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Indifference curve
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An example of an indifference map with three indifference curves represented
In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is indifferent. That is, the consumer has no preference for one combination or bundle of goods over a different combination on the same curve. One can also refer to each point on the indifference curve as rendering the same level of utility (satisfaction) for the consumer. In other words, an indifference curve is the locus of various points showing different combinations of two goods providing equal utility to the consumer. Utility is then a device to represent preferences rather than something from which preferences come.[1] The main use of indifference curves is in the representation of potentially observable demand patterns for individual consumers over commodity bundles.[2]
There are infinitely many indifference curves: one passes through each combination. A collection of (selected) indifference curves, illustrated graphically, is referred to as an indifference map.
Contents
[hide]
1 History
2 Map and properties of indifference curves
3 Assumptions of consumer preference theory
3.1 Application
3.2 Examples of indifference curves
4 Preference relations and utility
4.1 Preference relations
4.2 Formal link to utility theory
4.3 Examples
4.3.1 Linear utility
4.3.2 Cobb-Douglas utility
4.3.3 CES utility
4.3.4 Biology
5 See also
6 Notes
7 References
8 Further reading
9 External links
History[edit]
The theory of indifference curves was developed by Francis Ysidro Edgeworth, who explained in his 1881 book the mathematics needed for their drawing;[3] later on, Vilfredo Pareto was the first author to actually draw these curves, in his 1906 book.[4][5] The theory can be derived from William Stanley Jevons' ordinal utility theory, which posits that individuals can always rank any consumption bundles by order of preference.[6]
Map and properties of indifference curves[edit]
An example of how indifference curves are obtained as the level curves of a utility function
A graph of indifference curves for several utility levels of an individual consumer is called an indifference map. Points yielding different utility levels are each associated with distinct indifference curves and these indifference curves on the indifference map are like contour lines on a topographical map. Each point on the curve represents the same elevation. If you move "off" an indifference curve traveling in a northeast direction (assuming positive marginal utility for the goods) you are essentially climbing a mound of utility. The higher you go the greater the level of utility. The non-satiation requirement means that you will never reach the "top," or a "bliss point," a consumption bundle that is preferred to all others.
Indifference curves are typically[vague] represented[clarification needed] to be:
Defined only in the non-negative quadrant of commodity quantities (i.e. the possibility of having negative quantities of any good is ignored).
Negatively sloped. That is, as quantity consumed of one good (X) increases, total satisfaction would increase[clarification needed] if not offset by a decrease in the quantity consumed of the other good (Y). Equivalently, satiation, such that more of either good (or both) is equally preferred to no increase, is excluded.[clarification needed] (If utility U = f(x, y), U, in the third dimension, does not have a local maximum for any x and y values.)[clarification needed] The negative slope of the indifference curve reflects the assumption of the monotonicity of consumer's preferences, which generates monotonically increasing utility functions, and the assumption of non-satiation (marginal utility for all goods is always positive); an upward sloping indifference curve would imply that a consumer is indifferent between a bundle A and another bundle B because they lie on the same indifference curve, even in the case in which the quantity of both goods in bundle B is higher. Because of monotonicity of preferences and non-satiation, a bundle with more of both goods must be preferred to one with less of both, thus the first bundle must yield a higher utility, and lie on a different indifference curve at a higher utility level. The negative slope of the indifference curve implies that the marginal rate of substitution is always positive;
Complete, such that all points on an indifference curve are ranked equally preferred and ranked either more or less preferred than every other point not on the curve. So, with (2), no two curves can intersect (otherwise non-satiation would be violated).
Transitive with respect to points on distinct indifference curves. That is, if each point on I2 is (strictly) preferred to each point on I1, and each point on I3 is preferred to each point on I2, each point on I3 is preferred to each point on I1. A negative slope and transitivity exclude indifference curves crossing, since straight lines from the origin on both sides of where they crossed would give opposite and intransitive preference rankings.
(Strictly) convex. With (2), convex preferences[clarification needed] imply that the indifference curves cannot be concave to the origin, i.e. they will either be straight lines or bulge toward the origin of the indifference curve. If the latter is the case, then as a consumer decreases consumption of one good in successive units, successively larger doses of the other good are required to keep satisfaction unchanged.
Assumptions of consumer preference theory[edit]
Preferences are complete. The consumer has ranked all available alternative combinations of commodities in terms of the satisfaction they provide him.
