Friday, January 1, 2016

Hindu units of time

Vedic and Puranic texts describe units of Kala measurements, from Paramaṇu (about 17 microseconds) to Maha-Manvantara (311.04 trillion years). According to these texts, the creation and destruction of the universe is a cyclic process, which repeats itself forever. Each cycle starts with the birth and expansion (lifetime) of the universe equaling 311.04 trillion years, followed by its complete annihilation (which also prevails for the same duration). The current Universe was created in Padma kalpa, the last day Kalpa of 50th year of Brahma. This is 10.51 billion years ago. This figure is close to the modern estimated age of the universe in ≈ 13.79 billion years.
Various units of time are used across the Vedas, Puranas, Mahabharata, Suryasidhanta etc. Especially, Nimesha's multiple, it varies to 3, 10, 15, 18, 20, 27, 30, 45, 48, 60. At the lower end, these are pretty consistent. The Complete Hindu metrics of time (Kāla Vyavahāra) can be summarised as below.

Unit		Definition	Equivalence (Approx.)
Truti	त्रुटि		0.031 µs
Renu	रेणु	60 Truti	1.86 µs
Lava	लव	60 Renu	0.11 ms
Līkṣaka	लीक्षक	60 Lava	6.696 ms
Lipta	लिप्ता	60 Leekshaka	0.401 s
Vipala	विपल	60 Leekshaka	0.401 s
Pala	पल	60 Lipta	24.1056 s
Vighaṭi	विघटि
Vinādī	विनाडी
Ghaṭi	घटि	60 Vighaṭi	24 min
Nādī	नाडी
Danda	दण्ड
Muhūrta	मुहूर्त	2 Ghaṭi	48 min
Nakṣatra Ahorātram (Sidereal Day)	नक्षत्र अहोरात्रम्	60 Ghaṭī	24 h
Nakṣatra Ahorātram (Sidereal Day)	नक्षत्र अहोरात्रम्	30 Muhūrta	24 h

Alternate system
Unit	Definition	Equivalence (Approx.)
Truti		35.5 µs
Tatpara	100 Truti	3.55 ms
Nimesha	30 Tatpara	106.7 ms
Kāṣṭhā	30 Nimesha	3.2 s
Kalā	30 Kāṣṭhā	1.6 min
Muhūrta	30 Kalā	48 min
Nakṣatra Ahorātram (Sidereal Day)	30 Muhūrta	24 h

Small units of time used in the Vedas

Unit	Definition	Equivalence (Approx.)
Paramāṇu		26.3 µs
Aṇu	2 Paramāṇu	57.7 µs
Trasareṇu	3 Aṇu	158 µs
Truṭi	3 Trasareṇu	474 µs
Vedha	100 Truṭi	47.4 ms
Lava	3 Vedha	0.14 s
Nimeṣa	3 Lava	0.43 s
Kṣaṇa	3 Nimesha	1.28 s
Kāṣṭhā	5 Kṣaṇa	6.4 s
Laghu	15 Kāṣṭhā	1.6 min
Danda	15 Laghu	24 min
Muhūrta	2 Danda	48 min
Ahorātram (Day)	30 Muhūrta	24 h
Masa (Month)	30 Ahorātram	30 days
Ritu (Season)	2 Masa	2 months
Ayana	3 Rutu	6 months
Samvatsara (Year)	2 Ayana	360 days

Lunar metrics
A Tithi or lunar day is defined as the time it takes for the longitudinal angle between the moon and the Sun to increase by 12°.[1] Tithis begin at varying times of day and vary in duration from approximately 19 to approximately 26 hours.
A Paksa (also Pakṣa) or lunar fortnight consists of 15 tithis.
A Māsa or lunar month (approximately 29.5 days) is divided into 2 Pakṣas: the one between new moon and full moon (waxing) is called gaura or (bright) or Śukla Pakṣa; the one between full moon and new moon (waning) Kṛiṣhṇa (dark) paksha
A Ṛitu (or season) is 2 Māsa[2]
An Ayana is 3 Ṛitus
A year is two Ayanas [1][3]
Tropical metrics
A Yāma = 1/4 of a day (light) or night [ = 7½ Ghatis (घटि) = 3¾ Muhurtas = 3 Horas (होरा) ]
Four Yāmas make half of the day (either day or night)
Eight Yāmas make an Ahorātra (day + night)
An Ahorātra is a tropical day (Note: A day is considered to begin and end at sunrise, not midnight.)

