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1

AIEEE 2011

MCQ (Single Correct Answer)
A thermally insulated vessel contains an ideal gas of molecular mass $$M$$ and ratio of specific heats $$\gamma .$$ It is moving with speed $$v$$ and it's suddenly brought to rest. Assuming no heat is lost to the surroundings, Its temperature increases by:
A
$${{\left( {\gamma - 1} \right)} \over {2\gamma R}}M{v^2}K$$
B
$${{\gamma {M^2}v} \over {2R}}K$$
C
$${{\left( {\gamma - 1} \right)} \over {2R}}M{v^2}K$$
D
$${{\left( {\gamma - 1} \right)} \over {2\left( {\gamma + 1} \right)R}}M{v^2}K$$

Explanation

Here, work done is zero.
So, loss in kinetic energy $$=$$ change in internal energy of gas
$${1 \over 2}m{v^2} = n{C_v}\Delta T = n{R \over {\gamma - 1}}\Delta T$$
$${1 \over 2}m{v^2} = {m \over M}{R \over {\gamma - 1}}\Delta T$$
$$\therefore$$ $$\Delta T = {{M{v^2}\left( {\gamma - 1} \right)} \over {2R}}K$$
2

AIEEE 2011

MCQ (Single Correct Answer)
A Carnot engine operating between temperatures $${{T_1}}$$ and $${{T_2}}$$ has efficiency $${1 \over 6}$$. When $${T_2}$$ is lowered by $$62$$ $$K$$ its efficiency increases to $${1 \over 3}$$. Then $${T_1}$$ and $${T_2}$$ are, respectively:
A
$$372$$ $$K$$ and $$330$$ $$K$$
B
$$330$$ $$K$$ and $$268$$ $$K$$
C
$$310$$ $$K$$ and $$248$$ $$K$$
D
$$372$$ $$K$$ and $$310$$ $$K$$

Explanation

Efficiency of engine
$${1 \over 6} = 1 - {{{T_2}} \over {{T_1}}}$$ and $${\eta _2} = 1 - {{{T_2} - 62} \over {{T_1}}} = {1 \over 3}$$
$$\therefore$$ $${T_1} = 372\,K$$ and $${T_2} = {5 \over 6} \times 372 = 310\,K$$
3

AIEEE 2011

MCQ (Single Correct Answer)
Three perfect gases at absolute temperatures $${T_1},\,{T_2}$$ and $${T_3}$$ are mixed. The masses of molecules are $${m_1},{m_2}$$ and $${m_3}$$ and the number of molecules are $${n_1},$$ $${n_2}$$ and $${n_3}$$ respectively. Assuming no loss of energy, the final temperature of the mixture is:
A
$${{{n_1}{T_1} + {n_2}{T_2} + {n_3}{T_3}} \over {{n_1} + {n_2} + {n_3}}}$$
B
$${{{n_1}T_1^2 + {n_2}T_2^2 + {n_3}T_3^2} \over {{n_1}{T_1} + {n_2}{T_2} + {n_3}{T_3}}}$$
C
$${{n_1^2T_1^2 + n_2^2T_2^2 + n_3^2T_3^2} \over {{n_1}{T_1} + {n_2}{T_2} + {n_3}{T_3}}}$$
D
$${{\left( {{T_1} + {T_2} + {T_3}} \right)} \over 3}$$

Explanation

Number of moles of first gas $$ = {{{n_1}} \over {{N_A}}}$$
Number of moles of second gas $$ = {{{n_2}} \over {{N_A}}}$$
Number of moles of third gas $$ = {{{n_3}} \over {{N_A}}}$$
If there is no loss of energy then
$${P_1}{V_1} + {P_2}{V_2} + {P_3}{V_3} = PV$$
$${{{n_1}} \over {{N_A}}}R{T_1} + {{{n_2}} \over {{N_A}}}R{T_2} + {{{n_3}} \over {{N_A}}}R{T_3}$$
$$ = {{{n_1} + {n_2} + {n_3}} \over {{N_A}}}R{T_{mix}}$$
$$ \Rightarrow {T_{mix}} = {{{n_1}{T_1} + {n_2}{T_2} + {n_3}{T_3}} \over {{n_1} + {n_2} + {n_3}}}$$
4

AIEEE 2010

MCQ (Single Correct Answer)
A diatomic ideal gas is used in a Carnot engine as the working substance. If during the adiabatic expansion part of the cycle the volume of the gas increases from $$V$$ to $$32$$ $$V$$, the efficiency of the engine is
A
$$0.5$$
B
$$0.75$$
C
$$0.99$$
D
$$0.25$$

Explanation

$${T_1}{V^{\gamma - 1}} = {T_2}{\left( {32V} \right)^{\gamma - 1}}$$
$$ \Rightarrow {T_1} = {\left( {32} \right)^{\gamma - 1}}.{T_2}$$
For diatomic gas, $$\gamma = {7 \over 5}$$
$$\therefore$$ $$\gamma - 1 = {2 \over 5}$$
$$\therefore$$ $${T_1} = {\left( {32} \right)^{{2 \over 5}}}.{T_2} \Rightarrow {T_1} = 4{T_2}$$
Now, efficiency $$ = 1 - {{{T_2}} \over {{T_1}}}$$
$$ = 1 - {{{T_2}} \over {4{T_2}}}$$
$$ = 1 - {1 \over 4}$$
$$ = {3 \over 4}$$
$$ = 0.75.$$

Questions Asked from Heat and Thermodynamics

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