# Temperature and Heat

Hi,

Welcome to my 3rd post, Heat and temperature. Now, didn’t you want to know that how heat can travel? Simple awesome ways that even acrobats can’t do. Well, if you wanted to know more than this, this is the right place for you!

HEAT

Start this post with a question, ‘What is heat?’.

This is a big concept in Physics. What is heat?, What is the speed of heat?, What are the simple rules of thermodynamics?

Heat is well a type of energy which is calculated in a measures called ‘Joules’ or ‘J‘ which is a measure named by English scientist, James Prescott Joule. $\rm J = {}\rm \frac{kg \cdot m^2}{s^2} = N \cdot m = \rm Pa \cdot m^3={}\rm W \cdot s = C \cdot V$

kg- kilogram, m- metre, s- second, N- Newton, Pa- Pascal, W- Watt, V- Volt, C – Coulomb

James Prescott Joule (1818-1889) is also known for discovering the caloric theory. It states that heat contains a substance called caloric which travels from a hotter body to a cooler body. But, before this was discovered, there was another theory Phlogistion theory (φλογιστόν (phlogistón)-burning up, φλοξ (phlox)- flame) which involved a substance called phlogiston which contains inflammable substances and is released on combustion (burning). German physicist, Johanne Joachim Becher in 1667, created a book called Physica Subterranea which mentioned the phlogiston theory for the first time. The device used to measure the caloric in a particle of heat was called calorimeter. Thus, Joule could create the first joule-measuring machine.

THERMODYNAMICS

Thermodynamics is a branch of physics which deals with heat and temperature and their link to energy. Thermodynamics has 4 laws, namely –

ZEROTH LAW OF THERMODYNAMICS

“If 2 systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other”

Here, there is a system of thermodynamics mentioned here called a thermodynamic system. A thermodynamic system is a macroscopic volume in space, which has walls and surroundings, undergoes thermodynamic processes according to the principles of thermodynamics and it is a physical system which can be described by thermodynamic measures like entropy, internal energy, pressure, etc. Thermodynamic systems are in thermal equilibrium with each other if the natural molecular thermal energy is exchanged without causing a net change in the energy exchange. It acts more like a catalyst.

FIRST LAW OF THERMODYNAMICS

The increase in internal energy of a closed system is equal to the difference of the heat supplied to the system and the work done by it”

The thermodynamical potential of internal energy is mentioned. The increase of internal energy in the closed thermodynamic system can be explained in an equation like this- (W-work, Q- heat supplied in joules)-

ΔU = Q – W

SECOND LAW OF THERMODYNAMICS

“Heat cannot spontaneously flow from a colder location to a hotter location”

We have explained this in caloric theory. This net transfer of heat increases the entropy of heat. The equation for increase in entropy is shown below.

THIRD LAW OF THERMODYNAMICS

“As a system approaches absolute zero, the entropy of the system approaches a minimum value”

Absolute zero is a temperature that is equal to -273.15°C or is equal to 0 Kelvin. When a thermodynamic system reaches this value, the entropy of heat reaches a minimum value which decreases the net transfer of heat from a hotter object to a colder object.

SYSTEM MODELS

The thermodynamic system has many types of models, namely-

open, closed, adiabatically isolated, purely diathermically isolated, isolated

There are 5 basises on which the thermodynamic systems are classified, permeable to matter, permeable to energy but not to matter, adiabatic, adynamic and impermeable to matter, and isolation.

POTENTIALS

In thermodynamics, there are 5 thermodynamic potentials.(T– temperature, p-pressure, V-volume, μ- chemical potential, N– number of particles in a system, i-count of particle types in the system) Namely,

Internal energy

It is the energy contained within a thermodynamic system, which excludes the potential and kinetic energy in the system. It is denoted as U. $\int ( T dS - p dV + \sum_i \mu_i dN_i )$

Helmholtz free energy

It is a measure of the useful work obtainable from a closed thermodynamic system at a constant temperature. It is denoted as F $U-TS$

Enthalpy

Enthalpy is a potential shown by H. Enthalpy is simply written as- $H = U + pV$

Gibbs’ free energy

Gibbs’ free energy is a potential used to measure the usefulness or the process initiating work, obtainable from a thermodynamic system at a constant temperature and pressure. It is denoted by G. $U+pV-TS$

Landau (Grand) potential

It is the characteristic state function for the grand canonical ensemble, is a type of statistical ensemble (an idealization which has a many virtual copies of a thermodynamic system, considered all at once, each copy representing just the state of what the real system might be in) which is used to represent the possible states of a mechanical system of particles that is being maintained in thermal equilibrium with a reservoir. It is defined by- $

\Phi_{G} \ \stackrel{\mathrm{def}}{=}\ U - T S - \mu N

$

HEAT TRANSFERS

Mainly, there are 3 types of heat transfer, conduction, convection and radiation. Let’s discuss these types one by one-

