Write an Equation to Express the Relationship Between Pressure and Temperature

Gas Laws

Orin Flanigan , in Underground Gas Storage Facilities, 1995

Compressibility Factors

The ideal gas laws work well at relatively low pressures and relatively high temperatures. When the pressure and temperature depart from these ranges, significant error can result from the use of the ideal gas laws. At high pressures and low temperatures, for example, a gas will occupy a smaller volume than is predicted by the ideal gas law. One hypothesis has been advanced that as the gas molecules are crowded together the gravitational attraction between molecules becomes a factor and this attraction causes the gas volume to be less than calculated. At very high pressures, the reverse is true; the gas occupies a greater volume than is computed by the ideal gas law. One explanation for this is that as the gas molecules become crowded very close together, the physical size of the molecule begins to become a factor and the gas begins to become slightly incompressible. Regardless of the reasons, the ideal gas law does not accurately represent the behavior of gas at high pressures.

In order to study this problem, much research work has been done. The result of this research work is a term called the compressibility factor. This is also sometimes called the supercompressibility factor. The American Gas Association has sponsored research that defines these factors for all of the conditions to which natural gas is normally exposed. The factors have been found to be affected by temperature, pressure, gas specific gravity, and gas composition, particularly the inert content of the gas. This information is in the form of tables of values as well as the equations for calculating the factor values from the gas properties and conditions.

When the compressibility factor is incorporated into the ideal gas law equation, the result is:

where Z is the compressibility factor determined by whatever method is appropriate.

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Pressure Standards

A. BERMAN , in Total Pressure Measurements in Vacuum Technology, 1985

(iii) Failure to obey the ideal gas law

The ideal gas law PV = RT (for 1 mole) relates the measurable quantities P, V, and T of a perfect gas at low pressures. For pressures approaching the high range at which gas is admitted into the system and for real gases such as argon, hydrogen, and nitrogen, other relations more accurately approximate the behavior of the gas. Among these, the equation of state of Kamerling Ones and Holborn, expressing the product PV as a function of a power series in 1/V or in P with virial coefficients of state, offers the most accurate approach (Himmelblau, 1967, p. 142). Poulter (1977) expressed the relation which predicts the pressure P _1c in the calibration chamber after the first step by utilizing both Holborn's equation of state and the ideal gas law, for P = 100 kPa (760 Torr) and T = 273 K. He found that the values of P _1c calculated using the ideal gas law have to be multiplied by a correction factor in order to get those resulting from Holborn's equation. The value of the correction factor increases with the molecular mass of the gas used for the generation of the pressure points (e.g., 1.0004 for N₂ and 1.0066 for Xe).

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Vapor Processes

ByBethanie Joyce Hills Stadler , in Materials Processing, 2016

The ideal gas law can be used to find the number of molecules per volume and then the mean free path can be calculated:

$n_v=\frac{P}{kT}=\frac{P{N}_A}{RT}=\frac{(1\mathrm{atm})}{\left(0.08206\text{liter}\text{<mglyph src="https://sdfestaticassets-eu-west-1.sciencedirectassets.com/shared-assets/16/entities/rad"></mglyph>}\text{atm}/\text{mol}\text{<mglyph src="https://sdfestaticassets-eu-west-1.sciencedirectassets.com/shared-assets/16/entities/rad"></mglyph>}\mathrm{K}\right)(300\mathrm{K})}\frac{6.02\times {10}^{23}\text{molecules}/\text{mol}}{1000{\mathrm{cm}}^3/\text{liter}}=2.45\times {10}^{19}{\text{molecules}/\text{cm}}^3$

$l_{mfp}=\frac{1}{\sqrt{2}\pi d^2{n}_v}=\frac{1}{\sqrt{2}\pi {(3.1\times {10}^{-8}\mathrm{cm})}^2(2.45\times {10}^{19}{\text{molecules}/\text{cm}}^3)}=9.6\times {10}^{-6}\mathrm{cm}$

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Reservoir fluid properties

Abdus Satter , Ghulam M. Iqbal , in Reservoir Engineering, 2016

Ideal gas law

The ideal gas law states that the pressure, temperature, and volume of gas are related to each other. The following equation can be used to express the relationship:

(4.16) $pV=nRT$

where p = prevailing pressure, psia; V = volume of gas, ft.³; n = number of pound-moles of gas, lb_m-mol; R = gas law constant, (psia)(ft.³)/(°R)(lb_m-mol); T = prevailing absolute temperature, °R.

