $30 \, J$ $\Delta U$ $50 \, J$ $\Delta Q$ $150 \, J$ $10 \, J$
| $AB$ | $BC$ | $CA$ | |
|---|---|---|---|
| $\Delta W$ | $40 \, J$ | $30 \, J$ | |
| $\Delta U$ | $50 \, J$ | ||
| $\Delta Q$ | $150 \, J$ | $10 \, J$ |
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View SolutionOne mole of a monatomic ideal gas is taken through a cycle $ABCDA$ as shown in the $P-V$ diagram. Column $II$ gives the characteristics involved in the cycle. Match them with each of the processes given in Column $I$.
Column $I$ Column $II$ $(A)$ Process $A \rightarrow B$ $(p)$ Internal energy decreases. $(B)$ Process $B \rightarrow C$ $(q)$ Internal energy increases. $(C)$ Process $C \rightarrow D$ $(r)$ Heat is lost. $(D)$ Process $D \rightarrow A$ $(s)$ Heat is gained. $(t)$ Work is done on the gas.
| Column $I$ | Column $II$ |
| $(A)$ Process $A \rightarrow B$ | $(p)$ Internal energy decreases. |
| $(B)$ Process $B \rightarrow C$ | $(q)$ Internal energy increases. |
| $(C)$ Process $C \rightarrow D$ | $(r)$ Heat is lost. |
| $(D)$ Process $D \rightarrow A$ | $(s)$ Heat is gained. |
| $(t)$ Work is done on the gas. |
DifficultIIT 2011
View SolutionOne mole of a monatomic ideal gas undergoes the cyclic process $J \rightarrow K \rightarrow L \rightarrow M \rightarrow J$,as shown in the $P - T$ diagram. Match the quantities mentioned in $List-I$ with their values in $List-II$ and choose the correct option. [$R$ is the gas constant]
$List-I$ $List-II$ $(P)$ Work done in the complete cyclic process $(1)$ $R T_0 - 4 R T_0 \ln 2$ $(Q)$ Change in the internal energy of the gas in the process $JK$ $(2)$ $0$ $(R)$ Heat given to the gas in the process $KL$ $(3)$ $3 R T_0$ $(S)$ Change in the internal energy of the gas in the process $MJ$ $(4)$ $-2 R T_0 \ln 2$ $(5)$ $-3 R T_0 \ln 2$
| $List-I$ | $List-II$ |
| $(P)$ Work done in the complete cyclic process | $(1)$ $R T_0 - 4 R T_0 \ln 2$ |
| $(Q)$ Change in the internal energy of the gas in the process $JK$ | $(2)$ $0$ |
| $(R)$ Heat given to the gas in the process $KL$ | $(3)$ $3 R T_0$ |
| $(S)$ Change in the internal energy of the gas in the process $MJ$ | $(4)$ $-2 R T_0 \ln 2$ |
| $(5)$ $-3 R T_0 \ln 2$ |
For an ideal gas, a cyclic process $ABCA$ as shown in the $P-T$ diagram, when presented in a $P-V$ plot, would be:
MediumWBJEE 2025
View Solution$A$ given mass of a gas is compressed isothermally until its pressure is doubled. It is then allowed to expand adiabatically until its original volume is restored and its pressure is then found to be $0.75$ of its initial pressure. The ratio of the specific heats of the gas is approximately:
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