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# at constant pressure, how are the temperature and volume of a gas related?

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## Gas Laws

The pressure, volume, and temperature of most gases can be described with simple mathematical relationships that are summarized in one ideal gas law.

## Gas Laws

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## Summary

The basic gas laws for a constant amount of matter…

pressure-volume (constant temperature)

The pressure of a gas is inversely proportional to its volume when temperature is constant.

The product of pressure and volume is constant when temperature is constant.

This relationship is known as Boyle's law or Mariotte's law.

A constant temperature process is said to be isothermal.

∝  1  ( constant) 11 = 22 =

volume-temperature (constant pressure)

The volume of a gas is directly proportional to its temperature when pressure is constant.

The ratio of volume to temperature is constant when pressure is constant.

This relationship is known as Charles' law or Gay-Lussac's law.

a constant pressure process is said to be isobaric.

∝  ( constant) 1  =  2  = 1 2

pressure-temperature (constant volume)

The pressure of a gas is directly proportional to its temperature when volume is constant.

The ratio of pressure to temperature is constant when volume is constant.

This relationship is not associated with any particular scientist.

A constant volume process is said to be isochoric.

∝  ( constant) 1  =  2  = 1 2

The number of molecules in a given volume of gas at a given temperature is the same for all gases.

The ideal gas law (presented two ways)…

functional thermodynamics

= where…

=  absolute pressure

=  absolute temperature

=  volume and… =  number of moles

=  gas constant = 8.314 J/mol K

statistical thermodynamics

= where…

=  absolute pressure

=  absolute temperature

=  volume and…

=  number of particles

=  Boltzmann constant = 1.381 × 10−23J/K

Thermodynamic changes with special names…

An isobaric process is one that takes place without any change in pressure.

An isochoric process is one that takes place without any change in volume.

An isothermal process is one that takes place without any change in temperature.

Isothermal processes are often described as "slow".

The pressure of a gas is inversely proportional to its volume only if the change takes place isothermally.

An adiabatic process is one that takes place without any exchange of heat.

Adiabatic processes are often described as "fast".

The pressure of a gas is not inversely proportional to its volume if the change takes place adiabatically.

Source : physics.info

## 9.2 RELATING PRESSURE, VOLUME, AMOUNT, AND TEMPERATURE: THE IDEAL GAS LAW

### Learning Objectives

By the end of this section, you will be able to:

Identify the mathematical relationships between the various properties of gases

Use the ideal gas law, and related gas laws, to compute the values of various gas properties under specified conditions

During the seventeenth and especially eighteenth centuries, driven both by a desire to understand nature and a quest to make balloons in which they could fly (Figure 1), a number of scientists established the relationships between the macroscopic physical properties of gases, that is, pressure, volume, temperature, and amount of gas. Although their measurements were not precise by today’s standards, they were able to determine the mathematical relationships between pairs of these variables (e.g., pressure and temperature, pressure and volume) that hold for an ideal gas—a hypothetical construct that real gases approximate under certain conditions. Eventually, these individual laws were combined into a single equation—the ideal gas law—that relates gas quantities for gases and is quite accurate for low pressures and moderate temperatures. We will consider the key developments in individual relationships (for pedagogical reasons not quite in historical order), then put them together in the ideal gas law.

Figure 1. In 1783, the first (a) hydrogen-filled balloon flight, (b) manned hot air balloon flight, and (c) manned hydrogen-filled balloon flight occurred. When the hydrogen-filled balloon depicted in (a) landed, the frightened villagers of Gonesse reportedly destroyed it with pitchforks and knives. The launch of the latter was reportedly viewed by 400,000 people in Paris.

## PRESSURE AND TEMPERATURE: AMONTONS’S LAW

Imagine filling a rigid container attached to a pressure gauge with gas and then sealing the container so that no gas may escape. If the container is cooled, the gas inside likewise gets colder and its pressure is observed to decrease. Since the container is rigid and tightly sealed, both the volume and number of moles of gas remain constant. If we heat the sphere, the gas inside gets hotter (Figure 2) and the pressure increases.

Figure 2. The effect of temperature on gas pressure: When the hot plate is off, the pressure of the gas in the sphere is relatively low. As the gas is heated, the pressure of the gas in the sphere increases.

This relationship between temperature and pressure is observed for any sample of gas confined to a constant volume. An example of experimental pressure-temperature data is shown for a sample of air under these conditions in Figure 3. We find that temperature and pressure are linearly related, and if the temperature is on the kelvin scale, then P and T are directly proportional (again, when volume and moles of gas are held constant); if the temperature on the kelvin scale increases by a certain factor, the gas pressure increases by the same factor.

Figure 3. For a constant volume and amount of air, the pressure and temperature are directly proportional, provided the temperature is in kelvin. (Measurements cannot be made at lower temperatures because of the condensation of the gas.) When this line is extrapolated to lower pressures, it reaches a pressure of 0 at –273 °C, which is 0 on the kelvin scale and the lowest possible temperature, called absolute zero.

Guillaume Amontons was the first to empirically establish the relationship between the pressure and the temperature of a gas (~1700), and Joseph Louis Gay-Lussac determined the relationship more precisely (~1800). Because of this, the P–T relationship for gases is known as either Amontons’s law or Gay-Lussac’s law. Under either name, it states that the pressure of a given amount of gas is directly proportional to its temperature on the kelvin scale when the volume is held constant. Mathematically, this can be written:

P ∝ T or P = constant × T or P = k × T

P∝TorP=constant×TorP=k×T

where ∝ means “is proportional to,” and k is a proportionality constant that depends on the identity, amount, and volume of the gas.