CONDUCTION

Conduction is a type of heat transfer which can be done through certain types of metals only. These are called conducting materials. Thermal conduction is shown by the constant, k. The heat travels through the atoms of the metal which causes the atoms to vibrate. The vibration will continue on till the end of the metal. It is mostly defined by the equation-

k=W·1/m·1/K

W- watt, m-meter, K-kelvin

According to Fourier’s law- $\overrightarrow{q} = - k {\nabla} T$ $\overrightarrow{q}$ is the heat flux density, $-\nabla T$ is the temperature gradient

and when it is partially differentiated by x, $q_x = - k \frac{d T}{d x}$

and, $\frac{\partial Q}{\partial t} = -k \oint_S{{\nabla} T \cdot \,\overrightarrow{dA}}$ $\big. \frac{\partial Q}{\partial t}\big.$ is the amount of heat transferred per unit time in W, $\overrightarrow{dA}$ is an oriented surface in m2

CONVECTION

Convection also known as convective circulation is a process of heat transfer where the heat is transferred through the movement of fluids. There are two types of convection,

Natural convection where the fluids move due to the buoyancy forces (floating forces) which is caused by the density variations due to variation of thermal temperature in fluid. When the fluid comes in contact with a hot surface, the molecules will spread out and scatter due to the heat, making it less dense. So, the fluid is displaced and the cooler fluid which have molecules rigid, go down. This is a phenomenon called the convection/Bénard cell which happens due to the density differences between a body, liquid or gas .  Hence, the Second law of thermodynamics is fulfilled. The best example is sea breeze where the cold air from the sea goes to a hotter surface (land) and rises. Then, the fluid becomes cooler and sinks to the sea.

Forced convection where a fluid is forced to make a convection current using modern-day technology like fans, A/C’s etc.

Here also, the Fourier series applies. The solution to the Fourier series was created by Isaac Newton called the ‘Newton’s law of cooling’ ${ \frac{d Q}{d t} = h \cdot A \cdot ( T(t)-T_{\text{env}}) = h \cdot A \Delta T(t)\quad }$

is the heat transfer coefficient, is the heat transfer surface area, $T_{\text{env}}$ is the environment temperature

Isaac Newton gave a solution to this equation Rayleigh-Bénard convection is a natural convection which happens due to the fluid developing a continuous pattern of Bénard cells. Gravity and Buoyancy are the main reasons for this phenomena.

Radiation is a type of energy transfer where the energy transfers in the form of waves or particles. This type of energy transfer is divided into mainly types.

Ionizing radiation is a type of radiation which is done by highly-sufficient radiation to ionize atoms. In short, ionization is a process where the atom acquires a negative or positive charge so that an ion is created. Here, there is lower-energy damage actions happening like breaking chemical bonds between atoms. This ionization is done by knocking out electrons from the shell. If the source of the ionizing radiation comes from a radioactive material or nuclear processes, there is particle radiation (Radiation of energy by means of fast-moving subatomic particles). There are many types of ionizing radiation such as X-ray, Ultraviolet, γ (photon size-3×10−11 meters), β (emission of electrons and positrons), and α (helium 2 proton, 2 neutron).

Non-ionizing radiation has very small kinetic energy particles to do the ionization process. The main type of non-ionizing radiation is electromagnetic radiation which have the photons having only the sufficient energy to change the rotation, vibration, etc. A very common phenomena due to electromagnetic radiation is the spectrum. This observation first discovered by the ancient Greeks shows that white light comprises of the colors of the rainbow (VIBGYOR).

Information notes-

Ionization is another important topic which we need to learn if we want to learn all about radiation. We gave a simple definition of it. Another big question is the production of ions. They are produced by supplying sufficient energy to an electron which is bound to some other charged substance (positrons, ions, etc.). The amount of this energy is known as the ionization potential. Adiabatic ionization is a process where an electron is removed from/added to the molecule or atom in its lowest energy state and replaced with an ion in its lowest energy state. Townsend avalanche is a very good example of ionization where an electron is replaced with a positive ion.

