Sackur Tetrode Equation

5 Key excerpts on "Sackur Tetrode Equation"

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  eBook - ePub

  Biomolecular Thermodynamics

  From Theory to Application

  • Douglas Barrick(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  Equation 5.32 ).

  Each of the four energy differentials shown in Figure 5.4 leads to a unique pair of thermodynamic identities like those in Equation 5.32 (Table 5.1 ). The intensive variables T and p result from differentiation with respect to extensive quantities, whereas the extensive variables S and V result from differentiation with respect to intensive quantities.

  One particularly fundamental (and somewhat surprising) identity comes from the derivative of internal energy with respect to entropy:

  T = ( ∂ U ∂ S ) V	(5.33)

  This relationship states that the temperature, a quantity that we have an intuitive (and at least some mechanistic) understanding of, is equal to the extent to which the internal energy increases as the entropy increases.‡ At low temperature, a unit energy change corresponds to a large change in entropy. At high temperature, a unit energy change corresponds to a small change in entropy. This is consistent with the description of entropy in Chapter 4 , where we stated that at low temperature, heat produces a large increase in entropy compared to that at high temperature.

  This relationship can be illustrated with the four-particle energy ladder from Chapter 4 (Figure 5.6 ). Configurations are arranged in terms of energy (from left to right), with energy increasing by one unit from one column to the next. For each of these energy increases, the total number of microscopic configurations goes up, consistent with an entropy increase, and thus, from Equation 5.33 , a positive temperature value. However, as the total energy increase, the entropy increase (Δ S , calculated using the Boltzmann entropy formula from Chapter 4 ) per unit energy increase (Δ U ) goes down, consistent with a higher temperature. This relationship is shown in the bottom line of Figure 5.6 .

  Figure 5.6 The relationship between internal energy, entropy, and temperature for an energy ladder with equal spacings.

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  Thermodynamics For Dummies

  • Mike Pauken, Michael Pauken(Authors)
  • 2011(Publication Date)
  • For Dummies

    (Publisher)

  dS . Two important relationships, known as the Gibbs equations, are used to define relationships between entropy, heat transfer, and temperature.

  The energy equation developed in Chapter 5, written in differential form, gives a mathematical relationship between heat transfer (δ Q ), internal energy (dU ), and work (δ W ) for an internally reversible process as follows:

  δ Q rev = dU + δ W rev

  You get the first Gibbs equation by replacing δ Q rev with dU + δ W rev in the equation that defines entropy:

  The first Gibbs equation relates the change in entropy of a system (dS ) to internal energy (dU ) and boundary work where δ W rev = PdV , as follows:

  Internal energy is the energy in a material related to its molecular activity. I discuss internal energy in Chapter 2. Boundary work occurs when a boundary in a system moves, such as a piston moving in a cylinder (see Chapter 5).

  The second Gibbs equation relates the entropy change of a system to enthalpy and flow work. Enthalpy (H ) is a property that combines internal energy (U ) plus the product of pressure and specific volume (PV ) as I discuss in Chapter 2. Flow work is associated with the work done by flowing fluids in a process, such as a turbine or a compressor in a gas turbine engine (see Chapter 5). The second Gibbs equation is written as follows:

  I discuss how these relationships are used to calculate changes in entropy for several different thermodynamic systems in the following sections.

  Calculating Entropy Change

  In this section, I discuss how to use and modify the two Gibbs equations from the preceding section to determine the entropy change for processes involving solids, liquids, gases, saturated liquid-vapor mixtures, and ideal gases. You can find the enthalpy change of a process by integrating either of the Gibbs equations between the initial and final states of a process.