The value of gas law constant is 10.73 based on the units used in the above equation. It is also noted that T, °R = T,°F + 460.

Equation (4.16) is based on Boyle's law and Charles's law. The above relates the change in ideal gas volume to the changes in prevailing pressure and temperature, respectively. Furthermore, Equation (4.16) is referred to as the equation of state for an ideal gas.

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The Mechanics of Breathing

Joseph Feher , in Quantitative Human Physiology (Second Edition), 2017

Changes in Lung Volumes Produce the Pressure Differences That Drive Air Movement

The Ideal Gas Law describes the relation between pressure and volume in an ideal gas:

[6.1.3] $PV=nRT$

where P is the pressure measured in atmospheres or mmHg or Pa (=N   m⁻²), or some other appropriate unit, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in kelvin. When P is in atmospheres and V is in L, R=0.082   L   atm   mol⁻¹  K⁻¹. The inverse relation between pressure and volume is the basic principle responsible for pulmonary ventilation: increasing the volume of the thoracic cavity, with the enclosed lungs, decreases the pressure of the gas in the lungs and so air rushes in from the outside. Conversely, the reduction of the volume of the thoracic cavity increases the pressure of the gas in the lungs and so air moves from the lungs back out into the ambient air.

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Pneumatics and Hydraulics

Peter R.N. Childs , in Mechanical Design Engineering Handbook, 2014

18.4 Air Compressors and Receivers

Most pneumatic systems use air as the working fluid. The vast majority of pneumatic systems are open, with the air sourced from ambient atmosphere and vented back to the atmosphere.

The ideal gas law is given by

(18.8) $pV=nRT$

where

p  =   pressure (N/m²),

V  =   volume (m³),

n  =   number of moles of the gas (n  = m/M (m  =   mass, M  =   molar mass)),

R  =   the universal gas constant (=8.314510   J/mol   K (Cohen and Taylor, 1999)), and

T  =   temperature (K).

From Eqn (18.8), the general gas equation for a gas subjected to changes in pressure, volume, and temperature between two states 1 and 2 can be obtained.

(18.9) $\frac{p_1{V}_1}{T_1}=\frac{p_2{V}_2}{T_2}$

where

p  =   absolute pressure (N/m²),

T  =   temperature (K), and

V  =   volume (m³).

The two principal classes of air compressors are as follows:

•

Positive displacement compressors where a fixed volume of air is delivered with each rotation of the compressor shaft; and

•

Centrifugal and axial compressors.

The compressor must be selected such that it provides the pressure required as well as the volume of gas at the working pressure. A typical system will be designed to operate with the pressure in the receiver at a slightly higher pressure than that in the remainder of the circuit with a pressure regulator being used. A typical pneumatic system is illustrated in Figure 18.11.

Figure 18.11. Typical pneumatic system.

The use of an air receiver tends to enable a smaller air compressor to be specified. The air receiver needs to have a capacity that is sufficiently large to enable to provide the peak flow requirements for sufficient periods compatible with the application. The volume of the receiver tends to reduce fluctuations in pressure in comparison to a system directly connected to a compressor. Air exiting from a compressor will be hotter than the inlet air, and the receiver surface volume, sometimes finned, is used to dissipate the thermal energy by natural convection. Water will tend to condense within the receiver; regular drainage of this is necessary. Some applications require drying of air, in which case a simple water trap, Figure 18.12, refrigerator dryer, deliquescent dryer, or adsorption dryer may be necessary.

Figure 18.12. Air filter and water trap.

Figure courtesy of Parr (2011).

Typical regulating valves are illustrated in Figure 18.13.

Figure 18.13. (a) Relief valve, (b) nonrelieving pressure regulator, (c) relieving pressure regulator, and (d) pilot operated regulator.

Figures courtesy of Parr (2011).

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General Engineering and Science

In Standard Handbook of Petroleum and Natural Gas Engineering (Third Edition), 2016

2.7.6.6.2 Solution

By Henry's law, the partial pressure of solute i in the gas phase is P _i = H _i (T)x _i , where x _i is the mole fraction of i in solution. Data on Henry's law constant are obtained from Chapter 14 of Perry and Chilton's Chemical Engineers' Handbook [14] for gas–water systems at 20°C.

