Coin problem - Revision history

en>Joel B. Lewis: This is an article about math, not football -- this sort of arcana is totally off-point.

2014-12-13T14:14:42Z

This is an article about math, not football -- this sort of arcana is totally off-point.

← Older revision		Revision as of 16:14, 13 December 2014
Line 1:		Line 1:
	~~{{Continuum mechanics}}~~		The author's name is Christy Brookins. Some time in the past he selected to reside in North Carolina and he doesn't strategy on changing it. Invoicing is what I do. To play domino is something I truly enjoy doing.<br><br>Have a look at my website psychic phone ([http://medialab.zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- click through the following website])
	The '''combined gas law''' is a [[gas law]] which combines [[Charles's law]], [[Boyle's law]], and [[Gay-Lussac's law#Pressure-temperature law\|Gay-Lussac's law]]. There is no official founder for this law; it is merely an amalgamation of the three previously discovered laws. These laws each relate one [[Conjugate variables (thermodynamics)\|thermodynamic variable]] to another mathematically while holding everything else constant. Charles's ~~law states that volume and temperature are directly proportional to each other as long as pressure~~ is ~~held constant~~. Boyle's law asserts that pressure and volume are inversely proportional to each other at fixed temperature. Finally, Gay-Lussac's law introduces a direct proportionality between temperature and pressure as long as it is at a constant volume. The inter-dependence of these variables is shown in the combined gas law, which clearly states that:

	~~{{cquote\|The ratio between the pressure-volume product and the temperature of a system remains constant.}}~~

	~~This can be stated mathematically as~~

	~~:<math> \qquad \frac {pV}{T}= k </math>~~

	~~where:~~

	~~:''p'' is the [[pressure]]~~
	~~:''V '' is the [[volume]]~~
	~~:''T '' is the [[thermodynamic temperature\|temperature]] measured~~ in ~~[[kelvin]]~~
	~~:''k '' is a constant (with units of energy divided by temperature).~~

	~~For comparing~~ the ~~same substance under two different sets of conditions, the law can be written as:~~

	~~:<math> \qquad \frac {p_1V_1}{T_1}= \frac {p_2V_2}{T_2} </math>~~

	~~The addition of [[Avogadro's law]]~~ to ~~the combined gas law yields the [[ideal gas law]].~~

	~~==Derivation from the Gas Laws==~~

	~~{{main\|Gas Laws}}~~

	~~[[Boyle's Law]] states that the pressure-volume product is constant:~~
	~~:<math>PV = k_1 \qquad (1)</math>~~
	~~[[Charles's Law]] shows that the volume is proportional to [[absolute temperature]]:~~
	~~:<math>V = k_2T \qquad (2)</math>~~
	~~[[Gay-Lussac's Law]] says that the pressure is proportional to the absolute temperature:~~
	~~:<math>P = k_3T \qquad (3)</math>~~
	~~where P is the [[pressure]], V the volume and T the absolute temperature and of an [[ideal gas]].~~

	~~By combining (1) and either of (2) or (3) we can gain a new equation with P, V and T. Equation (2) is used~~ in ~~this example,~~ and ~~the arbitrary subscript on the constant is dropped so that k = k<sub>2</sub>.~~
	~~<br><math>PV = k_2T \Rightarrow \frac{PV}{T} = k</math>~~

	~~Substituting in [[Avogadro~~'~~s Law]] yields the [[ideal gas equation]].~~

	~~==Physical Derivation==~~

	~~A derivation of the combined gas law using only elementary algebra can contain surprises. For example, starting from the three empirical laws~~
	~~:<math> P = k_v\, T \,\!</math> ............(1) Gay-Lussac's Law, volume assumed constant~~
	~~:<math> V = k_p T \,\!</math> ............(2) Charles's Law, pressure assumed constant~~
	~~:<math> P V = k_t \,\!</math> ............(3) Boyle's Law, temperature assumed constant~~
	~~where k<sub>v</sub>, k<sub>p</sub>, and k<sub>~~t~~</sub> are the constants, one can multiply the three together to obtain~~
	~~:<math> PVPV = k_v T k_p T k_t \,\!</math>~~
	~~Taking the square root of both sides and dividing by T appears to produce of the desired result~~
	~~:<math> \frac {PV}{T} = \sqrt{k_p k_v k_t} \,\!</math>~~
	~~However, if before applying the above procedure, one merely rearranges the terms in Boyle's Law, k<sub>t</sub> = P V, then after canceling and rearranging, one obtains~~
	~~:<math> \frac{k_t}{k_v k_p} = T^2 \,\!</math>~~
	~~which is not very helpful if not misleading.~~

