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1

AIEEE 2005

MCQ (Single Correct Answer)
The figure shows a system of two concentric spheres of radii $${r_1}$$ and $${r_2}$$ are kept at temperatures $${T_1}$$ and $${T_2}$$, respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to
A
$$In\left( {{{{r_2}} \over {{r_1}}}} \right)$$
B
$${{\left( {{r_2} - {r_1}} \right)} \over {\left( {{r_1}{r_2}} \right)}}$$
C
$${\left( {{r_2} - {r_1}} \right)}$$
D
$${{{r_1}{r_2}} \over {\left( {{r_2} - {r_1}} \right)}}$$

Explanation


Consider a shell of thickness $$(dr)$$ and of radius $$(r)$$ and the temperature of inner and outer surfaces of this shell be $$T,(T-dT)$$
$$H = {{KA\left[ {\left( {T - dT} \right) - T} \right]} \over {dr}} = {{ - KAdT} \over {dr}}$$
$$H = - 4\pi K{r^2}{{dT} \over {dr}}$$ ( as $$A = 4\pi {r^2}$$ )
Then, $$\left( H \right)\int\limits_{{r_1}}^{{r^2}} {{1 \over {{r^2}}}} dr = - 4\pi K\int\limits_{{T_1}}^{{T_2}} {dT} $$
$$H\left[ {{1 \over {{r_1}}} - {1 \over {{r_2}}}} \right] = - 4\pi K\left[ {{T_2} - {T_1}} \right]$$
or $$H = {{ - 4\pi K{r_1}{r_2}\left( {{T_2} - {T_1}} \right)} \over {\left( {{r_2} - {r_1}} \right)}}$$
2

AIEEE 2005

MCQ (Single Correct Answer)
A system goes from $$A$$ to $$B$$ via two processes $$I$$ and $$II$$ as shown in figure. If $$\Delta {U_1}$$ and $$\Delta {U_2}$$ are the changes in internal energies in the processes $$I$$ and $$II$$ respectively, then
A
relation between $$\Delta {U_1}$$ and $$\Delta {U_2}$$ can not be determined
B
$$\Delta {U_1} = \Delta {U_2}$$
C
$$\Delta {U_2} < \Delta {U_1}$$
D
$$\Delta {U_2} > \Delta {U_1}$$

Explanation

Change in internal energy do not depend upon the path followed by the process. It only depends on initial and final states $$i.e.,$$ $$\Delta U{}_1 = \Delta {U_2}$$
3

AIEEE 2005

MCQ (Single Correct Answer)
A gaseous mixture consists of $$16$$ $$g$$ of helium and $$16$$ $$g$$ of oxygen. The ratio $${{Cp} \over {{C_v}}}$$ of the mixture is
A
$$1.62$$
B
$$1.59$$
C
$$1.54$$
D
$$1.4$$

Explanation

$${{{n_1} + {n_2}} \over {r - 1}} = {{{n_1}} \over {{r_1} - 1}} + {{{n_2}} \over {{r_2} - 1}}$$
$${{{{16} \over 4} + {{16} \over {32}}} \over {r - 1}} = {{16/4} \over {{5 \over 3} - 1}} + {{16/32} \over {1.4 - 1}}$$
$$\therefore$$ $$\gamma = 1.62$$
4

AIEEE 2004

MCQ (Single Correct Answer)
The temperature of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity $$K$$ and $$2K$$ and thickness $$x$$ and $$4x,$$ respectively, are $${T_2}$$ and $${T_1}\left( {{T_2} > {T_1}} \right).$$ The rate of heat transfer through the slab, in a steady state is $$\left( {{{A\left( {{T_2} - {T_1}} \right)K} \over x}} \right)f,$$ with $$f$$ equal to
A
$${2 \over 3}$$
B
$${1 \over 2}$$
C
$$1$$
D
$${1 \over 3}$$

Explanation

The thermal resistance
$${x \over {KA}} + {{4x} \over {2KA}} = {{3x} \over {KA}}$$
$$\therefore$$ $${{dQ} \over {dt}} = {{\Delta T} \over {{{3x} \over {KA}}}} = {{\left( {{T_2} - {T_1}} \right)KA} \over {3x}}$$
$$ = {1 \over 3}\left\{ {{{A\left( {{T_2} - {T_1}} \right)K} \over x}} \right\}$$
$$\therefore$$ $$f = {1 \over 3}$$

Questions Asked from Heat and Thermodynamics

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