Assume that there are two consumption bundles A and B each containing two commodities x and y. A consumer can unambiguously determine that one and only one of the following is the case:
A is preferred to B, formally written as A p B[7]
B is preferred to A, formally written as B p A[7]
A is indifferent to B, formally written as A I B[7]
This axiom precludes the possibility that the consumer cannot decide,[8] It assumes that a consumer is able to make this comparison with respect to every conceivable bundle of goods.[7]
Preferences are reflexive
This means that if A and B are identical in all respects the consumer will recognize this fact and be indifferent in comparing A and B
A = B ⇒ A I B[7]
Preferences are transitive[nb 1]
If A p B and B p C, then A p C.[7]
Also if A I B and B I C, then A I C.[7]
This is a consistency assumption.
Preferences are continuous
If A is preferred to B and C is sufficiently close to B then A is preferred to C.
A p B and C → B ⇒ A p C.
"Continuous" means infinitely divisible - just like there are infinitely many numbers between 1 and 2 all bundles are infinitely divisible. This assumption makes indifference curves continuous.
Preferences exhibit strong monotonicity
If A has more of both x and y than B, then A is preferred to B.
This assumption is commonly called the "more is better" assumption.
An alternative version of this assumption requires that if A and B have the same quantity of one good, but A has more of the other, then A is preferred to B.
It also implies that the commodities are good rather than bad. Examples of bad commodities can be disease, pollution etc. because we always desire less of such things.
Indifference curves exhibit diminishing marginal rates of substitution
The marginal rate of substitution tells how much 'y' a person is willing to sacrifice to get one more unit of 'x'.[clarification needed]
This assumption assures that indifference curves are smooth and convex to the origin.
This assumption also set the stage for using techniques of constrained optimization because the shape of the curve assures that the first derivative is negative and the second is positive.
Another name for this assumption is the substitution assumption. It is the most critical assumption of consumer theory: Consumers are willing to give up or trade-off some of one good to get more of another. The fundamental assertion is that there is a maximum amount that "a consumer will give up, of one commodity, to get one unit of another good, in that amount which will leave the consumer indifferent between the new and old situations"[9] The negative slope of the indifference curves represents the willingness of the consumer to make a trade off.[9]
Application[edit]
To maximise utility, a household should consume at (Qx, Qy). Assuming it does, a full demand schedule can be deduced as the price of one good fluctuates.
Consumer theory uses indifference curves and budget constraints to generate consumer demand curves. For a single consumer, this is a relatively simple process. First, let one good be an example market e.g., carrots, and let the other be a composite of all other goods. Budget constraints give a straight line on the indifference map showing all the possible distributions between the two goods; the point of maximum utility is then the point at which an indifference curve is tangent to the budget line (illustrated). This follows from common sense: if the market values a good more than the household, the household will sell it; if the market values a good less than the household, the household will buy it. The process then continues until the market's and household's marginal rates of substitution are equal.[10] Now, if the price of carrots were to change, and the price of all other goods were to remain constant, the gradient of the budget line would also change, leading to a different point of tangency and a different quantity demanded. These price / quantity combinations can then be used to deduce a full demand curve.[10] A line connecting all points of tangency between the indifference curve and the budget constraint is called the expansion path.[11]
Examples of indifference curves[edit]
Figure 1: An example of an indifference map with three indifference curves represented
Figure 2: Three indifference curves where Goods X and Y are perfect substitutes. The gray line perpendicular to all curves indicates the curves are mutually parallel.
Figure 3: Indifference curves for perfect complements X and Y. The elbows of the curves are collinear.
In Figure 1, the consumer would rather be on I3 than I2, and would rather be on I2 than I1, but does not care where he/she is on a given indifference curve. The slope of an indifference curve (in absolute value), known by economists as the marginal rate of substitution, shows the rate at which consumers are willing to give up one good in exchange for more of the other good. For most goods the marginal rate of substitution is not constant so their indifference curves are curved. The curves are convex to the origin, describing the negative substitution effect. As price rises for a fixed money income, the consumer seeks the less expensive substitute at a lower indifference curve. The substitution effect is reinforced through the income effect of lower real income (Beattie-LaFrance). An example of a utility function that generates indifference curves of this kind is the Cobb-Douglas function U ( x , y ) = x α y 1 − α , 0 ≤ α ≤ 1 {\displaystyle \scriptstyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha },0\leq \alpha \leq 1} . The negative slope of the indifference curve incorporates the willingness of the consumer to make trade offs.[9]
If two goods are perfect substitutes then the indifference curves will have a constant slope since the consumer would be willing to switch between at a fixed ratio. The marginal rate of substitution between perfect substitutes is likewise constant. An example of a utility function that is associated with indifference curves like these would be U ( x , y ) = α x + β y {\displaystyle \scriptstyle U\left(x,y\right)=\alpha x+\beta y} .