	A physical derivation, longer but more reliable, begins by realizing that the constant volume parameter in Gay-Lussac's law will change as the system volume changes. At constant volume V<sub>1</sub> the law might appear P = k<sub>1</sub> T while at constant volume V<sub>2</sub> it ~~might appear P= k<sub>2</sub> T~~ .
	~~Denoting this "variable constant volume" by k<sub>v</sub>(V), rewrite the law as~~
	~~:<math> P = k_v(V) \,T \,\!</math> ............(4)~~
	~~The same consideration applies to the constant in Charles's law which may rewritten~~
	~~:<math> V = k_p(P) \,T \,\!</math> ............(5)~~

	~~In seeking to find k<sub>v</sub>(V), one should not unthinkingly eliminate T between (4) and (5) since P is varying in the former while it~~ is ~~assumed constant in the latter. Rather it should first be determined in~~ what ~~sense these equations are compatible with one another~~. To ~~gain insight into this, recall that any two variables determine the third. Choosing P and V to be independent we picture the T values forming a surface above the PV plane~~. ~~A definite V~~<~~sub~~>0<~~/sub~~> ~~and P<sub>0</sub> define a T<sub>0</sub>,~~ a ~~point on that surface. Substituting these values in (4) and (5), and rearranging yields~~
	~~:<math> T_0 = \frac{P_0}{k_v(V_0)} \quad and \quad T_0 = \frac{V_0}{k_p(P_0)} </math>~~
	~~Since these both describe what is happening~~ at ~~the same point on the surface the two numeric expressions can be equated and rearranged~~
	~~:<math> \frac{k_v~~(~~V_0)}{k_p(P_0)} = \frac{P_0}{V_0}\,\!</math>............(6)~~
	~~The k<sub>v</sub>(V<sub>0</sub>) and k<sub>p</sub>(P<sub>0</sub>)are the slopes of orthogonal lines through that surface point. Their ratio depends only on P<sub>0</sub> / V<sub>0</sub> at that point.~~

	~~Note that the functional form of (6) did not depend on the particular point chosen. The same formula would have arisen for any other combination of P and V values. Therefore one can write~~
	~~:<math> \frac{k_v(V)}{k_p(P)} = \frac{P}{V} \quad\forall P, \forall V</math>............(7)~~
	This says each point on the surface has it own pair of orthogonal lines through it, with their slope ratio depending only on that point. Whereas (6) is a relation between specific slopes and variable values, (7) is a relation between slope functions and function variables. It holds true for any point on the surface, i.e. for any and all combinations of P and V values. To solve this equation for the function k<sub>v</sub>(V) first separate the variables, V on the left and P on the right.
	:~~<math> V\,k_v(V) = P\,k_p(P) <~~/~~math>~~
	~~Choose any pressure P<sub>1<~~/~~sub>~~. ~~the right side evaluates to some arbitrary value, call it k<sub>arb</sub>~~.
	~~:<math> V\,k_v(V) = k_{\text{arb}} \,\!<~~/~~math>............(8)~~
	~~This particular equation must now hold true, not just for one value of V but for '''all''' values of V. The only definition of k<sub>v<~~/~~sub>(V) that guarantees this for all V and arbitrary k<sub>arb</sub> is~~
	~~:<math> k_v(V) = \frac{k_{\text{arb}}}{V} </math>............(9)~~
	~~which may be verified by substitution in (8).~~

	~~Finally substituting (9) in Gay~~-~~Lussac's law (4) and rearranging produces the combined gas law~~
	~~:<math> \frac{PV}{T} = k_{\text{arb}} \,\!</math>~~

	Note that Boyle's law was not used in this derivation but is easily deduced from the result. Generally any two of the three starting laws are all that is needed in this type of derivation – all starting pairs lead to the same combined gas law.<ref>A similar derivation, one starting from Boyle's law, may be found in Raff, pp. 14–15</ref>

	~~==Applications==~~
	~~The combined gas law can be used to explain the mechanics where pressure, temperature, and volume are affected. For example: air conditioners, refrigerators and the formation of clouds.~~

	~~==See also==~~
	~~{{portal\|Underwater diving}}~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==Sources==~~
	*Raff, Lionel. Principles of Physical Chemistry. New Jersey: Prentice-~~Hall 2001~~

	~~==External links==~~
	* [http://www.webqc.org/combined_gas_law.html Combined gas law calculator]
	* [http://chair.pa.msu.edu/applets/pvt/a.htm Interactive Java applet on the ~~combined gas law] by Wolfgang Bauer~~

	~~{{DEFAULTSORT:Combined Gas Law}}~~

	~~{{Diving medicine, physiology and physics}}~~

	~~[[Category:Gas laws]~~]

en>Bkell: /* McNugget numbers */ replaced "the largest non-McNugget number is infinite" with "there is no largest non-McNugget number"