$\begin{array}{l}\hfill \begin{array}{cccc}\mathrm{Gas}\mathrm{i}:& {\mathrm{H}}_2\mathrm{S}& {\mathrm{CO}}_2& {\mathrm{N}}_2\\ {\mathrm{H}}_{\mathrm{i}}\text{(atm/mole fraction):}& 4.83\times 10^2& 1.42\times 10^3& 8.04\times 10^4\end{array}\end{array}$

Assuming ideal-gas law to hold, P _i = y _i P, where P=1   atm and y _i = mole fraction of i in the gas phase. The equilibrium mole fraction x _i of gas i in solution is then given by

$\begin{array}{l}\hfill {\mathrm{x}}_{\mathrm{i}}={\mathrm{P}}_{\mathrm{i}}/{\mathrm{H}}_{\mathrm{i}}={\mathrm{y}}_{\mathrm{i}}\mathrm{P}/{\mathrm{H}}_{\mathrm{i}}\end{array}$

_______________________________________________________________________________________________________
Gas		P i		n i
i	y i	(atm)	x i	(lb-moles)	V i (ft3)
H2S	0.20	0.20	4.14 × 10−4	2.30 × 10−2	8.86
CO2	0.30	0.30	2.11 × 10−4	1.17 × 10−2	4.51
N2	0.50	0.50	6.22 × 10−5	3.45 × 10−3	1.33

The table shows the rest of the required results for the number of moles n _i of each gas i present in the aqueous phase and the corresponding gas volume V _i dissolved in it. Since x _i ≪ 1, the total moles of liquid is $\mathrm{n}\stackrel{.}{\approx }(1000/18.01)$ lb-moles in 1000   lb of water and so n _i = x _i n can be calculated. At T=20°C=68°F=528°R and P=1   atm, the molal volume of the gas mixture is

$\begin{array}{l}\hfill \tilde{\mathrm{V}}=\frac{\mathrm{R}\mathrm{T}}{\mathrm{P}}=\frac{(0.7302)(528)}{(1)}=385.55{\mathrm{ft}}^3\text{/lb-mole}\end{array}$

The volume of each gas dissolved in 1,000   lb of water is then ${\mathrm{V}}_{\mathrm{i}}={\mathrm{n}}_{\mathrm{i}}\tilde{\mathrm{V}}$ .

Equilibrium Distribution Ratio or K factor. This is also termed distribution coefficient in the literature; it is a widely accepted method of describing vapor–liquid equilibria in nonideal systems. For any component i distributed between the vapor phase and liquid phase at equilibrium, the distribution coefficient or K factor is defined by

(2.7.42) $\begin{array}{l}\hfill {\mathrm{K}}_{\mathrm{i}}={\mathrm{y}}_{\mathrm{i}}/{\mathrm{x}}_{\mathrm{i}}={\mathrm{K}}_{\mathrm{i}}(\mathrm{T},\mathrm{P})\end{array}$

The dimensionless K _i is regarded as a function of system T and P only and not of phase compositions. It must be experimentally determined. Reference 64 provides charts of K _i (T, P) for a number of paraffinic hydrocarbons. K _i increases with an increase in system T and decreases with an increase in P. Away from the critical point, it is assumed that the K _i values of component i are independent of the other components present in the system. In the absence of experimental data, caution must be exercised in the use of K-factor charts for a given application. The term distribution coefficient is also used in the context of a solute (solid or liquid) distributed between two immiscible liquid phases; y _i and x _i are then the equilibrium mole fractions of solute i in each liquid phase.

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Design Basis

E. Shashi Menon Ph.D., P.E. , in Pipeline Planning and Construction Field Manual, 2011

Ideal Gases

An ideal gas is defined as a fluid in which the volume of the gas molecules is negligible when compared to the volume occupied by the gas. Such ideal gases are said to obey Boyle's law, Charles' law, and the ideal gas law or the perfect gas equation. We discuss ideal gases first, followed by real gases.

If M represents the molecular weight of a gas and the mass of a certain quantity of gas is m, the number of moles n is given by

(1.27) $n=\frac{m}{M}$

where n is the number that represents the number of moles in the given mass. For example, the molecular weight of methane is 16.043. Therefore, 50 lb of methane will contain approximately 3 moles.

The ideal gas law, also called the perfect gas equation, states that the pressure, volume, and temperature of the gas are related to the number of moles by the following equation:

(1.28) $PV=nRT$

where, in USCS units,

P – Absolute pressure, pounds per square inch absolute (psia)

V – Gas volume, ft³

n – Number of lb moles as defined in Eq. (1.27)

R – Universal gas constant, psia ft³/lb · mol·°R

T – Absolute temperature of gas, °R (°F + 460)

The universal gas constant R has a value of 10.73 psia ft³/lb · mol · °R in USCS units.