If two goods are perfect complements then the indifference curves will be L-shaped. Examples of perfect complements include left shoes compared to right shoes: the consumer is no better off having several right shoes if she has only one left shoe - additional right shoes have zero marginal utility without more left shoes, so bundles of goods differing only in the number of right shoes they include - however many - are equally preferred. The marginal rate of substitution is either zero or infinite. An example of the type of utility function that has an indifference map like that above is the Leontief function: U ( x , y ) = min { α x , β y } {\displaystyle \scriptstyle U\left(x,y\right)=\min\{\alpha x,\beta y\}} .
The different shapes of the curves imply different responses to a change in price as shown from demand analysis in consumer theory. The results will only be stated here. A price-budget-line change that kept a consumer in equilibrium on the same indifference curve:
in Fig. 1 would reduce quantity demanded of a good smoothly as price rose relatively for that good.
in Fig. 2 would have either no effect on quantity demanded of either good (at one end of the budget constraint) or would change quantity demanded from one end of the budget constraint to the other.
in Fig. 3 would have no effect on equilibrium quantities demanded, since the budget line would rotate around the corner of the indifference curve.[nb 2]
Preference relations and utility[edit]
Choice theory formally represents consumers by a preference relation, and use this representation to derive indifference curves showing combinations of equal preference to the consumer.
Preference relations[edit]
Let
A {\displaystyle A\;} be a set of mutually exclusive alternatives among which a consumer can choose.
a {\displaystyle a\;} and b {\displaystyle b\;} be generic elements of A {\displaystyle A\;} .
In the language of the example above, the set A {\displaystyle A\;} is made of combinations of apples and bananas. The symbol a {\displaystyle a\;} is one such combination, such as 1 apple and 4 bananas and b {\displaystyle b\;} is another combination such as 2 apples and 2 bananas.
A preference relation, denoted ⪰ {\displaystyle \succeq } , is a binary relation define on the set A {\displaystyle A\;} .
The statement
a ⪰ b {\displaystyle a\succeq b\;}
is described as ' a {\displaystyle a\;} is weakly preferred to b {\displaystyle b\;} .' That is, a {\displaystyle a\;} is at least as good as b {\displaystyle b\;} (in preference satisfaction).
The statement
a ∼ b {\displaystyle a\sim b\;}
is described as ' a {\displaystyle a\;} is weakly preferred to b {\displaystyle b\;} , and b {\displaystyle b\;} is weakly preferred to a {\displaystyle a\;} .' That is, one is indifferent to the choice of a {\displaystyle a\;} or b {\displaystyle b\;} , meaning not that they are unwanted but that they are equally good in satisfying preferences.
The statement
a ≻ b {\displaystyle a\succ b\;}
is described as ' a {\displaystyle a\;} is weakly preferred to b {\displaystyle b\;} , but b {\displaystyle b\;} is not weakly preferred to a {\displaystyle a\;} .' One says that ' a {\displaystyle a\;} is strictly preferred to b {\displaystyle b\;} .'
The preference relation ⪰ {\displaystyle \succeq } is complete if all pairs a , b {\displaystyle a,b\;} can be ranked. The relation is a transitive relation if whenever a ⪰ b {\displaystyle a\succeq b\;} and b ⪰ c , {\displaystyle b\succeq c,\;} then a ⪰ c {\displaystyle a\succeq c\;} .
For any element a ∈ A {\displaystyle a\in A\;} , the corresponding indifference curve, C a {\displaystyle {\mathcal {C}}_{a}} is made up of all elements of A {\displaystyle A\;} which are indifferent to a {\displaystyle a} . Formally,
C a = { b ∈ A : b ∼ a } {\displaystyle {\mathcal {C}}_{a}=\{b\in A:b\sim a\}} .
Formal link to utility theory[edit]
In the example above, an element a {\displaystyle a\;} of the set A {\displaystyle A\;} is made of two numbers: The number of apples, call it x , {\displaystyle x,\;} and the number of bananas, call it y . {\displaystyle y.\;}
In utility theory, the utility function of an agent is a function that ranks all pairs of consumption bundles by order of preference (completeness) such that any set of three or more bundles forms a transitive relation. This means that for each bundle ( x , y ) {\displaystyle \left(x,y\right)} there is a unique relation, U ( x , y ) {\displaystyle U\left(x,y\right)} , representing the utility (satisfaction) relation associated with ( x , y ) {\displaystyle \left(x,y\right)} . The relation ( x , y ) → U ( x , y ) {\displaystyle \left(x,y\right)\to U\left(x,y\right)} is called the utility function. The range of the function is a set of real numbers. The actual values of the function have no importance. Only the ranking of those values has content for the theory. More precisely, if U ( x , y ) ≥ U ( x ′ , y ′ ) {\displaystyle U(x,y)\geq U(x',y')} , then the bundle ( x , y ) {\displaystyle \left(x,y\right)} is described as at least as good as the bundle ( x ′ , y ′ ) {\displaystyle \left(x',y'\right)} . If U ( x , y ) > U ( x ′ , y ′ ) {\displaystyle U\left(x,y\right)>U\left(x',y'\right)} , the bundle ( x , y ) {\displaystyle \left(x,y\right)} is described as strictly preferred to the bundle ( x ′ , y ′ ) {\displaystyle \left(x',y'\right)} .