2013-09-30T04:41:36Z

McNugget numbers: replaced "the largest non-McNugget number is infinite" with "there is no largest non-McNugget number"

← Older revision		Revision as of 06:41, 30 September 2013
Line 1:		Line 1:
	~~Andrew Simcox~~ is the ~~name his mothers~~ and ~~fathers gave him~~ and ~~he completely enjoys this name~~. ~~She works~~ as a ~~travel agent but soon she~~'ll be ~~on her own~~. ~~Alaska~~ is ~~exactly~~ where he's ~~usually been living~~. ~~She~~ is ~~truly fond of caving~~ but ~~she doesn~~'t have the ~~time lately~~.<br><br>~~Here~~ is ~~my web blog~~ - ~~clairvoyance [~~[http://~~afeen~~.~~fbho~~.~~net~~/v2/~~index~~.~~php?do=~~/~~profile-210~~/~~info~~/ ~~just click~~ the ~~next web site~~]]		{{Continuum mechanics}}
			The '''combined gas law''' is a [[gas law]] which combines [[Charles's law]], [[Boyle's law]], and [[Gay-Lussac's law#Pressure-temperature law\|Gay-Lussac's law]]. There is no official founder for this law; it is merely an amalgamation of the three previously discovered laws. These laws each relate one [[Conjugate variables (thermodynamics)\|thermodynamic variable]] to another mathematically while holding everything else constant. Charles's law states that volume and temperature are directly proportional to each other as long as pressure is held constant. Boyle's law asserts that pressure and volume are inversely proportional to each other at fixed temperature. Finally, Gay-Lussac's law introduces a direct proportionality between temperature and pressure as long as it is at a constant volume. The inter-dependence of these variables is shown in the combined gas law, which clearly states that:

			{{cquote\|The ratio between the pressure-volume product and the temperature of a system remains constant.}}

			This can be stated mathematically as

			:<math> \qquad \frac {pV}{T}= k </math>

			where:

			:''p'' is the [[pressure]]
			:''V '' is the [[volume]]
			:''T '' is the [[thermodynamic temperature\|temperature]] measured in [[kelvin]]
			:''k '' is a constant (with units of energy divided by temperature).

			For comparing the same substance under two different sets of conditions, the law can be written as:

			:<math> \qquad \frac {p_1V_1}{T_1}= \frac {p_2V_2}{T_2} </math>

			The addition of [[Avogadro's law]] to the combined gas law yields the [[ideal gas law]].

			==Derivation from the Gas Laws==

			{{main\|Gas Laws}}

			[[Boyle's Law]] states that the pressure-volume product is constant:
			:<math>PV = k_1 \qquad (1)</math>
			[[Charles's Law]] shows that the volume is proportional to [[absolute temperature]]:
			:<math>V = k_2T \qquad (2)</math>
			[[Gay-Lussac's Law]] says that the pressure is proportional to the absolute temperature:
			:<math>P = k_3T \qquad (3)</math>
			where P is the [[pressure]], V the volume and T the absolute temperature and of an [[ideal gas]].

			By combining (1) and either of (2) or (3) we can gain a new equation with P, V and T. Equation (2) is used in this example, and the arbitrary subscript on the constant is dropped so that k = k<sub>2</sub>.
			<br><math>PV = k_2T \Rightarrow \frac{PV}{T} = k</math>

			Substituting in [[Avogadro's Law]] yields the [[ideal gas equation]].

			==Physical Derivation==

			A derivation of the combined gas law using only elementary algebra can contain surprises. For example, starting from the three empirical laws
			:<math> P = k_v\, T \,\!</math> ............(1) Gay-Lussac's Law, volume assumed constant
			:<math> V = k_p T \,\!</math> ............(2) Charles's Law, pressure assumed constant
			:<math> P V = k_t \,\!</math> ............(3) Boyle's Law, temperature assumed constant
			where k<sub>v</sub>, k<sub>p</sub>, and k<sub>t</sub> are the constants, one can multiply the three together to obtain
			:<math> PVPV = k_v T k_p T k_t \,\!</math>
			Taking the square root of both sides and dividing by T appears to produce of the desired result
			:<math> \frac {PV}{T} = \sqrt{k_p k_v k_t} \,\!</math>
			However, if before applying the above procedure, one merely rearranges the terms in Boyle's Law, k<sub>t</sub> = P V, then after canceling and rearranging, one obtains
			:<math> \frac{k_t}{k_v k_p} = T^2 \,\!</math>
			which is not very helpful if not misleading.