Name		Definition	Equivalence
Yama	याम	¼ th of a day (light) or night	3 hours
Sāvana Ahorātram	सावन अहोरात्रम्	8 Yamas	1 Solar day

Reckoning of time among other entities.
Among the Pitṛs (forefather) in their Dimension (Loka):
1 human fortnight (15 days) = 1 day (light) or night of the Pitṛs.
1 human month (30 days) = 1 day (light) and night of the Pitṛs.
30 days of the Pitṛs = 1 month of the Pitṛs = (30 × 30 = 900 human days).
12 months of the Pitṛs = 1 year of the Pitṛs = (12 months of Pitṛs × 900 human days = 10800 human days).
The lifespan of the Pitṛs is 100 years of the Pitṛs (= 36,000 Pitṛ days = 1,080,000 human days = 3000 human years)
1 day of the Devas = 1 human year
1 month of the Devas = 30 days of the Devas
1 year of the Devas (1 divine year) = 12 months of the Devas
Among the Devas in their Dimension (Loka):
The life span of any Hindu deva spans nearly (or more than) 4.5 million years. Statistically, we can also look it as:
12000 Deva Years = Life Span of Devas = 1 Mahā-Yuga.
The Viṣṇu Purāṇa Time measurement section of the Viṣṇu Purāṇa Book I Chapter III explains the above as follows:
2 Ayanas (6-month periods, see above) = 1 human year or 1 day of the devas
4,000 + 400 + 400 = 4,800 divine years (= 1,728,000 human years) = 1 Satya Yuga
3,000 + 300 + 300 = 3,600 divine years (= 1,296,000 human years) = 1 Tretā Yuga
2,000 + 200 + 200 = 2,400 divine years (= 864,000 human years) = 1 Dvāpara Yuga
1,000 + 100 + 100 = 1,200 divine years (= 432,000 human years) = 1 Kali Yuga
12,000 divine year = 4 Yugas (= 4,320,000 human years) = 1 Mahā-Yuga (also is equaled to 12000 Daiva (divine) Yuga)
[2*12,000 = 24,000 divine year = 12000 revolutions of sun around its dual]

For Brahma in their Dimension (Loka):
1000 Mahā-Yugas = 1 Kalpa = 1 day (day only) of Brahma
(2 Kalpas constitute a day and night of Brahma, 8.64 billion human years)
30 days of Brahma = 1 month of Brahma (259.2 billion human years)
12 months of Brahma = 1 year of Brahma (3.1104 trillion human years)
50 years of Brahma = 1 Parārdha
2 parardhas = 100 years of Brahma = 1 Para = 1 Mahā-Kalpa (the lifespan of Brahma)(311.04 trillion human years)
One day of Brahma is divided into 1000 parts called charaṇas. The charaṇas are divided as follows:

The Four Yugas
4 charaṇas (1,728,000 solar years)	Satya Yuga
3 charaṇas (1,296,000 solar years)	Treta Yuga
2 charaṇas (864,000 solar years)	Dvapara Yuga
1 charaṇas (432,000 solar years)	Kali Yuga

The cycle repeats itself, so altogether there are 1,000 cycles of Mahā-Yuga in one day of Brahma.