Consider a particular bundle ( x 0 , y 0 ) {\displaystyle \left(x_{0},y_{0}\right)} and take the total derivative of U ( x , y ) {\displaystyle U\left(x,y\right)} about this point:
d U ( x 0 , y 0 ) = U 1 ( x 0 , y 0 ) d x + U 2 ( x 0 , y 0 ) d y {\displaystyle dU\left(x_{0},y_{0}\right)=U_{1}\left(x_{0},y_{0}\right)dx+U_{2}\left(x_{0},y_{0}\right)dy}
or, without loss of generality,
d U ( x 0 , y 0 ) d x = U 1 ( x 0 , y 0 ) .1 + U 2 ( x 0 , y 0 ) d y d x {\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=U_{1}(x_{0},y_{0}).1+U_{2}(x_{0},y_{0}){\frac {dy}{dx}}} (Eq. 1)
where U 1 ( x , y ) {\displaystyle U_{1}\left(x,y\right)} is the partial derivative of U ( x , y ) {\displaystyle U\left(x,y\right)} with respect to its first argument, evaluated at ( x , y ) {\displaystyle \left(x,y\right)} . (Likewise for U 2 ( x , y ) . {\displaystyle U_{2}\left(x,y\right).} )
The indifference curve through ( x 0 , y 0 ) {\displaystyle \left(x_{0},y_{0}\right)} must deliver at each bundle on the curve the same utility level as bundle ( x 0 , y 0 ) {\displaystyle \left(x_{0},y_{0}\right)} . That is, when preferences are represented by a utility function, the indifference curves are the level curves of the utility function. Therefore, if one is to change the quantity of x {\displaystyle x\,} by d x {\displaystyle dx\,} , without moving off the indifference curve, one must also change the quantity of y {\displaystyle y\,} by an amount d y {\displaystyle dy\,} such that, in the end, there is no change in U:
d U ( x 0 , y 0 ) d x = 0 {\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0} , or, substituting 0 into (Eq. 1) above to solve for dy/dx:
d U ( x 0 , y 0 ) d x = 0 ⇔ d y d x = − U 1 ( x 0 , y 0 ) U 2 ( x 0 , y 0 ) {\displaystyle {\frac {dU\left(x_{0},y_{0}\right)}{dx}}=0\Leftrightarrow {\frac {dy}{dx}}=-{\frac {U_{1}(x_{0},y_{0})}{U_{2}(x_{0},y_{0})}}} .
Thus, the ratio of marginal utilities gives the absolute value of the slope of the indifference curve at point ( x 0 , y 0 ) {\displaystyle \left(x_{0},y_{0}\right)} . This ratio is called the marginal rate of substitution between x {\displaystyle x\,} and y {\displaystyle y\,} .
Examples[edit]
Linear utility[edit]
If the utility function is of the form U ( x , y ) = α x + β y {\displaystyle U\left(x,y\right)=\alpha x+\beta y} then the marginal utility of x {\displaystyle x\,} is U 1 ( x , y ) = α {\displaystyle U_{1}\left(x,y\right)=\alpha } and the marginal utility of y {\displaystyle y\,} is U 2 ( x , y ) = β {\displaystyle U_{2}\left(x,y\right)=\beta } . The slope of the indifference curve is, therefore,
d x d y = − β α . {\displaystyle {\frac {dx}{dy}}=-{\frac {\beta }{\alpha }}.}
Observe that the slope does not depend on x {\displaystyle x\,} or y {\displaystyle y\,} : the indifference curves are straight lines.
