b. Find the minimum temperature to which the first object can be lowered. · OlympiadBench #1435

Physics · thermodynamics

Problem

b. Find the minimum temperature

T_{L}

to which the first object can be lowered. Given constants:

g=9.8 \mathrm{~N} / \mathrm{kg}

k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}

c=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}

N_{\mathrm{A}}=6.02 \times 10^{23}(\mathrm{~mol})^{-1}

\sigma=5.67 \times 10^{-8} \mathrm{~J} /\left(\mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{K}^{4}\right)

1 \mathrm{eV}=1.602 \times 10^{-19} \mathrm{~J}

m_{e}=9.109 \times 10^{-31} \mathrm{~kg}=0.511 \mathrm{MeV} / \mathrm{c}^{2}

m_{p}=1.673 \times 10^{-27} \mathrm{~kg}=938 \mathrm{MeV} / \mathrm{c}^{2}

\sin \theta \approx \theta-\frac{1}{6} \theta^{3}

for

|\theta| \ll 1

\begin{aligned} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\ & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} \\ & k_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K} \\ & R=N_{\mathrm{A}} k_{\mathrm{B}}=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) \\ & e=1.602 \times 10^{-19} \mathrm{C} \\ & h=6.63 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.14 \times 10^{-15} \mathrm{eV} \cdot \mathrm{s} \\ & (1+x)^{n} \approx 1+n x \text { for }|x| \ll 1 \\ & \ln (1+x) \approx x \text { for }|x| \ll 1 \\ & \cos \theta \approx 1-\frac{1}{2} \theta^{2} \text { for }|\theta| \ll 1 \end{aligned}

\begin{array}{lrl} \text { velocity parameter } & \beta & =\frac{v}{c} \\ \text { Lorentz factor } & \gamma & =\frac{1}{\sqrt{1-\beta^{2}}} \\ \text { relativistic momentum } & p & =\gamma \beta m c \\ \text { relativistic energy } & E & =\gamma m c^{2} \\ \text { relativistic doppler shift } & \frac{f}{f_{0}} & =\sqrt{\frac{1-\beta}{1+\beta}} \end{array}

Extra Supplementary Reading Materials: Consider two objects with equal heat capacities

C

and initial temperatures

T_{1}

and

T_{2}

. A Carnot engine is run using these objects as its hot and cold reservoirs until they are at equal temperatures. Assume that the temperature changes of both the hot and cold reservoirs is very small compared to the temperature during any one cycle of the Carnot engine. Context question: a. Find the final temperature

T_{f}

of the two objects, and the total work

W

done by the engine. Context answer:

T_{f}=\sqrt{T_{1} T_{2}}

W=C\left[T_{1}+T_{2}-2 \sqrt{T_{1} T_{2}}\right]

Extra Supplementary Reading Materials: Now consider three objects with equal and constant heat capacity at initial temperatures

T_{1}=100 \mathrm{~K}, T_{2}=300 \mathrm{~K}

, and

T_{3}=300 \mathrm{~K}

. Suppose we wish to raise the temperature of the third object. To do this, we could run a Carnot engine between the first and second objects, extracting work

W

. This work can then be dissipated as heat to raise the temperature of the third object. Even better, it can be stored and used to run a Carnot engine between the first and third object in reverse, which pumps heat into the third object. Assume that all work produced by running engines can be stored and used without dissipation.

Related theory

thermodynamics

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