In SI units, the perfect gas equation is as follows:

(1.29) $PV=nRT$

where

P – Absolute pressure, kPa

V – Gas volume, m³

n – Number of kg moles as defined in Eq. (1.27)

R – Universal gas constant, kPa · m³/kg · mol · K

T – Absolute temperature of gas, K (°C + 273)

The universal gas constant R has a value of 8.314 J/mol · K in SI units.

We can combine Eq. (1.27) with Eq. (1.28) and express the ideal gas equation as follows:

(1.30) $PV=\frac{mRT}{M}$

The constant R is the same for all ideal gases and hence it is called the universal gas constant.

It has been found that the ideal gas equation is correct only at low pressures close to the atmospheric pressure (14.7 psia or 101 kPa). Since gas pipelines generally operate at pressures higher than atmospheric pressures, we must modify Eq. (1.30) to take into account the effect of compressibility. The latter is accounted for by using a term called the compressibility factor or gas deviation factor. We discuss real gases and the compressibility factor under the heading Real Gases.

It must be noted that in the ideal gas equation (Eq. [1.30]), the pressures and temperatures must be in absolute units. Absolute pressure is defined as the gauge pressure (as measured by a pressure gauge) plus the local atmospheric pressure at the specific location. Therefore,

(1.31) $P_{\text{abs}}=P_{\text{gauge}}+P_{\text{atm}}$

Thus, if the gas pressure is 200 psig (measured by a pressure gauge) and the atmospheric pressure is 14.7 psia, the absolute pressure of the gas is

$P_{\text{abs}}=200+14.7=214.7\text{\hspace{0.17em}psia}$

Absolute pressure is expressed as psia while the gauge pressure is referred to as psig. The adder to the gauge pressure, which is the local atmospheric pressure, is also called the base pressure. In SI units, 500 kPa gauge pressure is equal to 601 kPa absolute pressure if the base pressure is 101 kPa. Pressure in USCS units is stated in pounds per square inch (lb/in²) or psi. In SI units, pressure is expressed in kilopascal (kPa), megapascal (MPa), or bar. Refer to Appendix 1 for unit conversion tables.

The absolute temperature of a gas is measured above a certain datum. In USCS units, the absolute scale of temperature is designated as degree Rankin (°R) and is obtained by adding the constant 460 to the gas temperature in °F. In SI units, the absolute temperature scale is referred to as Kelvin (K). Absolute temperature in K is equal to (°C + 273).

Therefore,

(1.32) $°\text{R}=°\text{F}+460$

(1.33) $\text{K}=°\text{C}+273$

Note that unlike temperatures in degree Rankin (°R), there is no degree symbol for absolute temperature in Kelvin (K).

Ideal gases also obey Boyle's law and Charles' law. Boyle's law relates the pressure and volume of a given quantity of gas when the temperature is kept constant. Constant temperature is called isothermal condition. According to Boyle's law, for a given quantity of gas under isothermal conditions, the pressure is inversely proportional to the volume. In other words, the volume of a gas will double, if its pressure is halved and vice versa. Since density and volume are inversely related, Boyle's law also means that the pressure is directly proportional to the density at a constant temperature. Thus, a given quantity of gas at a fixed temperature will double in density when the pressure is doubled. Similarly, a 10% reduction in pressure will cause the density to also decrease by the same amount. Boyle's law may be expressed as follows:

(1.34) $\frac{P_1}{P_2}=\frac{V_2}{V_1}\text{\hspace{1em}\hspace{0.17em}or}{P}_1{V}_1=P_2{V}_2$

where P ₁ and V ₁ are the pressure and volume of the gas at condition 1 and P ₂ and V ₂ are the corresponding value at some other condition 2, where the temperature is the same.

Charles' law states that for constant pressure, the gas volume is directly proportional to its temperature. Similarly, if volume is kept constant, the pressure varies directly as the temperature, as indicated by the following equations:

(1.35) $\frac{V_1}{V_2}=\frac{T_1}{T_2}\text{\hspace{0.17em}at\hspace{0.17em}constant\hspace{0.17em}pressure}$

(1.36) $\frac{P_1}{P_2}=\frac{T_1}{T_2}\text{\hspace{0.17em}at\hspace{0.17em}constant\hspace{0.17em}volume}$

where T ₁ and V ₁ are the temperature and volume of the gas at condition 1 and T ₂ and V ₂ are the corresponding values at some other condition 2, where the pressure is the same. Similarly, at constant volume, T ₁ and P ₁ and T ₂ and P ₂ are the temperatures and pressures of the gas at conditions 1 and 2, respectively.