Cobb-Douglas utility[edit]
If the utility function is of the form U ( x , y ) = x α y 1 − α {\displaystyle U\left(x,y\right)=x^{\alpha }y^{1-\alpha }} the marginal utility of x {\displaystyle x\,} is U 1 ( x , y ) = α ( x / y ) α − 1 {\displaystyle U_{1}\left(x,y\right)=\alpha \left(x/y\right)^{\alpha -1}} and the marginal utility of y {\displaystyle y\,} is U 2 ( x , y ) = ( 1 − α ) ( x / y ) α {\displaystyle U_{2}\left(x,y\right)=(1-\alpha )\left(x/y\right)^{\alpha }} .Where α < 1 {\displaystyle \alpha <1} . The slope of the indifference curve, and therefore the negative of the marginal rate of substitution, is then
d x d y = − 1 − α α ( x y ) . {\displaystyle {\frac {dx}{dy}}=-{\frac {1-\alpha }{\alpha }}\left({\frac {x}{y}}\right).}
CES utility[edit]
A general CES (Constant Elasticity of Substitution) form is
U ( x , y ) = ( α x ρ + ( 1 − α ) y ρ ) 1 / ρ {\displaystyle U(x,y)=\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{1/\rho }}
where α ∈ ( 0 , 1 ) {\displaystyle \alpha \in (0,1)} and ρ ≤ 1 {\displaystyle \rho \leq 1} . (The Cobb-Douglas is a special case of the CES utility, with ρ → 0 {\displaystyle \rho \rightarrow 0\,} .) The marginal utilities are given by
U 1 ( x , y ) = α ( α x ρ + ( 1 − α ) y ρ ) ( 1 / ρ ) − 1 x ρ − 1 {\displaystyle U_{1}(x,y)=\alpha \left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}x^{\rho -1}}
and
U 2 ( x , y ) = ( 1 − α ) ( α x ρ + ( 1 − α ) y ρ ) ( 1 / ρ ) − 1 y ρ − 1 . {\displaystyle U_{2}(x,y)=(1-\alpha )\left(\alpha x^{\rho }+(1-\alpha )y^{\rho }\right)^{\left(1/\rho \right)-1}y^{\rho -1}.}
Therefore, along an indifference curve,
d x d y = − 1 − α α ( x y ) 1 − ρ . {\displaystyle {\frac {dx}{dy}}=-{\frac {1-\alpha }{\alpha }}\left({\frac {x}{y}}\right)^{1-\rho }.}
These examples might be useful for modelling individual or aggregate demand.
Biology[edit]
As used in biology, the indifference curve is a model for how animals 'decide' whether to perform a particular behavior, based on changes in two variables which can increase in intensity, one along the x-axis and the other along the y-axis. For example, the x-axis may measure the quantity of food available while the y-axis measures the risk involved in obtaining it. The indifference curve is drawn to predict the animal's behavior at various levels of risk and food availability.
See also[edit]
Budget constraint
Community indifference curve
Consumer theory
Convex preferences
Endowment effect
Indifference price
Level curve
Microeconomics
Rationality
Utility–possibility frontier
Notes[edit]
Jump up ^ The transitivity of weak preferences is sufficient for most indifference-curve analyses: If A is weakly preferred to B, meaning that the consumer likes A at least as much as B, and B is weakly preferred to C, then A is weakly preferred to C.[8]
Jump up ^ Indifference curves can be used to derive the individual demand curve. However, the assumptions of consumer preference theory do not guarantee that the demand curve will have a negative slope.[12]
References[edit]
Jump up ^ Geanakoplos, John (1987). "Arrow-Debreu model of general equilibrium". The New Palgrave: A Dictionary of Economics. 1. pp. 116–124 [p. 117].
Jump up ^ Böhm, Volker; Haller, Hans (1987). "Demand theory". The New Palgrave: A Dictionary of Economics. 1. pp. 785–792 [p. 785].
Jump up ^ Francis Ysidro Edgeworth (1881). Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. London: C. Kegan Paul and Co.
Jump up ^ Vilfredo Pareto (1919). Manuale di Economia Politica — con una Introduzione alla Scienza Sociale [Manual of Political Economy]. Piccola Biblioteca Scientifica. 13. Milano: Societa Editrice Libraria.
Jump up ^ http://www.policonomics.com/indifference-curves/
Jump up ^ http://www.policonomics.com/william-stanley-jevons/
^ Jump up to: a b c d e f g Binger; Hoffman (1998). Microeconomics with Calculus (2nd ed.). Reading: Addison-Wesley. pp. 109–117. ISBN 0-321-01225-9.
^ Jump up to: a b Perloff, Jeffrey M. (2008). Microeconomics: Theory & Applications with Calculus. Boston: Addison-Wesley. p. 62. ISBN 978-0-321-27794-7.
^ Jump up to: a b c Silberberg; Suen (2000). The Structure of Economics: A Mathematical Analysis (3rd ed.). Boston: McGraw-Hill. ISBN 0-07-118136-9.