For a confined, constant volume of gas, the ratio

P T PT

is therefore constant (i.e.,

P T = k PT=k

). If the gas is initially in “Condition 1” (with P = P1 and T = T1), and then changes to “Condition 2” (with P = P2 and T = T2), we have that

P 1 T 1 = k P1T1=k and P 2 T 2 = k P2T2=k , which reduces to P 1 T 1 = P 2 T 2 P1T1=P2T2

. This equation is useful for pressure-temperature calculations for a confined gas at constant volume. Note that temperatures must be on the kelvin scale for any gas law calculations (0 on the kelvin scale and the lowest possible temperature is called absolute zero). (Also note that there are at least three ways we can describe how the pressure of a gas changes as its temperature changes: We can use a table of values, a graph, or a mathematical equation.)

Source : opentextbc.ca

## 6.3: Relationships among Pressure, Temperature, Volume, and Amount

Last updated Sep 24, 2015 6.2: Gas Pressure

6.4 The Ideal Gas Law

To understand the relationships among pressure, temperature, volume, and the amount of a gas.

Early scientists explored the relationships among the pressure of a gas (P) and its temperature (T), volume (V), and amount (n) by holding two of the four variables constant (amount and temperature, for example), varying a third (such as pressure), and measuring the effect of the change on the fourth (in this case, volume). The history of their discoveries provides several excellent examples of the scientific method.

## The Relationship between Pressure and Volume: Boyle's Law

As the pressure on a gas increases, the volume of the gas decreases because the gas particles are forced closer together. Conversely, as the pressure on a gas decreases, the gas volume increases because the gas particles can now move farther apart. Weather balloons get larger as they rise through the atmosphere to regions of lower pressure because the volume of the gas has increased; that is, the atmospheric gas exerts less pressure on the surface of the balloon, so the interior gas expands until the internal and external pressures are equal.

The Irish chemist Robert Boyle (1627–1691) carried out some of the earliest experiments that determined the quantitative relationship between the pressure and the volume of a gas. Boyle used a J-shaped tube partially filled with mercury, as shown in Figure

6.3.1 6.3.1

. In these experiments, a small amount of a gas or air is trapped above the mercury column, and its volume is measured at atmospheric pressure and constant temperature. More mercury is then poured into the open arm to increase the pressure on the gas sample. The pressure on the gas is atmospheric pressure plus the difference in the heights of the mercury columns, and the resulting volume is measured. This process is repeated until either there is no more room in the open arm or the volume of the gas is too small to be measured accurately. Data such as those from one of Boyle’s own experiments may be plotted in several ways (Figure

6.3.2 6.3.2 ). A simple plot of V V versus P P

gives a curve called a hyperbola and reveals an inverse relationship between pressure and volume: as the pressure is doubled, the volume decreases by a factor of two. This relationship between the two quantities is described as follows:

PV=constant (6.3.1) (6.3.1)PV=constant

Figure 6.3.1 6.3.1

: Boyle’s Experiment Using a J-Shaped Tube to Determine the Relationship between Gas Pressure and Volume. (a) Initially the gas is at a pressure of 1 atm = 760 mmHg (the mercury is at the same height in both the arm containing the sample and the arm open to the atmosphere); its volume is V. (b) If enough mercury is added to the right side to give a difference in height of 760 mmHg between the two arms, the pressure of the gas is 760 mmHg (atmospheric pressure) + 760 mmHg = 1520 mmHg and the volume is V/2. (c) If an additional 760 mmHg is added to the column on the right, the total pressure on the gas increases to 2280 mmHg, and the volume of the gas decreases to V/3.

Dividing both sides by

P P

gives an equation illustrating the inverse relationship between

P P and V V : V= const. P =const.( 1 P ) (6.3.2)

(6.3.2)V=const.P=const.(1P)

or V∝ 1 P (6.3.3) (6.3.3)V∝1P

where the ∝ symbol is read “is proportional to.” A plot of V versus 1/P is thus a straight line whose slope is equal to the constant in Equation 6.2.1 and Equation 6.2.3. Dividing both sides of Equation 6.2.1 by V instead of P gives a similar relationship between P and 1/V. The numerical value of the constant depends on the amount of gas used in the experiment and on the temperature at which the experiments are carried out. This relationship between pressure and volume is known as Boyle’s law, after its discoverer, and can be stated as follows: At constant temperature, the volume of a fixed amount of a gas is inversely proportional to its pressure.

Figure 6.3.2 6.3.2

: Plots of Boyle’s Data. (a) Here are actual data from a typical experiment conducted by Boyle. Boyle used non-SI units to measure the volume (in.3 rather than cm3) and the pressure (in. Hg rather than mmHg). (b) This plot of pressure versus volume is a hyperbola. Because PV is a constant, decreasing the pressure by a factor of two results in a twofold increase in volume and vice versa. (c) A plot of volume versus 1/pressure for the same data shows the inverse linear relationship between the two quantities, as expressed by the equation V = constant/P.

Boyle’s Law: https://youtu.be/lu86VSupPO4

Source : chem.libretexts.org

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James 11 month ago

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