Therefore, according to Charles' law for an ideal gas at constant pressure, the volume will change in the same proportion as its temperature. Thus, a 20% increase in temperature will cause a 20% increase in volume as long as the pressure does not change. Similarly, if volume is kept constant, a 20% increase in temperature will result in the same percentage in increase in gas pressure. Constant pressure is also known as isobaric condition.

Example Problem 1.7 (USCS)

An ideal gas occupies a tank volume of 400 ft³ at a pressure of 200 psig and a temperature of 100°F.

1.

What is the gas volume at standard conditions of 14.73 psia and 60°F? Assume atmospheric pressure is 14.6 psia.

2.

If the gas is cooled to 80°F, what is the gas pressure?

Solution

1.

Initial conditions

$\begin{array}{ll}{P}_1\hfill & =200+14.6=214.6\text{\hspace{0.17em}psia}\hfill \\ V_1\hfill & =400{\text{\hspace{0.17em}ft}}^3\hfill \\ T_1\hfill & =100+460=560°\text{R}\hfill \end{array}$

Final conditions

$P_2=14.73\text{\hspace{0.17em}psia}$

V ₂ is to be calculated.

$T_2=60+460=520°\text{R}$

Using the ideal gas equation (Eq. [1.30]), we can state that

$\frac{214.6\times 400}{560}=\frac{14.73\times V_2}{520}$

$V_2=5411.{3\text{\hspace{0.17em}ft}}^3$

2.

When the gas is cooled to 80°F, the final conditions are to be determined.

$\begin{array}{cc}{T}_2\hfill & =80+460=540°\text{R}\hfill \\ V_2\hfill & =400{\text{\hspace{0.17em}ft}}^3\hfill \end{array}$

P ₂ is to be calculated.

The initial conditions are

$\begin{array}{c}{P}_1=200+14.6=214.6\text{\hspace{0.17em}psia}\hfill \\ V_1=400{\text{\hspace{0.17em}ft}}^3\hfill \\ T_1=100+460=560°\text{R}\hfill \end{array}$

It can be seen that the volume of gas is constant (tank volume) and the temperature reduces from 100°F to 80°F. Therefore, using the Charles' law equation (Eq. [1.36]), we can calculate the final pressure as follows:

$\frac{214.6}{P_2}=\frac{560}{540}$

Solving for P ₂, we get

$P_2=206.94\text{\hspace{0.17em}psia}=206.94-14.6=192.34\text{\hspace{0.17em}psig}$

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Physical Properties

E. Shashi Menon , in Transmission Pipeline Calculations and Simulations Manual, 2015

15 Ideal Gases

An ideal gas is defined as a gas in which the volume of the gas molecules is negligible compared to the volume occupied by the gas. Also, the attraction or repulsion between the individual gas molecules and the container are negligible. Further, for an ideal gas, the molecules are considered to be perfectly elastic and there is no internal energy loss resulting from collision between the molecules. Such ideal gases are said to obey several classical equations such as the Boyle's law, Charles's law and the ideal gas equation or the perfect gas equation. We will first discuss the behavior of ideal gases and then follow it up with the behavior of real gases.

If M represents the molecular weight of a gas and the mass of a certain quantity of gas is m, the number of moles is given by

(3.42) $\mathrm{n}=\mathrm{m}/\mathrm{M}$

where n is the number that represents the number of moles in the given mass.

As an example, the molecular weight of methane is 16.043. Therefore, 50   lb of methane will contain approximately 3   mol.

The ideal gas law, sometimes referred to as the perfect gas equation simply states that the pressure, volume, and temperature of the gas are related to the number of moles by the following equation.

(3.43) $\mathrm{PV}=\mathrm{nRT}$

where

P – Absolute pressure, psia

V – Gas volume, ft³

n – Number of lb moles as defined in Equation (3.42)

R – Universal gas constant

T – Absolute temperature of gas, °R (°F   +   460).