^ Jump up to: a b Lipsey, Richard G. (1975). An Introduction to Positive Economics (Fourth ed.). Weidenfeld & Nicolson. pp. 182–186. ISBN 0-297-76899-9.
Jump up ^ Salvatore, Dominick (1989). Schaum's Outline of Theory and Problems of Managerial Economics. McGraw-Hill. ISBN 0-07-054513-8.
Jump up ^ Binger; Hoffman (1998). Microeconomics with Calculus (2nd ed.). Reading: Addison-Wesley. pp. 141–143. ISBN 0-321-01225-9.
Further reading[edit]
Beattie, Bruce R.; LaFrance, Jeffrey T. (2006). "The Law of Demand versus Diminishing Marginal Utility". Appl. Econ. Perspect. Pol. 28 (2): 263–271. doi:10.1111/j.1467-9353.2006.00286.x.
External links[edit]
Wikimedia Commons has media related to Indifference curves.
Anatomy of Cobb-Douglas Type Utility Functions in 3D
Anatomy of CES Type Utility Functions in 3D
Retrieved from "https://en.wikipedia.org/w/index.php?title=Indifference_curve&oldid=809978518"
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values: {\displaystyle \frac{dU\left(x_0,y_0\right)}{dx} = 0\Leftrightarrow\frac{dy}{dx}=-\frac{U_1(x_0,y_0)}{U_2(x_0,y_0)}}

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values: {\displaystyle U(x,y)=\left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{1/\rho}}

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Charles's law
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Charles's law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles's law is:
When the pressure on a sample of a dry gas is held constant, the Kelvin temperature and the volume will be directly related.[1]
This directly proportional relationship can be written as:
V ∝ T {\displaystyle V\propto T}
or
V T = k , {\displaystyle {\frac {V}{T}}=k,}
where:
V is the volume of the gas,
T is the temperature of the gas (measured in kelvins),
k is a constant.
This law describes how a gas expands as the temperature increases; conversely, a decrease in temperature will lead to a decrease in volume. For comparing the same substance under two different sets of conditions, the law can be written as:
V 1 T 1 = V 2 T 2 or V 2 V 1 = T 2 T 1 or V 1 T 2 = V 2 T 1 . {\displaystyle {\frac {V_{1}}{T_{1}}}={\frac {V_{2}}{T_{2}}}\qquad {\text{or}}\qquad {\frac {V_{2}}{V_{1}}}={\frac {T_{2}}{T_{1}}}\qquad {\text{or}}\qquad V_{1}T_{2}=V_{2}T_{1}.}
The equation shows that, as absolute temperature increases, the volume of the gas also increases in proportion.
Contents
[hide]
1 Discovery and naming of the law
2 Relation to absolute zero
3 Relation to kinetic theory
4 See also
5 References
6 Further reading
7 External links
Discovery and naming of the law[edit]
The law was named after scientist Jacques Charles, who formulated the original law in his unpublished work from the 1780s.
In two of a series of four essays presented between 2 and 30 October 1801,[2] John Dalton demonstrated by experiment that all the gases and vapours that he studied expanded by the same amount between two fixed points of temperature. The French natural philosopher Joseph Louis Gay-Lussac confirmed the discovery in a presentation to the French National Institute on 31 Jan 1802,[3] although he credited the discovery to unpublished work from the 1780s by Jacques Charles. The basic principles had already been described by Guillaume Amontons[4] and Francis Hauksbee[5] a century earlier.
Dalton was the first to demonstrate that the law applied generally to all gases, and to the vapours of volatile liquids if the temperature was well above the boiling point. Gay-Lussac concurred.[6] With measurements only at the two thermometric fixed points of water, Gay-Lussac was unable to show that the equation relating volume to temperature was a linear function. On mathematical grounds alone, Gay-Lussac's paper does not permit the assignment of any law stating the linear relation. Both Dalton's and Gay-Lussac's main conclusions can be expressed mathematically as:
V 100 − V 0 = k V 0 {\displaystyle V_{100}-V_{0}=kV_{0}\,}
where V100 is the volume occupied by a given sample of gas at 100 °C; V0 is the volume occupied by the same sample of gas at 0 °C; and k is a constant which is the same for all gases at constant pressure. This equation does not contain the temperature and so has nothing to do with what became known as Charles' Law. Gay-Lussac's value for k (1⁄2.6666), was identical to Dalton's earlier value for vapours and remarkably close to the present-day value of 1⁄2.7315. Gay-Lussac gave credit for this equation to unpublished statements by his fellow Republican citizen J. Charles in 1787. In the absence of a firm record, the gas law relating volume to temperature cannot be named after Charles. Dalton's measurements had much more scope regarding temperature than Gay-Lussac, not only measuring the volume at the fixed points of water, but also at two intermediate points. Unaware of the inaccuracies of mercury thermometers at the time, which were divided into equal portions between the fixed points, Dalton, after concluding in Essay II that in the case of vapours, “any elastic fluid expands nearly in a uniform manner into 1370 or 1380 parts by 180 degrees (Fahrenheit) of heat”, was unable to confirm it for gases.
