Riemann surface
matlab

Describe briefly three different instruments that can be used for the accurate measurement of temperature and state roughly the temperature range in which they are useful and one important advantage of each instrument. Include at least one instrument that is capable of measuring temperatures down to $1 \mathrm{~K}$.
(Wisconsin)

.

Solution:

Magnetic thermometer: Its principle is Curie’s law $\chi=C / T$, where $\chi$ is the susceptibility of the paramagnetic substance used, $T$ is its absolute temperature and $C$ is a constant. Its advantage is that it can measure temperatures below $1 \mathrm{~K}$.

Optical pyrometer: It is based on the principle that we can find the temperature of a hot body by measuring the energy radiated from it, using the formula of radiation. While taking measurements, it does not come into direct contact with the measured body. Therefore, it is usually used to measure the temperatures of celestial bodies.

Vapor pressure thermometer: It is a kind of thermometer used to measure low temperatures. Its principle is as follows. There exists a definite relation between the saturation vapor pressure of a chemically pure material and its boiling point. If this relation is known, we can determine temperature by measuring vapor pressure. It can measure temperatures greater than $14 \mathrm{~K}$, and is the thermometer usually used to measure low temperatures.

A solid object has a density $\rho$, mass $M$, and coefficient of linear expansion $\alpha$. Show that at pressure $p$ the heat capacities $C_p$ and $C_v$ are related by
$$C_p-C_v=3 \alpha M p / \rho$$

Solution:
From the first law of thermodynamics $d Q=d U+p d V$ and $\left(\frac{d U}{d T}\right)p \approx$ $\left(\frac{d U}{d T}\right)_v$ (for solid), we obtain $$C_p-C_v=\left(\frac{d Q}{d T}\right)_p-\left(\frac{d U}{d T}\right)_v=p \frac{d V}{d T} .$$ From the definition of coefficient of linear expansion $\alpha=\alpha{\mathrm{solid}} / 3=\frac{1}{3 V} \frac{d V}{d T}$, we obtain
$$\frac{d V}{d T}=3 \alpha V=3 \alpha \frac{M}{\rho} .$$
Substituting this in (*), we find
$$C_p-C_v=3 \alpha \frac{M}{\rho} p$$

E-mail: help-assignment@gmail.com  微信:shuxuejun

help-assignment™是一个服务全球中国留学生的专业代写公司