The universal gas constant R has a value of 10.732   psia ft³/lb mole °R in USCS units. We can combine Eqn (3.42) with Eqn (3.43) and express the ideal gas equation as follows

(3.44) $\mathrm{PV}=\mathrm{mRT}/\mathrm{M}$

where all symbols have been defined previously. It has been found that the ideal gas equation is correct only at low pressures close to the atmospheric pressure. Because gas pipelines generally operate at pressures higher than atmospheric pressures, we must modify Eqn (3.44) to take into account the effect of compressibility. The latter is accounted for by using a term called the compressibility factor or gas deviation factor. We will discuss the compressibility factor later in this chapter.

In the perfect gas Eqn (3.44), the pressures and temperatures must be in absolute units. Absolute pressure is defined as the gauge pressure (as measured by a gauge) plus the local atmospheric pressure. Therefore

(3.45) ${\mathrm{P}}_{\mathrm{abs}}={\mathrm{P}}_{\mathrm{gauge}}+{\mathrm{P}}_{\mathrm{atm}}$

Thus if the gas pressure is 20   psig and the atmospheric pressure is 14.7   psia, we get the absolute pressure of the gas as

${\mathrm{P}}_{\mathrm{abs}}=20+14.7=34.7\mathrm{psia}$

Absolute pressure is expressed as psia, whereas the gauge pressure is referred to as psig. The adder to the gauge pressure, which is the local atmospheric pressure, is also called the base pressure. In SI units, 500   kPa gauge pressure is equal to 601   kPa absolute pressure if the base pressure is 101   kPa.

The absolute temperature is measured above a certain datum. In USCS units, the absolute scale of temperatures is designated as degree Rankin (°R) and is equal to the sum of the temperature in °F and the constant 460. In SI units, the absolute temperature scale is referred to as degree Kelvin (K). Absolute temperature in K is equal to °C   +   273.

Therefore,

Absolute temperature, °R   =   Temp   °F   +   460.

Absolute temperature, K   =   Temp   °C   +   460.

It is customary to drop the degree symbol for absolute temperature in Kelvin.

Ideal gases also obey Boyle's law and Charles's law. Boyle's law is used to relate the pressure and volume of a given quantity of gas when the temperature is kept constant. Constant temperature is also called isothermal condition. Boyle's law is as follows

${\mathrm{P}}_1/{\mathrm{P}}_2={\mathrm{V}}_2/{\mathrm{V}}_1$

or

(3.46) ${\mathrm{P}}_1{\mathrm{V}}_1={\mathrm{P}}_2{\mathrm{V}}_2$

where P₁ and V₁ are the pressure and volume at condition 1 and P₂ and V₂ are the corresponding value at some other condition 2 where the temperature is not changed.

Charles's law states that for constant pressure, the gas volume is directly proportional to the gas temperature. Similarly, if volume is kept constant, the pressure varies directly as the temperature. Therefore we can state the following.

(3.47) ${\mathrm{V}}_1/{\mathrm{V}}_2={\mathrm{T}}_1/{\mathrm{T}}_2\text{at}\text{constant}\text{pressure}$

(3.48) ${\mathrm{P}}_1/{\mathrm{P}}_2={\mathrm{T}}_1/{\mathrm{T}}_2\text{at}\text{constant}\text{volume}$

Example Problem 3.8

A certain mass of gas has a volume of 1000   ft³ at 60   psig. If temperature is constant and the pressure increases to 120   psig, what is the final volume of the gas? The atmospheric pressure is 14.7   psi.

Solution

Boyle's law can be applied because the temperature is constant. Using Eqn(3.46), we can write.

${\mathrm{V}}_2={\mathrm{P}}_1{\mathrm{V}}_1/{\mathrm{P}}_2$

or

${\mathrm{V}}_2=(60+14.7)\times 1000/(120+14.7)=554.57\mathrm{f}{\mathrm{t}}^3$

Example Problem 3.9

At 75   psig and 70   °F, a gas has a volume of 1000   ft³. If the volume is kept constant and the gas temperature increases to 120   °F, what is the final pressure of the gas? For constant pressure at 75   psig, if the temperature increases to 120   °F, what is the final volume? Use 14.7   psi for the base pressure.

Solution

Because the volume is constant in the first part of the problem, Charles's law applies.

$(75+14.7)/({\mathrm{P}}_2)=(70+460)/(120+460)$

Solving for P₂ we get

${\mathrm{P}}_2=98.16\mathrm{psia}\mathrm{or}88.46\mathrm{psig}$

For the second part, the pressure is constant and Charles's law can be applied.

