Ideal gas law derivation

Deriving empirical gas laws from the ideal gas law Starting from the ideal gas law, derive the relationship between any two variables. (EXAMPLE 5.5)... 

EXAMPLE 7.1 The ideal gas law derived from the lattice modeL Take the definition of pressure, pjT = dSjdV)y,u, from Equation (7.6). Into this expression, insert the function 5(V) from the lattice model in Example 2.2. For a lattice of M sites with N particles, use the Boltzmann expression for entropy and Equation (2.3) to get... 

The area for an ideal gas film is, thus, given by the ideal gas law derived in Example 4.3, which per molecule can be written using the Boltzmann constant ( b) ... 

The last quantity that we discuss is the mean repulsive force / exerted on the wall. For a single chain this is defined taking the derivative of the logarithm of the chain partition function with respect to the position of the wall (in the —z direction). In the case of a semi-infinite system exposed to a dilute solution of polymer chains at polymer density one can equate the pressure on the wall to the pressure in the bulk which is simply given by the ideal gas law The conclusion then is that [74]... 

The ideal gas law is readily applied to problems of this type. A relationship between the variables involved is derived from this law. In this case, pressure and temperature change, while n and V remain constant. 

Similar two-point equations can be derived from the ideal gas law to solve any problem of this type. 

The quantities n, V, and (3 /m) T are thus the first five (velocity) moments of the distribution function. In the above equation, k is the Boltzmann constant the definition of temperature relates the kinetic energy associated with the random motion of the particles to kT for each degree of freedom. If an equation of state is derived using this equilibrium distribution function, by determining the pressure in the gas (see Section 1.11), then this kinetic theory definition of the temperature is seen to be the absolute temperature that appears in the ideal gas law. 

The ideal gas law says that the molar density is determined by pressure and temperature and is thus known and constant in the reactor. Setting the time derivative of molar density to zero gives an expression for Qom at steady state. The result is... 

Rearranging this equation so that we have mass divided by volume, we can derive an ideal gas law equation that will allow us to calculate density of any ideal gas, given the temperature, pressure and MW, per... 

Vapor density The vapor density of a substance is defined as the ratio of the mass of vapor per unit volume. An equation for estimating vapor density is readily derived from a varied form of the ideal gas law ... 

With the acceptance of the atomic view of the world - accompanied by the necessity to explain reactions in extremely dilute gases (where the continuum theory fails) - the kinetic gas theory was developed. Using this it is possible not only to derive the ideal gas law in another manner but also to calculate many other quantities involved with the kinetics of gases - such as collision rates, mean free path lengths, monolayer formation time. 

We utilize the laboratory, which is not a separate course, in the process of introducing higher level mathematics. For instance, the first day of laboratory is given to mathematics exercises that review simple integrals and derivatives, and the chain rule. This is also where partial derivatives are introduced using the ideal gas law and the van der Waals equation as object lessons. It is here that we also introduce the triangle derivative rule for partial derivatives, Eqn 4. 

Kistiakowsky (1923) utilized the Clapeyron equation and the ideal gas law to derive an expression to estimate each individual compound s in which the... 

The material covered in this chapter is self-contained, and is derived from well-known relationships such as Newton s second law and the ideal gas law. Some quantum mechanical results and the statistical thermodynamics definition of entropy are given without rigorous derivation. The end result will be a number of practical formulas that can be used to calculate thermodynamic properties of interest. 

See Table 3 1.4). In the derivation of Equation 11.35, the ideal gas law was used. For high-pressure GC and situations in which carrier gases which deviate from ideality are used, this equation may not be valid. 

Density is defined as the concentration of matter, measured by the mass per unit volume [1]. The molar volume, Vm, is defined as the volume occupied by 1 mol of a substance. The molar volume of an ideal gas is 22.4140dm3mol-1 (22.4140liter mol-1) at 1 atm pressure and 0°C. Vapor densities pv are derived through rearrangement of the ideal gas law equation as... 

In meteorological calculations, the ideal gas law is a satisfactory approximation for the derivation of formulas for the mixture of gases that constitute the atmosphere. The derivation is ... 

It should be quite obvious that, although the model provided in the form of the ideal gas law does a reasonable job at lower pressures, it rapidly deviates as the pressure increases and the volume decreases. We can see this more clearly in Figure 2.5, where we compare the real data with that derived from the ideal gas law in a scatter plot of p versus 1/v. We can see from our plot that the experimental data, shown as solid circles, are modelled reasonably well by a linear (straight line) function, but only for pressures less than 50 atm. The Boyle model is clearly of limited applicability in this case. 

Beginning with these assumptions, it s possible not only to understand the behavior of gases but also to derive quantitatively the ideal gas law (though we ll not do so here). For example, let s look at how the individual gas laws follow from the five postulates of kinetic-molecular theory ... 

The ideal gas law can be derived solely from theoretical principles by making a few assumptions about the nature of gases and the meaning of temperature. The derivation can be found in any physical chemistry textbook. 

The ideal gas law can be derived from the combined gas law. One interpretation of the combined gas law is that volume is inversely proportional to pressure and directly proportional to absolute temperature and to moles. That is, the product of pressure and volume, divided by the product of absolute temperature and moles will equal a constant value. If we state the combined gas law for a single set of conditions, we can obtain the ideal gas law ... 

FIGURE 7.3 Simple model system for derivation of the ideal gas law. We start by assuming there is only one molecule in a cube of length L, and it is moving directly along the y-axis. We will generalize these results in the next section. 

Gas molecules do not actually bounce off the wall of a container (or your skin) as if it were a uniform massive structure, the way we sketched it in Figure 7.2 they collide with individual atoms at the wall surface, which are also moving because of vibrations. If the temperature of the wall and the gas are the same, on average the gas kinetic energy is as likely to increase or decrease as a result of any single collision. Thus a more accurate statement of the assumption required to derive the ideal gas law is that the walls and gas molecules are at the same temperature, so there is no average energy flow between the two. 

Assume this laser pulse is completely absorbed by a black wall. Use the relation E = cp to calculate the momentum it transfers to the wall. By analogy with the calculations we did to derive the ideal gas law, calculate the radiation pressure the pulse exerts on the 10 /rm spot while it is on. 

For a macroscopic box (say L = 0.1 m) and realistic molecular masses, the separation between these levels is far less than ks T. As a result, the distribution of energy levels appears virtually continuous, and quantum corrections to (for example) the ideal gas law are generally extremely small. However, it is possible to use Equation 8.6 to derive the ideal gas law from quantum mechanics instead of using the kinetic theory of gases (see Problem 8-10). 

Chapter 7 covers the kinetic theory of gases. Diffusion and the one-dimensional velocity distribution were moved to Chapter 4 the ideal gas law is used throughout the book. This chapter covers more complex material. I have placed this material later in this edition, because any reasonable derivation of PV = nRT or the three-dimensional speed distribution really requires the students to understand a good deal of freshman physics. There is also significant coverage of dimensional analysis determining the correct functional form for the diffusion constant, for example. 

The ideal gas law will also be derived from the point of view of statistical mechanics in Chapter 5. 

Derive the Clausius-Clapeyron equation [Eq. (44)] from Eq. (40) by neglecting the volume of the condensed phase and using the ideal gas law for the vapor. 

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