Tunnel ionization is an ionization which happens due to quantum tunneling. But, unlike normal ionization, this allows the electron to just pass through the potential barrier instead of having sufficient energy. The probability of this type of passage ‘exponentially’ depends on its width. A high energy electron passing can make through it easily but the barrier becomes thinner. This can be done when an atom/molecule interacts with the near infrared strong near laser pauses. This process is understood as a  process which has a bounded electron, through the absorption of more than 1 photon from the laser field is ionized. This big picture is called multiphoton ionization (MPI). The rate of MPI on an atom which has ionization potential $E_i$ in a linearly polarized laser with angular frequency $\omega$ $\gamma= \frac{\omega F}{\sqrt {2E_i}}$ is the Keldysh (Leonid Keldysh) adiabaticity parameter, $n^{*}=\sqrt{2E_i}/Z^{2}$is the peak electric field of laser, $l^{*}=n^{*}-1$, and the coefficients of this equation – $f_{lm}= \frac{(2l+1)(l+|m|)^{!}}{2^{m}|m|^{!}(l-|m|)^{!}}$ $g(\gamma)=\frac{3}{2\gamma} (1+\frac{1}{2\gamma^{2}}\sinh^{-1}(\gamma)-\frac{\sqrt{1+\gamma^{2}}}{2\gamma})$ $|C_{n^{*}l^{*}}|^{2}= \frac{2^{2n^{*}}}{n^{*}\Gamma(n^{*}+l^{*}+1)\Gamma(n^{*}l^{*})}$ $A_{m}(\omega, \gamma)=\frac{4}{3\pi}\frac{1}{|m|^{!}}\frac{\gamma^{2}}{1+\gamma^{2}}\sum_{n>v}^{\infty}w_{m}(\sqrt{\frac{2\gamma}{\sqrt{1+\gamma^{2}}}(n-v)}e^{-(n-v)\alpha(\gamma)})$, and also, $w_{m}(x)=e^{-x^{2}}\int_0^x (x^2-y^2)^m e^{y^2}\,dy$ $\alpha(\gamma)= 2(\sinh^{-1}(\gamma)-\frac{\gamma}{\sqrt{1+\gamma^{2}}})$ $v= \frac{E_i}{\omega}(1+\frac{2}{\gamma^{2}})$
The equation is- $W_{PPT}=|C_{n^{*}l^{*}}|^{2}\sqrt{\frac{6}{\pi}}f_{lm}E_{i}(2(2E_i)^{\frac{3}{2}}/F)^{n^{*2}-|m|-3/2}(1+\gamma)^{2})^{|m/2|+3/4}A_{m}(\omega, \gamma)e^{-(2(2E_i)^{\frac{3}{2}}/F)g(\gamma)}$

Like that, we have the quasi-static tunnel ionization shown by the ADK model. It is shown by- $W_{ADK}=|C_{n^{*}l^{*}}|^{2}\sqrt{\frac{6}{\pi}}f_{lm}E_{i}(2(2E_i)^{\frac{3}{2}}/F)^{n^{*2}-|m|-3/2}e^{-(2(2E_i)^{\frac{3}{2}}/3F)}$

The normal ionization rate is shown by the PPT model- $U_p$ is the ponderomotive energy, $J_{n} (u, v)$ is the double Bessel function, $n_{i} =E_i/\omega$, $n_\mathrm{osc} = U_{p} / \omega$ $N=[n_i + n_\mathrm{osc}]$ is the min. no. of photons necessary to ionize the atoms, $p = \sqrt{ 2 \omega (n-n_\mathrm{osc}- n_i)}$, $n_{f} = 2 \sqrt { n_\mathrm{osc} / \omega} p cos(\theta)$ $\theta$ is the angle between momentum of electron, and the electric field of the laser, FT, 3D Fourier transform. $I_{KAR} = (\frac {2 Z^2}{n^2 F r})^n$ incorporates the Coulomb correction in the SFA model. The equation is- $W_{KRA}= \sum_{n=N}^{n=\infty} 2 \pi \omega^{2} p (n-n_\mathrm{osc})^2 \int d \Omega |FT (I_{KAR} \Psi ( \mathbf{r}))|^2 J_n^2 (n_f, \frac{n_\mathrm{osc}}{2})$

Thermal radiation is a non-ionizing radiation which is the emission of electromagnetic waves from all matter which has temperature greater the absolute zero temperature (0k). This has the kinetic energy of the random movement of molecules and atoms in matter which contain charged particles which also have kinetic interaction with each other. There are 4 main properties for thermal radiation-

1. Thermal radiation emitted by a body at different temperatures has a wide range of frequencies. This frequency distribution is shown by Planck’s law of black body radiation which states that if $B_\nu$ is the spectral radiance of the body, is the absolute temperature, kB is the Boltzmann constant, is the speed of light, and is the Planck constant- ( $\nu$ is the frequency) $B_\nu(\nu, T) = \frac{ 2 h \nu^3}{c^2} \frac{1}{e^\frac{h\nu}{k_\mathrm{B}T} - 1}$

and in terms of wavelength λ- $B_\lambda(\lambda, T) =\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{\lambda k_\mathrm{B}T}} - 1}$

There are different forms in which this can be portrayed.

2. The frequency range of the emitted radiation shifts to higher frequencies when the temperature of the emitter increases. The term white hot due to further heating of the emitter causing to change to high frequencies (color) and it appears white to the human eye. Even if an object is white hot, 99% of it will be infrared just like red hot substances. This is explained by the Wien’s displacement law which states that the black body radiation curve for different temperatures peaks at a wavelength (λ) inversely proportional to the temperature. That wavelength λmax is given by- $\lambda_\text{max} = \frac{b}{T}$

Where is the Wien’s displacement constant.