			A physical derivation, longer but more reliable, begins by realizing that the constant volume parameter in Gay-Lussac's law will change as the system volume changes. At constant volume V<sub>1</sub> the law might appear P = k<sub>1</sub> T while at constant volume V<sub>2</sub> it might appear P= k<sub>2</sub> T .
			Denoting this "variable constant volume" by k<sub>v</sub>(V), rewrite the law as
			:<math> P = k_v(V) \,T \,\!</math> ............(4)
			The same consideration applies to the constant in Charles's law which may rewritten
			:<math> V = k_p(P) \,T \,\!</math> ............(5)

			In seeking to find k<sub>v</sub>(V), one should not unthinkingly eliminate T between (4) and (5) since P is varying in the former while it is assumed constant in the latter. Rather it should first be determined in what sense these equations are compatible with one another. To gain insight into this, recall that any two variables determine the third. Choosing P and V to be independent we picture the T values forming a surface above the PV plane. A definite V<sub>0</sub> and P<sub>0</sub> define a T<sub>0</sub>, a point on that surface. Substituting these values in (4) and (5), and rearranging yields
			:<math> T_0 = \frac{P_0}{k_v(V_0)} \quad and \quad T_0 = \frac{V_0}{k_p(P_0)} </math>
			Since these both describe what is happening at the same point on the surface the two numeric expressions can be equated and rearranged
			:<math> \frac{k_v(V_0)}{k_p(P_0)} = \frac{P_0}{V_0}\,\!</math>............(6)
			The k<sub>v</sub>(V<sub>0</sub>) and k<sub>p</sub>(P<sub>0</sub>)are the slopes of orthogonal lines through that surface point. Their ratio depends only on P<sub>0</sub> / V<sub>0</sub> at that point.

			Note that the functional form of (6) did not depend on the particular point chosen. The same formula would have arisen for any other combination of P and V values. Therefore one can write
			:<math> \frac{k_v(V)}{k_p(P)} = \frac{P}{V} \quad\forall P, \forall V</math>............(7)
			This says each point on the surface has it own pair of orthogonal lines through it, with their slope ratio depending only on that point. Whereas (6) is a relation between specific slopes and variable values, (7) is a relation between slope functions and function variables. It holds true for any point on the surface, i.e. for any and all combinations of P and V values. To solve this equation for the function k<sub>v</sub>(V) first separate the variables, V on the left and P on the right.
			:<math> V\,k_v(V) = P\,k_p(P) </math>
			Choose any pressure P<sub>1</sub>. the right side evaluates to some arbitrary value, call it k<sub>arb</sub>.
			:<math> V\,k_v(V) = k_{\text{arb}} \,\!</math>............(8)
			This particular equation must now hold true, not just for one value of V but for '''all''' values of V. The only definition of k<sub>v</sub>(V) that guarantees this for all V and arbitrary k<sub>arb</sub> is
			:<math> k_v(V) = \frac{k_{\text{arb}}}{V} </math>............(9)
			which may be verified by substitution in (8).

			Finally substituting (9) in Gay-Lussac's law (4) and rearranging produces the combined gas law
			:<math> \frac{PV}{T} = k_{\text{arb}} \,\!</math>

			Note that Boyle's law was not used in this derivation but is easily deduced from the result. Generally any two of the three starting laws are all that is needed in this type of derivation – all starting pairs lead to the same combined gas law.<ref>A similar derivation, one starting from Boyle's law, may be found in Raff, pp. 14–15</ref>

			==Applications==
			The combined gas law can be used to explain the mechanics where pressure, temperature, and volume are affected. For example: air conditioners, refrigerators and the formation of clouds.

			==See also==
			{{portal\|Underwater diving}}

			==Notes==
			{{reflist}}

			==Sources==
			*Raff, Lionel. Principles of Physical Chemistry. New Jersey: Prentice-Hall 2001

			==External links==
			* [http://www.webqc.org/combined_gas_law.html Combined gas law calculator]
			* [http://chair.pa.msu.edu/applets/pvt/a.htm Interactive Java applet on the combined gas law] by Wolfgang Bauer

			{{DEFAULTSORT:Combined Gas Law}}

			{{Diving medicine, physiology and physics}}

			[[Category:Gas laws]]

en>Staszek Lem at 01:57, 29 August 2012

2012-08-29T01:57:08Z

New page

Andrew Simcox is the name his mothers and fathers gave him and he completely enjoys this name. She works as a travel agent but soon she'll be on her own. Alaska is exactly where he's usually been living. She is truly fond of caving but she doesn't have the time lately.<br><br>Here is my web blog - clairvoyance [[http://afeen.fbho.net/v2/index.php?do=/profile-210/info/ just click the next web site]]