${\mathrm{V}}_1/{\mathrm{V}}_2={\mathrm{T}}_1/{\mathrm{T}}_2$

$1000/{\mathrm{V}}_2=(70+460)/(120+460)$

Solving for V₂ we get

${\mathrm{V}}_2=1094.34\mathrm{f}{\mathrm{t}}^3$

Example Problem 3.10

An ideal gas occupies a tank volume of 250   ft³ at a pressure of 80   psig and temperature of 110   °F.

1.

What is the gas volume at standard conditions of 14.73   psia and 60   °F? Assume atmospheric pressure is 14.6   psia.

2.

If the gas is cooled to 90   °F, what is the gas pressure?

Solution

1.

Using the ideal gas Eqn (3.43), we can state that

${\mathrm{P}}_1{\mathrm{V}}_1/{\mathrm{T}}_1={\mathrm{P}}_2{\mathrm{V}}_2/{\mathrm{T}}_2$

${\mathrm{P}}_1=80+14.6=94.6\mathrm{psia}$

${\mathrm{V}}_1=250\mathrm{f}{\mathrm{t}}^3$

${\mathrm{T}}_1=110+460=570°\mathrm{R}$

${\mathrm{P}}_2=14.73$

V₂ is to be calculated

and

${\mathrm{T}}_2=60+460=520°\mathrm{R}$

$94.6\times 250/570=14.73\times {\mathrm{V}}_2/520$

${\mathrm{V}}_2=\mathrm{1,464.73}\mathrm{f}{\mathrm{t}}^3$

2.

When the gas is cooled to 90   °F, the final conditions are:

${\mathrm{T}}_2=90+460=550°\mathrm{R}$

${\mathrm{V}}_2=250\mathrm{f}{\mathrm{t}}^3$

P₂ is to be calculated.

The initial conditions are:

${\mathrm{P}}_1=80+14.6=94.6\mathrm{psia}$

${\mathrm{V}}_1=250\mathrm{f}{\mathrm{t}}^3$

${\mathrm{T}}_1=110+460=570°\mathrm{R}$

It can be seen that the volume of gas is constant and the temperature reduces from 110   °F to 90   °F. Therefore using Charles's law, we can calculate as follows

${\mathrm{P}}_1/{\mathrm{P}}_2={\mathrm{T}}_1/{\mathrm{T}}_2$

$94.6/{\mathrm{P}}_2=570/550$

${\mathrm{P}}_2=94.6\times 550/570=91.28\mathrm{psia}\mathrm{or}\mathrm{the}91.28-14.6=76.68\mathrm{psig}$

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Thermometry

C.A. Swenson , T.J. Quinn , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.B.1 Gas Thermometry

The ideal-gas law [Eq. (5)] is valid experimentally for a real gas only in the low-pressure limit, with higher-order terms (the virial coefficients, not defined here) effectively causing R to be both pressure and temperature dependent for most experimental conditions. While these terms can be calculated theoretically, most gas thermometry data are taken for a variety of pressures, and the ideal-gas limit, and, hence, the ideal-gas temperature, is achieved through an extrapolation to P=0. The slope of this extrapolation gives the virial coefficients, which are useful not only for experimental design, but also for comparison with theory. The following discussion of ideal-gas thermometry is concerned, first, with conventional gas thermometry, then with the measurement of sound velocities, and, finally, with the use of capacitance or interferometric techniques. Each of these instruments should give comparable results, although the "virial coefficients" will have different forms.

Gas thermometry in the past 20 years or so has benefited from a number of innovations that have improved the accuracy of the results. Pressures are measured using free piston (dead weight) gauges that are more flexible and easier to use than mercury manometers. The thermometric gas (usually helium) is separated from the pressure-measuring system by a capacitance diaphragm gauge, which gives an accurately defined room-temperature volume and a separation of the pressure-measurement system from the working gas. In addition, residual-gas analyzers can determine when the thermometric volume has been sufficiently degassed to minimize desorption effects.

In isothermal gas thermometry, absolute measurements of the pressure, volume, and quantity of a gas (number of moles) are used with the gas constant to determine the temperature directly from Eq. (5). Data are taken isothermally at several pressures, and the results are extrapolated to P  =   0 to obtain the ideal-gas temperature as well as the virial coefficients. A measurement at 273.16   K gives the gas constant.