3. The total amount of radiation increases as the frequency rises. It grows as T4, where is the absolute temperature in Kelvin. The Stefan-Boltzmann law states that when $j^{\star}$ is the total energy radiated per unit surface area of a black body at all wavelengths per unit time, and σ is the Stefan Boltzmann constant- $j^{\star} = \sigma T^{4}.$

4. The rate of electromagnetic radiation emitted at a given frequency is proportional to the amount of absorption that it would experience by the source. This is very applicable to all properties of wave.

The radiative energy of heat (black body) is shown by an equation if $F_{A \rarr B}$ is the proportion of radiation which leaves and strikes B $\dot{Q}_{1 \rightarrow 2} = \sigma A_{1}F_{1 \rightarrow 2}(T_1^4-T_2^4) \!$

And, for a grey body, if $\epsilon$ (with a base A) are the emmisitives of a surface A, $\dot{Q}= \dfrac{\sigma(T_1^4-T_2^4)}{\dfrac{1-\epsilon_1}{A_1\epsilon_1}+ \dfrac{1}{A_1F_{1 \rightarrow 2}}+ \dfrac{1-\epsilon_2}{A_2\epsilon_2}}$

Combustion

Another important topic in Thermodynamics is combustion more commonly known as burning. It happens like this-

2H2(g)+O2(g)→2H2O (g)

Temperature

Well?! What’s temperature got to do with it? Everyone knows that ‘It is the scale of how hot or cold an object is.’. But, there are different scales of temperature that you need to know. The Fahrenheit scale (used in US, Carribean) was discovered by Polish born German physicist Daniel Gabriel Fahrenheit. Also, there was another physicist called Anders from Sweden (UPPSALAN) who created the Celsius scale. Yeah, Anders Celsius. He was an astronomer, mathematician and physicist. An English Physicist called William Thomson created the K scale. You’re right! The Kelvin created by the 1st  Baron Kelvin, William Thomson. This thing had kind of circulated worldwide, another Danish physicist called Ole Christensen Rømer created the Rømer scale which was inaccurate and Fahrenheit had to multiply those values by 4 to get the Fahrenheit measurements. But, Ole’s thermometer was better in case of safety.

ALCOHOL AND MERCURY THERMOMETERS

Before understanding this, you must understand the NFPA 704 Table for chemical substances. There are basises on which each chemical is categorized.

The BLUE Part of the diamond denotes health. The number indicates that there is no health hazard and as the number increases to 4, the health hazards increase.

The RED part denotes Inflammability. The number shows less inflammability and as it increases to 4, inflammability increases.

The YELLOW part denotes Instability/Reactivity. shows less instability and reactivity.

The WHITE part is a place for special symbols like-

OX- It is an oxidizer which makes it burn easily without air supply

W- The chemical reacts with water in a strange manner.

SA Simple Asphyxiant gas which reduces Oxygen and is non-toxic

COR- Corrosive

ACID, ALK (Alkaline) or BIO- Biological Hazard

POI- Poisonous

RAD or -Radioactive

CRYO- Cryogenic

Now, we learn about the chemicals we use in thermometers-

Mercury (Hg) or quicksilver is a chemical which has an NFPA 704 diamond of- It can measure temperatures from -38.87to 356.58C. It doesn’t stick to glass. It is silvery, grey in color. It stays at liquid for a wide range of temperatures, too. Another chemical used is Ethanol (C2H6O) which is a group of chemicals called alcohol. Although it can only measure upto 78 C, it can achieve even lower temperatures than alcohol like -115C. Alcohol is more commonly used in lab thermometers due to the red color. These lab thermometers can measure from -10C, to 110C. They consist of a large, thin glass tube which protects the alcohol. Whereas clinical thermometer can measure only from 94F (35C) to 106F (42C) and is mainly mercury. It has a construction called kink which is shaped like a circle and prevents the mercury from falling back down so, the people can measure it. A major invention by Sir James Dewar from Scotland was the thermos flask which prevented heat loss by all 3 modes of heat transfer. It has insulating materials for the outer cover to prevent conduction heat loss. There is also vacuum which does this. It has no air molecules in it so heat loss by convection is prevented. The highly reflective surface disables radiation. HOTS Question

1. Arun has 2 thermometers before him. He wants the safer one. Suggest and give reason. The thermometers are given below.  2. Explain more how the thermos flask prevents heat loss.

3. Why does your doctor say ‘Handle carefully‘ when he keeps a thermometer in your mouth to check fever?

4. Explain convection in sea breeze and the thermodynamic law that is observable in this.

5. Why do the different colors of spectrum go in different directions?