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10-1.Thermometry, Thermal Expansion and Calorimetry
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The specific heat of a metal at low temperatures varies according to $S = aT^3$ where a is a constant and $T$ is the absolute temperature. The heat energy needed to raise unit mass of the metal from $T = 1 K$ to $T = 2 K$ is
A
$3 a$
B
$\frac{{15\,a}}{4}$
C
$\frac{{2\,a}}{3}$
D
$\frac{{12\,a}}{5}$
Solution
At temperature $t$ the heat energy required to raise temperature of unit mass by $d t$ is $d q=a t^{3} \times 1 \times d t$
So heat required to raise temperature from
$1 K$ to $2 K$ is $\int_{0}^{Q} d q=\int_{1}^{2} a t^{3} d t \Rightarrow Q=\left.a \frac{t^{4}}{4}\right|_{1} ^{2}=a(16-1) / 4$
$\Rightarrow Q=15 a / 4$
Standard 11
Physics