Relation to absolute zero[edit]
Charles' law appears to imply that the volume of a gas will descend to zero at a certain temperature (−266.66 °C according to Gay-Lussac's figures) or −273.15 °C. Gay-Lussac was clear in his description that the law was not applicable at low temperatures:
but I may mention that this last conclusion cannot be true except so long as the compressed vapours remain entirely in the elastic state; and this requires that their temperature shall be sufficiently elevated to enable them to resist the pressure which tends to make them assume the liquid state.[3]
At absolute zero temperature the gas possesses zero amount of energy and hence the molecules restrict motion. Gay-Lussac had no experience of liquid air (first prepared in 1877), although he appears to believe (as did Dalton) that the "permanent gases" such as air and hydrogen could be liquified. Gay-Lussac had also worked with the vapours of volatile liquids in demonstrating Charles' law, and was aware that the law does not apply just above the boiling point of the liquid:
I may however remark that when the temperature of the ether is only a little above its boiling point, its condensation is a little more rapid than that of atmospheric air. This fact is related to a phenomenon which is exhibited by a great many bodies when passing from the liquid to the solid state, but which is no longer sensible at temperatures a few degrees above that at which the transition occurs.[3]
The first mention of a temperature at which the volume of a gas might descend to zero was by William Thomson (later known as Lord Kelvin) in 1848:[7]
This is what we might anticipate, when we reflect that infinite cold must correspond to a finite number of degrees of the air-thermometer below zero; since if we push the strict principle of graduation, stated above, sufficiently far, we should arrive at a point corresponding to the volume of air being reduced to nothing, which would be marked as −273° of the scale (−100/.366, if .366 be the coefficient of expansion); and therefore −273° of the air-thermometer is a point which cannot be reached at any finite temperature, however low.
However, the "absolute zero" on the Kelvin temperature scale was originally defined in terms of the second law of thermodynamics, which Thomson himself described in 1852.[8] Thomson did not assume that this was equal to the "zero-volume point" of Charles' law, merely that Charles' law provided the minimum temperature which could be attained. The two can be shown to be equivalent by Ludwig Boltzmann's statistical view of entropy (1870).
However, Charles also stated:
The volume of a fixed mass of dry gas increases or decreases by 1⁄273 times the volume at 0 °C for every 1 °C rise or fall in temperature. Thus:
V T = V 0 + ( 1 273 × V 0 ) × T {\displaystyle V_{T}=V_{0}+({\tfrac {1}{273}}\times V_{0})\times T}
V T = V 0 ( 1 + T 273 ) {\displaystyle V_{T}=V_{0}(1+{\tfrac {T}{273}})}
where VT is the volume of gas at temperature T, V0 is the volume at 0 °C.
Relation to kinetic theory[edit]
The kinetic theory of gases relates the macroscopic properties of gases, such as pressure and volume, to the microscopic properties of the molecules which make up the gas, particularly the mass and speed of the molecules. In order to derive Charles' law from kinetic theory, it is necessary to have a microscopic definition of temperature: this can be conveniently taken as the temperature being proportional to the average kinetic energy of the gas molecules, Ek:
T ∝ E k ¯ . {\displaystyle T\propto {\bar {E_{\rm {k}}}}.\,}
Under this definition, the demonstration of Charles' law is almost trivial. The kinetic theory equivalent of the ideal gas law relates PV to the average kinetic energy:
P V = 2 3 N E k ¯ {\displaystyle PV={\frac {2}{3}}N{\bar {E_{\rm {k}}}}\,}
See also[edit]
Boyle's law
Combined gas law
Gay-Lussac's law
Avogadro's law
Ideal gas law
Hand boiler
References[edit]
Jump up ^ Fullick, P. (1994), Physics, Heinemann, pp. 141–42, ISBN 0-435-57078-1 .
Jump up ^ J. Dalton (1802), "Essay II. On the force of steam or vapour from water and various other liquids, both in vacuum and in air" and Essay IV. "On the expansion of elastic fluids by heat," Memoirs of the Literary and Philosophical Society of Manchester, vol. 8, pt. 2, pages 550–574 and pages 595–602.
^ Jump up to: a b c Gay-Lussac, J. L. (1802), "Recherches sur la dilatation des gaz et des vapeurs" [Researches on the expansion of gases and vapors], Annales de chimie, 43: 137–175 . English translation (extract).