One cycle of the above four Yugas is one Mahā-Yuga (4.32 million solar years)
as is confirmed by the Gītā Śloka 8.17 (statement) "sahasra-yuga-paryantam ahar yad brahmaṇo viduḥ rātriṁ yuga-sahasrāntāṁ te 'ho-rātra-vido janāḥ", meaning, a day of brahma is of 1000 Mahā-Yuga. Thus a day of Brahma, Kalpa, is of duration: 4.32 billion solar years. Two Kalpas constitute a day and night (Adhi Sandhi) of Brahma.
A Manvantara consists of 71 Mahā-Yuga (306,720,000 solar years). Each Manvantara is ruled by a Manu.
After each Manvantara follows one Saṃdhi Kāla of the same duration as a Kṛta Yuga (1,728,000 = 4 Charaṇas). (It is said that during a Saṃdhi Kāla, the entire earth is submerged in water.)
A Kalpa consists of a period of 4.32 Billion solar years followed by 14 Manvataras and Saṃdhi Kalas.
A day of Brahma equals

(14 times 71 Mahā-Yuga) + (15 × 4 Charaṇas)

= 994 Mahā-Yuga + (15 * 4800)

= 994 Mahā-Yuga + (72,000 years)[deva years] / 6 = 12,000[deva years] viz. one maha yuga.

= 994 Mahā-Yuga + 6 Mahā-Yuga

= 1,000 Mahā-Yuga

The Surya Siddhanta definition of timescales

The Surya Siddhanta [Chapter 14 Mānādhyāyah (मानाध्यायः)], documents a comprehensive model of nine divisions of time called māna (मान) which span from very small time units (Prāņa [प्राण] - 4 seconds) to very large time scales (Para [पर] - 311.04 Trillion solar years).

What does formation energy mean?

I had read some theoretical paper about defect in GaN. They analysis the defect using formation energies as a function of Fermi level. If the formation energy is too big, this defect is very unlikely to be seen. So what does this “formation energy” mean?
The formation energy is the energy required to produce a defect (a vacancy) into the perfect crystal structure.In the case of GaN, the formation energy of each defect is a function of the Fermi energy and the difference of the chemical potentials between Ga and N.
The formation energy is the cost of creating a defect into an otherwise perfect solid. For a vacancy it is calculated as the energy needed to remove an atom from the bulk and take it to infinity. Formation energies are very important because the concentration of defects in a solid in thermodynamic equilibrium depends exponentially on it.

What's the differences among the concepts: binding energy, cohesive energy and formation energy?

Formation energy is the change in energy when a material is formed from it's constituent elements in their reference states. For example, the formation energy of alumina (Al2O3) is the change in energy when fcc aluminum and O2 gas combine to make Al2O3.
Cohesive energy is the amount of energy it takes to break something up into isolated atoms. This is also called the atomization energy.
Binding energy in general means the amount of energy to split something up, and can mean different things depending on the context. For example, if you're talking about a molecule, it can refer to atomization energy.

Something to keep in mind regarding sign conventions is that formation energies are typically given as negative values, whereas cohesive energies and binding energies are typically positive values.

Standard enthalpy of formation

The standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the compound from its constituent elements, with all substances in their standard states at 1 atmosphere (1 atm or 101.3 kPa). Its symbol is ΔH_f^O or Δ_fH^O. The superscript theta (zero) on this symbol indicates that the process has occurred under standard conditions at the specified temperature (usually 25 degrees Celsius or 298.15 K). Standard states are as follows:

For a gas: the standard state is a pressure of exactly 1 atm
For a solute present in an ideal solution: a concentration of exactly one mole/liter (M) at a pressure of 1 atm
For a pure substance or a solvent in a condensed state (a liquid or a solid): the standard state is the pure liquid or solid under a pressure of 1 atm
For an element: the form in which the element is most stable under 1 atm of pressure. One exception is phosphorus, for which the most stable form at 1 atm is black phosphorus, but white phosphorus is chosen as the standard reference state for zero enthalpy of formation.^[1]

For example, the standard enthalpy of formation of carbon dioxide would be the enthalpy of the following reaction under the conditions above:

C_(s,graphite) + O_2(g) → CO_2(g)

All elements are written in their standard states, and one mole of product is formed. This is true for all enthalpies of formation.

The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol⁻¹), but also in calorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). In physics the energy per particle is often expressed inelectronvolts which corresponds to about 100 kJ mol⁻¹.

All elements in their standard states (oxygen gas, solid carbon in the form of graphite, etc.) have a standard enthalpy of formation of zero, as there is no change involved in their formation.

