Pressure of an ideal gas

Most thermometry using the KTTS direcdy requites a thermodynamic instmment for interpolation. The vapor pressure of an ideal gas is a thermodynamic function, and a common device for reali2ing the KTTS is the helium gas thermometer. The transfer function of this thermometer may be chosen as the change in pressure with change in temperature at constant volume, or the change in volume with change in temperature at constant pressure. It is easier to measure pressure accurately than volume thus, constant volume gas thermometry is the usual choice (see Pressure measurement). 

We may therefore sum up the results in the statement that the laws of osmotic pressure of a dilute solution are formally identical with the laws of gas pressure of an ideal gas (van t Hoff s Gaseous Theory of Solution). 

Figure 3-1. The pressure of an ideal gas as a function of log(volume) and temperature.

At the same temperature, how would the pressure of an ideal gas differ from that of a gas with mutually attractive particles How would it differ from that of a gas with mutually repulsive... 

In contrast to the pressure of an ideal gas, the fugacity is not only a function of the amount of substance and temperature, but also of the composition (types and amounts of gaseous compounds present) of the gaseous system and of die total pressure. The fugacitiy of a gaseous compound is, however, closely related to its partial pressure. To account for the nonideality of the gas, one can relate these terms by using a fugacity coefficient, 0ig ... 

What happens to the composition of the equilibrium mixture if we increase the pressure by decreasing the volume (Recall from Sections 9.2 and 9.3 that the pressure of an ideal gas is inversely proportional to the volume at constant temperature and constant number of moles of gas P = nRT/V.) According to Le Chatelier s principle, net reaction will occur in the direction that relieves the stress of the increased pressure, which means that the number of moles of gas must decrease. Therefore, we predict that the net reaction will proceed from left to right because the forward reaction converts 4 mol of gaseous reactants to 2 mol of gaseous products. In general, Le Chatelier s principle predicts that... 

If a gas deviates from ideality, this equation ceases to apply. However, it is desired to preserve this simple form of expression as much as possible for non-ideal, real gases. The above equation shows that the free energy G is a linear function of the logarithm of the pressure of an ideal gas. Now, let s introduce a function which when used in place of the real pressure ensures linearity between G and the logarithm of this function in any state of any gas. This function is called fugacity (/), a sort of corrected pressure. 

Example 1. Find an expression for the internal pressure of an ideal gas. 

The observed pressure of a real gas is less than the pressure of an ideal gas. There are large number of particles in any sample of a gas for which the average velocity must be zero, whereas the average velocity will always be final. 

Given the component partial pressures of an ideal gas mixture and the total gas pressure, determine the mixture composition expressed in either mole fractions (or mole percents), volume fractions (or % v/v), or mass fractions (or % w/w). 

The relationship between the density p (mass/volume), temperature, and pressure of an ideal gas can be obtained by first relating the specific molar volume, V (volume/mole), to the density. Using a specific set of units for illustration. 

The last equality is true because AH = 0 when the pressure of an ideal gas is changed at constant temperature. The entropy change for an ideal gas in an isothermal process was calculated in Section 13.5 ... 

Loosely speaking, pressure of an ideal gas is the random translational kinetic energy per volume. Pressure is an intensive state function. Pressure and volume are discussed in depth in Chemistry Lecture 2 and Physics Lecture 5. 

The number of network strands per unit volume (number density of strands) is p = njV. In the last equality, p is the network density (mass per unit volume), Mg is the number-average molar mass of a network strand, and IZ is the gas constant. The network modulus increases with temperat-iire hpnaiisp its origin is entropic, analogous to the pressure of an ideal gas p = nkTjV. The modulus also increases linearly with the number density of network strands u = n/V = pJ fEquation (7.31) states that the modulus of any network polymer is kT per strand. 

From kinetic theory, the pressure of an ideal gas can have the form... 

If the gas is behaving like an ideal gas, a doubling of the Kelvin temperature doubles the pressure. If the temperature decreases to 50% of the original Kelvin temperature, the pressure decreases to 50% of the original pressure. This relationship can be expressed by saying that the pressure of an ideal gas is directly proportional to the Kelvin temperature of the gas if the volume and the number of gas particles are constant. This relationship is sometimes called Gay-Lussac s Law. 

We discovered in Section 13.1 that pressure of an ideal gas is direcdy proportional to the number of gas particles (expressed in moles), direcdy proportional to temperature, and inversely proportional to the volume of the container. 

Gay-Lussac s Law The pressure of an ideal gas is directly proportional to the Kelvin temperature of the gas if the volume and the number of gas particles are constant. 

From the discussion of Sec. 3.3 the internal energy change and pressure of an ideal gas are... 

In otherwords, the osmotic pressure of an Ideal solution (ideal means that it obeys = AcRT) is the same as the pressure of an Ideal gas at the same pressure and temperature. 

We begin by observing that the pressure of an ideal gas can be written as the product of the number of particles N and k T/V, the pressure of a system consisting of a single particle ... 

Recall that the partial pressure of an ideal gas in a mixture is given by ... 

As the pressure of an ideal gas increases, its volume decreases proportionately. Charles law describes the relationship between the volume and temperature of an ideal gas at constant pressure ... 

Of course, in the long run, all the work transforms into heat which is dissipated in the surroundings. How do we know Well, if there were no surrounding medium to take the heat away, both a gas when compressed and a polymer when stretched would get warmer (see also below Section 7.11). Does that mean elasticity of a polymer chain depends on the environment which absorbs the heat Well, we know that the pressure of an ideal gas does not depend on the type of the environment, so maybe there is something similar for a polymer ... 

Thus, E turns out to be the same as the pressure of an ideal gas whose molecular concentration is 3z/ (i.e. three times the concentration of the cross-links). It means that the more cross-links there are in a highly elastic sample, the less elastic it is. Therefore, the value of E does not indicate a specific polymer. It varies dramatically depending on the density of the cross-links. 

Note that although the extensive volumes are all the same V = Vpure intensive volumes differ, v < Upm-e, because each pure gas necessarily contains fewer molecules than the mixture. Equation (4.1.19) states that the pressure of an ideal-gas mixture is the sum of the pure component pressures, when N,- molecules of each pme i are confined to a vessel having the same extensive volume V as that of the mixture vessel and each pure is at the mixture temperature T. Since each pure component is an ideal gas, we can substitute the ideal-gas law (4.1.1) into the rhs of (4.1.19) and find the same equation of state as for pure gases. 

A colligative property depends on the number of molecules, not on their nature. The main examples of colligative properties are the pressure of an ideal gas (pV = nRT), the depression of the freezing point of a solvent in the presence of a solute, the elevation of the boiling point by a solute and also the osmotic pressure. The latter three are used for Mn determination. 

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