What law states that the pressure of a gas is inversely proportional to its volume at constant temperature?

Boyle's law states that pressure is inversely proportional to volume, for a fixed amount of gas at constant temperature.

Índice Show

Figure 1: Boyle’s Law: Pressure vs. Volume Graph
Figure2: Pressure vs. Inverse Volume Graph
Figure 3: Inverse Pressure versus Volume
Conclusions:

$p\propto {1\over V}$

$p$ is pressure (Pa)
$V$ is volume (m3)

$p_1V_1=p_2V_2$

Boyle's law can be verified by measuring the pressure of a fixed mass of gas as its volume is changed. The graph of p vs V shows an inverse relationship.

The pressure of a gas increases when the volume is made smaller. This is because the molecules are closer to the walls and so collide with the walls more often. If the temperature is constant, so is the kinetic energy of the molecules and the force exerted by each collision.

Use quizzes to practise application of theory.

START QUIZ!

Online tutorials to help you solve original problems

Gases have various properties that can be observed in the laboratory, including temperature, pressure, mass and volume. With experimentation, scientists have found that these variables are related to each other. Robert Boyle is a scientist known for studying the relationship between pressure and volume of a confined gas held at constant temperature. Boyle found that when he manipulated the pressure, the volumeof the gas responded in the opposite direction. Today, Boyle’s Law states that pressure of a gas is inversely proportional to its volume. The mathematical expression for this relationship is P1V1= P2V2, assuming temperature to be constant.

Purpose:

The purpose of this exercise is to verify that pressure and volume indeed are inversely related through graphical analysis based on data collected by Robert Boyle in 1662.

Analysis:

In order to analyze a table of data presented by Robert Boyle in 1662, a graph has been constructed using Excel to demonstrate his findings. According to this graph, shown in Figure 1, as volume increases, pressure decreases. The R2 value of 0.9999 indicates the power trendline fits the regression almost perfectly, showing that there is a precisely inverse relationship between the pressure and volume of a gas at constant temperature. The equation of the trendline, y=1400.9x-0.996x indicates that indeed the relationship observed is a constant of 1400.9 times approximately 1/x, reinforcing the inverse relationship between temperature and volume of a gas.

Secondly, to see the inverse relationship more clearly, a second graph was constructed. This graph of pressure versus inverse volume should demonstrate a linear relationship, if indeed the two variables, pressure and volume, are inversely related. As the graph in Figure 2 illustrates, the relationship between pressure and inverse volume is linear, with high certainty illustrated by the R2 of 0.9999. The relationship is approximately equal to y = 1403x, a linear relationship.

To further analyze this finding, a last graph was constructed to demonstrate the relationship between the inverse of pressure versus volume. The expected linear relationship was demonstrated with high certainty again with an R2 approximately equal to 1. The trend is linear, with an equation of about y = 0.0007x, as shown in Figure 3.

Figure 1: Boyle’s Law: Pressure vs. Volume Graph

Figure2: Pressure vs. Inverse Volume Graph

Figure 3: Inverse Pressure versus Volume

Conclusions:

As demonstrated by this report, the graphical analysis of Robert Boyle’s data supports his findings that the pressure of a gas is inversely proportional to the volume at constant temperature. This is illustrated by the graph in Figure 1 above that indicates that as volume increases, pressure decreases. It is further confirmed by the following two graphs, Figures 2 and 3, which both demonstrate that the relationship between one variable and the inverse of the other is indeed a linear function. Each graph has a high correlation efficient, at about 1, which demonstrates with certainty that there is truly an inverse relationship between volume and pressure.

The gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases.

In 1662 Robert Boyle studied the relationship between volume and pressure of a gas of fixed amount at constant temperature. He observed that volume of a given mass of a gas is inversely proportional to its pressure at a constant temperature. Boyle's law, published in 1662, states that, at constant temperature, the product of the pressure and volume of a given mass of an ideal gas in a closed system is always constant. It can be verified experimentally using a pressure gauge and a variable volume container. It can also be derived from the kinetic theory of gases: if a container, with a fixed number of molecules inside, is reduced in volume, more molecules will strike a given area of the sides of the container per unit time, causing a greater pressure.

A statement of Boyle's law is as follows:

The volume of a given mass of a gas is inversely related to pressure when the temperature is constant.

The concept can be represented with these formulae:

$V ∝ 1 P {\displaystyle V\propto {\frac {1}{P}}}$ , meaning "Volume is inversely proportional to Pressure", or
$P ∝ 1 V {\displaystyle P\propto {\frac {1}{V}}}$ , meaning "Pressure is inversely proportional to Volume", or
$P V = k 1 {\displaystyle PV=k_{1}}$ , or

where

P

is the pressure, and

V

is the volume of a gas, and

k1

is the constant in this equation (and is not the same as the proportionality constants in the other equations in this article).

Charles's law, or the law of volumes, was found in 1787 by Jacques Charles. It states that, for a given mass of an ideal gas at constant pressure, the volume is directly proportional to its absolute temperature, assuming in a closed system. The statement of Charles's law is as follows: the volume (V) of a given mass of a gas, at constant pressure (P), is directly proportional to its temperature (T). As a mathematical equation, Charles's law is written as either:

V ∝ T {\displaystyle V\propto T\,}

, or

V / T = k 2 {\displaystyle V/T=k_{2}}

, or

V 1 / T 1 = V 2 / T 2 {\displaystyle V_{1}/T_{1}=V_{2}/T_{2}}

where "V" is the volume of a gas, "T" is the absolute temperature and k2 is a proportionality constant (which is not the same as the proportionality constants in the other equations in this article).