A major problem in isothermal gas thermometry is determining the quantity of gas in the thermometer, since this ultimately requires the accurate measurement of a small difference between two large masses. Most often, this problem is bypassed by "filling" the thermometer to a known pressure at a standard temperature, with relative quantities of gas for subsequent fillings determined by division at this temperature between volumes that have a known ratio. The standard temperature may involve a fixed point or, for temperatures near the ice point, an SPRT that has been calibrated at the triple point of water. Since the volume of the gas for a given filling is constant for data taken on several subsequent isotherms, and the mass ratios are known very accurately, the absolute quantity of gas needs to be known only approximately. Excellent secondary thermometry is very important to reproduce the isotherm temperatures for subsequent gas thermometer fillings. The results for the isotherms (virial coefficients and temperatures) then are referenced to this standard "filling temperature."

The procedure for constant-volume gas thermometry is very much the same as that for isotherm thermometry, but detailed bulb pressure data are taken as a function of temperature for one (and possibly more) "filling" of the bulb at the standard temperature. To first order, pressure ratios are equal to temperature ratios, with thermodynamic temperatures calculated using known virial coefficients. In practice, the virial coefficients vary slowly with temperature, so a relatively few isotherm determinations can be sufficient to allow the detailed investigation of a secondary thermometer to be carried out using many data points in a constant-volume gas thermometry experiment. If the constant-volume gas thermometer is to be used in an interpolating gas thermometer mode (as for the ITS-90), the major corrections are due to the nonideality of the gas. When a nonideality correction is made using known values for the viral coefficients, the gas thermometer can be calibrated at three fixed points (near 4 and at 13.8 and 24.6   K) to give a quadratic pressure–temperature relation that corresponds to T within roughly 0.1   mK.

The velocity of sound in an ideal gas is given by

(6) ${\text{c}}^2=\left({\text{C}}_{\text{P}}/{\text{C}}_{\text{V}}\right)\text{<mi mathvariant="italic" is="true">RT</mi>}/\text{M},$

where the heat capacity ratio (C _P/C _V) is 5/3 for a monatomic gas such as helium. Since times and lengths can be measured very accurately, the measurement of acoustic velocities by the detection of successive resonances in a cylindrical cavity (varying the length at constant frequency) appears to offer an ideal way to measure temperature. This is not completely correct, however, since boundary (wall and edge) effects that affect the velocity of sound are important even for the simplest case in which only one mode is present in the cavity (frequencies of a few kilohertz). These effects unfortunately become larger as the pressure is reduced. An excellent theory relates the attenuation in the gas to these velocity changes, but the situation is very complex and satisfactory results are possible only with complete attention to detail. An alternative configuration uses a spherical resonator in which the acoustic motion of the gas is perpendicular to the wall, thus eliminating viscosity boundary layer effects. The most reliable recent determination of the gas constant, R, is based on very careful sound velocity measurements in argon as a function of pressure at 273.16   K, using a spherical resonator.

The dielectric constant and index of refraction of an ideal gas also are density dependent through the Clausius–Mossotti equation,

(7) $\left({ϵ}_{\text{r}}-1\right)/\left({ϵ}_{\text{r}}+2\right)=\alpha /{\text{V}}_{\text{m}}=\alpha \text{<mi mathvariant="italic" is="true">RT</mi>}/\text{P},$

in which ε_r(= ε/ε₀) is the dielectric constant and α is the molar polarizability. Equation (7) suggests that an isothermal measurement of the dielectric constant as a function of pressure should be equivalent to an isothermal gas thermometry experiment, while an experiment at constant pressure is equivalent to a constant-volume gas thermometry experiment. The dielectric constant, which is very close to unity, is most easily determined in terms of the ratio of the capacitance of a stable capacitor that contains gas at the pressure P to its capacitance when evacuated. The results that are obtained when this ratio is measured using a three-terminal ratio transformer bridge are comparable in accuracy with those from conventional gas thermometry. An advantage is that the quantity of gas in the experiment need never be known, although care must be taken in cell design to ensure that the nonnegligible changes in cell dimensions with pressure can be understood in terms of the bulk modulus of the (copper) cell construction material.

At high frequencies (those of visible light), the dielectric constant is equal to the square of the index of refraction of the gas (ε_r  = n ²), so an interferometric experiment should also be useful as a primary thermometer. No results for this type of experiment have been reported, however.

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Write an Equation to Express the Relationship Between Pressure and Temperature

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