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JEE Advance - Physics (2019 - Paper 2 Offline - No. 17)

In a thermodynamic process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by T$$\Delta $$X where T is 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, $$X = {3 \over 2}R\,\ln \left( {{T \over {{T_A}}}} \right) + R\,\ln \left( {{V \over {{V_A}}}} \right)$$

Here, R is gas constant, V is volume of gas, TA and VA are constants.

The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

In a thermodynamic process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by T$$\Delta $$X where T is 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, $$X = {3 \over 2}R\,\ln \left( {{T \over {{T_A}}}} \right) + R\,\ln \left( {{V \over {{V_A}}}} \right)$$

Here, R is gas constant, V is volume of gas, TA and VA are constants.

The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

In a thermodynamic process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by T$$\Delta $$X where T is 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, $$X = {3 \over 2}R\,\ln \left( {{T \over {{T_A}}}} \right) + R\,\ln \left( {{V \over {{V_A}}}} \right)$$

Here, R is gas constant, V is volume of gas, TA and VA are constants.

The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.

If the process on one mole of monatomic ideal gas is as shown in the TV-diagram with $${P_0}{V_0} = {1 \over 3}R{T_0}$$, the correct match is,

JEE Advanced 2019 Paper 2 Offline Physics - Heat and Thermodynamics Question 39 English
I $$ \to $$ P, II $$ \to $$ R, III $$ \to $$ T, IV $$ \to $$ S
I $$ \to $$ P, II $$ \to $$ T, III $$ \to $$ Q, IV $$ \to $$ T
I $$ \to $$ S, II $$ \to $$ T, III $$ \to $$ Q, IV $$ \to $$ U
I $$ \to $$ P, II $$ \to $$ R, III $$ \to $$ T, IV $$ \to $$ P

Explanation

1-2 process is isothermal and 2-3 process is isochoric.

(I) $${W_{1 \to 2}} = nRI\,\ln {{{V_f}} \over {{V_i}}} = 1 \times R{{{T_0}} \over 3}\,\ln {{{V_2}} \over {{V_1}}}$$

W2$$ \to $$3 = 0 (Isochoric process)

$${W_{1 \to 2 \to 3}}$$ = $${W_{1 \to 2}}$$ + $${W_{2 \to 3}}$$

= $${{R{T_0}} \over 3}$$ ln 2(I$$ \to $$P)

(II) $$\Delta U = {f \over 2}nR({T_f} - {T_i})$$

$$\Delta {U_{1 \to 2 \to 3}} = {3 \over 2}\left[ {{T_0} - {{{T_0}} \over 3}} \right]$$

$$ = R{T_0}$$ (II$$ \to $$R)

(III) $${Q_{1 \to 2 \to 3}} = \Delta {U_{1 \to 2 \to 3}} + {W_{1 \to 2 \to 3}}$$

(First law of thermodynamics)

$$ = R{T_0} + {{R{T_0}} \over 3}\ln 2$$

$${{R{T_0}} \over 3}[3 + \ln 2]$$ (III$$ \to $$T)

(IV) $${Q_{1 \to 2}} = \Delta {U_{1 \to 2}} + {W_{1 \to 2}} = 0 + {{R{T_0}} \over 3}ln2 = {{R{T_0}} \over 3}ln2$$ (IV$$ \to $$P)

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