On page 157, Gay-Lussac mentions the unpublished findings of Charles: "Avant d'aller plus loin, je dois prévenir que quoique j'eusse reconnu un grand nombre de fois que les gaz oxigène, azote, hydrogène et acide carbonique, et l'air atmosphérique se dilatent également depuis 0° jusqu'a 80°, le cit. Charles avait remarqué depuis 15 ans la même propriété dans ces gaz ; mais n'avant jamais publié ses résultats, c'est par le plus grand hasard que je les ai connus." (Before going further, I should inform [you] that although I had recognized many times that the gases oxygen, nitrogen, hydrogen, and carbonic acid [i.e., carbon dioxide], and atmospheric air also expand from 0° to 80°, citizen Charles had noticed 15 years ago the same property in these gases; but having never published his results, it is by the merest chance that I knew of them.)
Jump up ^ See:
Amontons, G. (presented 1699, published 1732) "Moyens de substituer commodément l'action du feu à la force des hommes et des chevaux pour mouvoir les machines" (Ways to conveniently substitute the action of fire for the force of men and horses in order to power machines), Mémoires de l’Académie des sciences de Paris (presented 1699, published 1732), 112–126; see especially pages 113–117.
Amontons, G. (presented 1702, published 1743) "Discours sur quelques propriétés de l'Air, & le moyen d'en connoître la température dans tous les climats de la Terre" (Discourse on some properties of air and on the means of knowing the temperature in all climates of the Earth), Mémoires de l’Académie des sciences de Paris, 155–174.
Review of Amontons' findings: "Sur une nouvelle proprieté de l'air, et une nouvelle construction de Thermométre" (On a new property of the air and a new construction of thermometer), Histoire de l'Academie royale des sciences, 1–8 (submitted: 1702 ; published: 1743).
Jump up ^
Englishman Francis Hauksbee (1660–1713) independently also discovered Charles' law: Francis Hauksbee (1708) "An account of an experiment touching the different densities of air, from the greatest natural heat to the greatest natural cold in this climate," Philosophical Transactions of the Royal Society of London 26(315): 93–96.
Jump up ^ Gay-Lussac (1802), from page 166:
"Si l'on divise l'augmentation totale de volume par le nombre de degrés qui l'ont produite ou par 80, on trouvera, en faisant le volume à la température 0 égal à l'unité, que l'augmentation de volume pour chaque degré est de 1 / 223.33 ou bien de 1 / 266.66 pour chaque degré du thermomètre centrigrade."
If one divides the total increase in volume by the number of degrees that produce it or by 80, one will find, by making the volume at the temperature 0 equal to unity (1), that the increase in volume for each degree is 1 / 223.33 or 1 / 266.66 for each degree of the centigrade thermometer.
From page 174:
" … elle nous porte, par conséquent, à conclure que tous les gaz et toutes les vapeurs se dilatent également par les mêmes degrés de chaleur."
… it leads us, consequently, to conclude that all gases and all vapors expand equally [when subjected to] the same degrees of heat.
Jump up ^ Thomson, William (1848), "On an Absolute Thermometric Scale founded on Carnot's Theory of the Motive Power of Heat, and calculated from Regnault's Observations", Philosophical Magazine: 100–6 .
Jump up ^ Thomson, William (1852), "On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule's equivalent of a Thermal Unit, and M. Regnault's Observations on Steam", Philosophical Magazine, 4 . Extract.
Further reading[edit]
Krönig, A. (1856), "Grundzüge einer Theorie der Gase", Annalen der Physik, 99: 315–22, Bibcode:1856AnP...175..315K, doi:10.1002/andp.18561751008 . Facsimile at the Bibliothèque nationale de France (pp. 315–22).
Clausius, R. (1857), "Ueber die Art der Bewegung, welche wir Wärme nennen", Annalen der Physik und Chemie, 176: 353–79, Bibcode:1857AnP...176..353C, doi:10.1002/andp.18571760302 . Facsimile at the Bibliothèque nationale de France (pp. 353–79).
Joseph Louis Gay-Lussac – Liste de ses communications, archived from the original on October 23, 2005 . (in French)
External links[edit]
The Wikibook School Science has a page on the topic of: Making Charles' law tubes
Charles's law simulation from Davidson College, Davidson, North Carolina
Charles's law demonstration by Prof. Robert Burk, Carleton University, Ottawa, Canada
Charles's law animation from the Leonardo Project (GTEP/CCHS, UK)
Charles’s law Calculator from EngineeringUnits.com, USA, UK)

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