Gay-Lussac's law, Amontons' law or the pressure law was found by Joseph Louis Gay-Lussac in 1808. It states that, for a given mass and constant volume of an ideal gas, the pressure exerted on the sides of its container is directly proportional to its absolute temperature.

As a mathematical equation, Gay-Lussac's law is written as either:

P ∝ T {\displaystyle P\propto T\,}

, or

P / T = k {\displaystyle P/T=k}

, or

P 1 / T 1 = P 2 / T 2 {\displaystyle P_{1}/T_{1}=P_{2}/T_{2}}

,where P is the pressure, T is the absolute temperature, and k is another proportionality constant.

Avogadro's law (hypothesized in 1811) states that at a constant temperature and pressure, the volume occupied by an ideal gas is directly proportional to the number of molecules of the gas present in the container. This gives rise to the molar volume of a gas, which at STP (273.15 K, 1 atm) is about 22.4 L. The relation is given by

V ∝ n {\displaystyle V\propto n\,}

, or

V 1 n 1 = V 2 n 2 {\displaystyle {\frac {V_{1}}{n_{1}}}={\frac {V_{2}}{n_{2}}}\,}

where n is equal to the number of molecules of gas (or the number of moles of gas).

Relationships between Boyle's, Charles's, Gay-Lussac's, Avogadro's, combined and ideal gas laws, with the Boltzmann constant kB = RNA = n RN (in each law, properties circled are variable and properties not circled are held constant)

The Combined gas law or General Gas Equation is obtained by combining Boyle's Law, Charles's law, and Gay-Lussac's Law. It shows the relationship between the pressure, volume, and temperature for a fixed mass (quantity) of gas:

P V = k 5 T {\displaystyle PV=k_{5}T}

This can also be written as:

P 1 V 1 T 1 = P 2 V 2 T 2 {\displaystyle {\frac {P_{1}V_{1}}{T_{1}}}={\frac {P_{2}V_{2}}{T_{2}}}}

With the addition of Avogadro's law, the combined gas law develops into the ideal gas law:

P V = n R T {\displaystyle PV=nRT}

where

P is pressure
V is volume
n is the number of moles
R is the universal gas constant
T is temperature (K)

The proportionality constant, now named R, is the universal gas constant with a value of 8.3144598 (kPa∙L)/(mol∙K).

An equivalent formulation of this law is:

P V = N k B T {\displaystyle PV=Nk_{\text{B}}T}

where

P is the pressure
V is the volume
N is the number of gas molecules
kB is the Boltzmann constant (1.381×10−23J·K−1 in SI units)
T is the temperature (K)

These equations are exact only for an ideal gas, which neglects various intermolecular effects (see real gas). However, the ideal gas law is a good approximation for most gases under moderate pressure and temperature.

This law has the following important consequences:

If temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of gas.
If the temperature and volume remain constant, then the pressure of the gas changes is directly proportional to the number of molecules of gas present.
If the number of gas molecules and the temperature remain constant, then the pressure is inversely proportional to the volume.
If the temperature changes and the number of gas molecules are kept constant, then either pressure or volume (or both) will change in direct proportion to the temperature.

Graham's law states that the rate at which gas molecules diffuse is inversely proportional to the square root of the gas density at constant temperature. Combined with Avogadro's law (i.e. since equal volumes have equal number of molecules) this is the same as being inversely proportional to the root of the molecular weight. Dalton's law of partial pressures states that the pressure of a mixture of gases simply is the sum of the partial pressures of the individual components. Dalton's law is as follows:

P total = P 1 + P 2 + P 3 + ⋯ + P n ≡ ∑ i = 1 n P i , {\displaystyle P_{\textrm {total}}=P_{1}+P_{2}+P_{3}+\cdots +P_{n}\equiv \sum _{i=1}^{n}P_{i},}

and all component gases and the mixture are at the same temperature and volume where Ptotal is the total pressure of the gas mixture Pi is the partial pressure, or pressure of the component gas at the given volume and temperature. Amagat's law of partial volumes states that the volume of a mixture of gases (or the volume of the container) simply is the sum of the partial volumes of the individual components. Amagat's law is as follows:

V total = V 1 + V 2 + V 3 + ⋯ + V n ≡ ∑ i = 1 n V i , {\displaystyle V_{\textrm {total}}=V_{1}+V_{2}+V_{3}+\cdots +V_{n}\equiv \sum _{i=1}^{n}V_{i},}

and all component gases and the mixture are at the same temperature and pressure where Vtotal is the total volume of the gas mixture, or the volume of the container, Vi is the partial volume, or volume of the component gas at the given pressure and temperature. Henry's law states that At constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid.

p = k H c {\displaystyle p=k_{\rm {H}}\,c}

Real gas law formulated by Johannes Diderik van der Waals (1873).

Castka, Joseph F.; Metcalfe, H. Clark; Davis, Raymond E.; Williams, John E. (2002). Modern Chemistry. Holt, Rinehart and Winston. ISBN 0-03-056537-5.
Guch, Ian (2003). The Complete Idiot's Guide to Chemistry. Alpha, Penguin Group Inc. ISBN 1-59257-101-8.
Zumdahl, Steven S (1998). Chemical Principles. Houghton Mifflin Company. ISBN 0-395-83995-5.