Q. The heat $(Q)$ supplied to a solid, which is otherwise thermally isolated from its surroundings, is plotted as a function of its absolute temperature, $\theta$. It is found that they are related by the equation, $Q=a \theta^{2}+b \theta^{4}$. ( $a, b$ are constants). The heat capacity of the solid is :

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A

$a \frac{\theta^{3}}{3}+b \frac{\theta^{5}}{5}$

B

$a \theta+b \theta^{3}$

C

$a \frac{\theta}{3}+b \frac{\theta^{5}}{5}$

D

$2 a \theta+4 b \theta^{3}$

Solution:

The heat capacity of the solid is :
$\frac{d Q}{d \theta}=2 a \theta+4 b \theta^{3}$

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$a \frac{\theta^{3}}{3}+b \frac{\theta^{5}}{5}$

$a \theta+b \theta^{3}$

$a \frac{\theta}{3}+b \frac{\theta^{5}}{5}$

$2 a \theta+4 b \theta^{3}$

The heat capacity of the solid is : $\frac{d Q}{d \theta}=2 a \theta+4 b \theta^{3}$

Questions from Thermal Properties of Matter

1. If the temperature difference on the two sides of a wall increases from $ 100{}^\circ C $ to $ 200{}^\circ C $ , its thermal conductivity

2. A wall is made of equally thick layers $A$ and $B$ of different materials. Thermal conductivity of $A$ is twice that of $B$. In the steady state, the temperature difference across the wall is $36^{\circ}C$. The temperature difference across the layer $A$ is

3. The radiation emitted by a star $A$ is $10,000$ times that of the sun. If the surface temperatures of the sun and the star $A$ are $8000 \, K$ and $2000 \, K$ respectively. The ratio of the radii of the star $A$ and the sun is

4. A black body at $1373$ $℃$ emits maximum energy corresponding to a wavelength of $1.78$ microns. The temperature of the moon for which $\lambda _{m}=14$ micron would be

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1. If $E$ and $G$ respectively denote energy and gravitational constant, then $\frac{ E }{ G }$ has the dimensions of:

2. The de Broglie wavelength of an electron moving with kinetic energy of $144 \,eV$ is nearly

3. A car starts from rest and accelerates at $5\, m / s ^{2}$ At $t=4\, s$, a ball is dropped out of a window by a person sitting in the car. What is the velocity and acceleration of the ball at $t =6\, s$ ? (Take $\left. g =10\, m / s ^{2}\right)$

4. A wheel with $20$ metallic spokes each $1\, m$ long is rotated with a speed of $120\, rpm$ in a plane perpendicular to a magnetic field of $0.4\, G$. The induced emf between the axle and rim of the wheel will be, $(1 G = 10^{-4}\, T)$

5. A barometer is constructed using a liquid (density $= 760 \,kg/m^{3})$ What would be the height of the liquid column, when a mercury barometer reads $76\, cm$ ? (density of mercury $= 13600 kg/m^{3})$

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The heat capacity of the solid is :
$\frac{d Q}{d \theta}=2 a \theta+4 b \theta^{3}$

5. A barometer is constructed using a liquid (density $= 760 \,kg/m^{3})$ What would be the height of the liquid column, when a mercury barometer reads $76\, cm$ ?
(density of mercury $= 13600 kg/m^{3})$