The formation reaction is a constant pressure and constant temperature process. Since the pressure of the standard formation reaction is fixed at 1 atm, the standard formation enthalpy or reaction heat is a function of temperature. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol ΔH_f298^O .

The standard enthalpy of formation is equivalent to the sum of many separate processes included in the Born-Haber cycle of synthesis reactions. For example, to calculate the standard enthalpy of formation of sodium chloride, we use the following reaction:

Na_(s) + (1/2)Cl_2(g) → NaCl_(s)

This process is made of many separate sub-processes, each with its own enthalpy. Therefore, we must take into account:

The standard enthalpy of atomization of solid sodium
The first ionization energy of gaseous sodium
The standard enthalpy of atomization of chlorine gas
The electron affinity of chlorine atoms
The lattice enthalpy of sodium chloride

The sum of all these values will give the standard enthalpy of formation of sodium chloride.

Additionally, applying Hess's Law shows that the sum of the individual reactions corresponding to the enthalpy change of formation for each substance in the reaction is equal to the enthalpy change of the overall reaction, regardless of the number of steps or intermediate reactions involved. This is because enthalpy is a state function. In the example above the standard enthalpy change of formation for sodium chloride is equal to the sum of the standard enthalpy change of formation for each of the steps involved in the process. This is especially useful for very long reactions with many intermediate steps and compounds.

Chemists may use standard enthalpies of formation for a reaction that is hypothetical. For instance carbon and hydrogen will not directly react to form methane, yet the standard enthalpy of formation for methane is determined to be −74.8 kJ mol⁻¹ from using other known standard enthalpies of reaction with Hess's law. That it is negative shows that the reaction, if it were to proceed, would be exothermic; that is, it is enthalpically more stable than hydrogen gas and carbon.

It is possible to predict heat of formations for simple unstrained organic compounds with the Heat of formation group additivity method.

(State: g = gaseous; l = liquid; s = solid; aq = aqueous)

Standard Enthalpies of Formation (at 25°C, 298 K)

Chemical Compound	Phase (matter)	Chemical formula	Δ H_f⁰ in kJ/mol
Acetone	l	C₃H₆O	−248.4
Acetylene	g	C₂H₂	+227.4
Ammonia	g	NH₃	−46.1
Ammonia (Ammonium Hydroxide)	aq	NH₃ (NH₄OH)	−80.8
Ammonium nitrate	s	NH₄NO₃	−365.6
Benzene	l	C₆H₆	+49.1
Bromine	l	Br₂	0
Bromine	g	Br₂	+31
Bromine	g	Br	+111.9
Calcium	s	Ca	0
Calcium carbonate	s	CaCO₃	−1207.6
Calcium oxide	s	CaO	−634.9
Carbon	s	C (graphite)	0
Carbon	s	C (diamond)	+1.88
Carbon dioxide	g	CO₂	−393.5
Carbon monoxide	g	CO	−110.5
Chlorine	g	Cl₂	0
Chlorine	g	Cl	+121.3
Copper(II) sulfate	aq	CuSO₄	−769.98
Ethane	g	C₂H₆	−84.68
Ethanol	l	C₂H₅OH	−277.6
Ethylene	g	C₂H₄	+52.4
Fluorine	g	F₂	0
Fluorine	g	F	+79.38
Glucose	s	C₆H₁₂O₆	−1273.3
Hydrogen	g	H₂	0
Hydrogen bromide	g	HBr	−36.3
Hydrogen chloride	g	HCl	−92.3
Hydrogen fluoride	g	HF	−273.3
Iodine	s	I₂	0
Iodine	g	I₂	+62
Isopropanol	g	C₃H₇OH	−318.1
Methane	g	CH₄	−74.87
Methanol	l	CH₃OH	−238.6
Nitric oxide	g	NO	+91.3
Nitrogen	g	N₂	0
Nitrogen dioxide	g	NO₂	+33.2
Oxygen	g	O₂	0
Ozone	g	O₃	+142.7
Propane	g	C₃H₈	−103.85
Silica	s	SiO₂	−911
Silver	s	Ag	0
Silver chloride	s	AgCl	−127.0
Sodium	s	Na	0
Sodium	g	Na	+107.5
Sodium bicarbonate	s	NaHCO₃	−950.8
Sodium carbonate	s	Na₂CO₃	−1131
Sodium chloride (table salt)	aq	NaCl	−407
Sodium chloride (table salt)	s	NaCl	−411.12
Sodium chloride (table salt)	l	NaCl	−385.92
Sodium chloride (table salt)	g	NaCl	−181.42
Sodium hydroxide	aq	NaOH	−470.1
Sodium hydroxide	s	NaOH	−426.7
Sodium nitrate	aq	NaNO₃	−446.2
Sodium nitrate	s	NaNO₃	−424.8
Sucrose	s	C₁₂H₂₂O₁₁	−2226.1
Sulfur (monoclinic)	s	S₈	0.3
Sulfur (rhombic)	s	S₈	0
Sulfur dioxide	g	SO₂	−296.8
Sulfur trioxide	g	SO₃	−395.7
Sulfuric acid	l	H₂SO₄	−814
Water	l	H₂O	−285.8
Water vapor	g	H₂O	−241.82
Zinc sulfate	s	ZnSO₄	−980.14

Saturday, December 19, 2015

Boltzmann constant

Values of k	Units	Comments
1.38064852(79)×10⁻²³	J/K	SI units, 2010 CODATA value, J/K = m²⋅kg/(s²⋅K) in SI base units^[1]
8.6173324(78)×10⁻⁵	eV/K	2010 CODATA value^[1] 1 electronvolt = 1.602176565(35)×10⁻¹⁹ J^[1] 1/k = 11604.519(11) K/eV
2.0836618(19)×10¹⁰	Hz/K	2010 CODATA value^[1] 1 Hz⋅h = 6.62606957(29)×10⁻³⁴ J^[1]
3.1668114(29)×10⁻⁶	E_H/K	E_H = 2R_∞hc = 4.35974434(19)×10⁻¹⁸ J^[1] = 6.579683920729(33) Hz⋅h^[1]
1.0	Atomic units	by definition
1.38064852(79)×10⁻¹⁶	erg/K	CGS system, 1 erg = 1×10⁻⁷ J
3.2976230(30)×10⁻²⁴	cal/K	1 steam table calorie = 4.1868 J
1.8320128(17)×10⁻²⁴	cal/°R	1 degree Rankine = 5/9 K
5.6573016(51)×10⁻²⁴	ft lb/°R	1 foot-pound force = 1.3558179483314004 J
0.69503476(63)	cm⁻¹/K	2010 CODATA value^[1] 1 cm⁻¹⋅hc = 1.986445683(87)×10⁻²³ J
0.0019872041(18)	kcal/(mol⋅K)	per mole form often used in statistical mechanics—using thermochemical calorie = 4.184 joule
0.0083144621(75)	kJ/(mol⋅K)	per mole form often used in statistical mechanics
4.10	pN⋅nm	kT in piconewton nanometer at 24 °C, used in biophysics
−228.5991678(40)	dBW/(K⋅Hz)	in decibel watts, used in telecommunications (see Johnson–Nyquist noise)
1.442 695 041...	Sh	in shannons (logarithm base 2), used in information entropy (exact value 1/ln(2))
1	nat	in nats (logarithm base e), used in information entropy (see Planck units, below)

The Boltzmann constant (k_B or k), named after Ludwig Boltzmann, is a physical constant relating energy at the individual particle level with temperature. It is the gas constant R divided by the Avogadro constant N_A:

k={\frac {R}{N_{\text{A}}}}.\,

The Boltzmann constant has the dimension energy divided by temperature, the same as entropy. The accepted value in SI units is1.38064852(79)×10⁻²³ J/K.

Since k is a physical constant of proportionality between temperature and energy, its numerical value depends on the choice of units for energy and temperature. The small numerical value of the Boltzmann constant in SI units means a change in temperature by 1 K only changes a particle's energy by a small amount. A change of 1 °C is defined to be the same as a change of 1 K. The characteristic energy kT is a term encountered in many physical relationships.

THOUGHT CRACKERS

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