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  eBook - ePub

  BIOS Instant Notes in Physical Chemistry

  • Gavin Whittaker, Andy Mount, Matthew Heal(Authors)
  • 2000(Publication Date)
  • Taylor & Francis

    (Publisher)

  For the special case of a monatomic gas, the only contribution to the partitionfunction results from translational energy levels. This ultimately yields theSackur-Tetrode equation for the entropy of a perfect

  monatomic gas of mass, m , at a pressure, p :

  Heat Capacity

  Partition functions allow calculation of the heat capacity of a system. The following discussion of heat capacity applies to the constant volume heat capacity , from which the constant pressure heat capacity may be easily calculated (Topic B1 ). For a gas, substitution of

  qtrans

  into the expression for U yields

  Therefore the molar translational heat capacity is given by Ctrans =dEtrans /dT= 3R/2, and

  qrot

  and

  qvib

  can be likewise treated. It is found that, for a diatomic gas, both the molar quantities Crot and

  Cvib

  vary between 0 and R depending upon the ratio of kT to the difference between energy levels, hv . For

  Crot

  or

  Cvib

  , when , the heat capacity is zero, rising to a molar value of R when . Generally, for a translation or a rotation, the maximum heat capacity is equal to R/2 for each degree of freedom , that is, each independent mode of motion. Thus, a gaseous diatomic molecule may have three translational degrees of freedom (one for each orthogonal direction of motion), two rotational degrees of freedom (from rotation about each of two axes perpendicular to the main axis of the molecule and to one another). Any vibration contributes R to the molar heat capacity, R /2 each from the kinetic and potential energy components of the vibration.

  The difference between energy levels follows the trend , and this means that at low temperature only the translational motion makes a significant contribution to the heat capacity. As temperature increases, the heat capacity progressively increases also, as first the rotational modes, and then the vibrational modes contribute to the heat capacity (Fig. 2 ).

  Fig. 2

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  BIOS Instant Notes in Chemistry for Biologists

  • J Fisher, J.R.P. Arnold, Julie Fisher, John Arnold(Authors)
  • 2020(Publication Date)
  • Taylor & Francis

    (Publisher)

  If we add a third molecule, the probability of finding all three in V 2 is 0.125 (i.e. 0.5 3). Thus, when we get to more realistic numbers of molecules, co for finding all the molecules in V 1 becomes extremely small. So, if the volume of the cylinder is reduced to V 1 and then increased to V 2, it is highly unlikely that the gas molecules will all remain in V 1. This analysis shows how probability is an important factor in describing a system, and predicting how it will behave. It is therefore reasonable to associate entropy with probability. The appropriate expression involves the natural logarithm of the probability: This is known as the Boltzmann equation, and k is the Boltzmann's constant. (Boltzmann was a 19th century physicist whose ideas were not generally accepted until after his death in 1906.) If we have a process that proceeds from state 1 (which has a probability of ω 1) to state 2, which has a probability of ω 2, then the change in entropy is: This expression may be modified, as probabilities may be expressed in terms of volumes, eventually leading to; where n is the number of moles of the gas present, R is the ideal gas constant. It should be noted that this expression holds at constant temperature and also that it is not necessary to specify the manner in which change has been brought about. This is because entropy is another function of state. The Boltzmann equation leads us to the third law of thermodynamics. This states that the entropy of a perfect crystal is zero at a temperature of absolute zero. That is to say: at this temperature the crystal is perfectly ordered, so only one arrangement of the constituent atoms is possible (neglecting any interchanging of identical atoms), and so its probability is 1, and ln(ω) is zero. This is a somewhat abstract concept, since a temperature of absolute zero has never been achieved experimentally

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  Compressors

  Selection and Sizing

  • Royce N. Brown(Author)
  • 2011(Publication Date)
  • Gulf Professional Publishing

    (Publisher)

  By establishing a starting point at P 1, and T 1, and taking a path of constant entropy to P 2, the isentropic T 2 (ideal) value can be read from the diagram. For a gas mixture or gas with no convenient Mollier diagram available, the problem becomes more acute. There are two alternatives: one is to use an equation of state and the other is to use a method suggested by Edmister and McGarry [2]. The latter is somewhat tedious, making the equation of state the preferred method. Real Gas Tools A good tool to have when working with gases outside the practical range of the Perfect Gas Law are computer codes that will generate gas physical properties from a catalog of equations of state. In itself this would be useful when working with single gas streams. To make this tool even more useful, the capability to solve for properties of mixtures is needed. While there are numerous methods available for computing gas mixtures, one of the most basic is Kay’s Rule, which for pseudocritical temperatures uses a simple mole-fraction average [5]. The final element that should be included is a database of gases. Unfortunately, this type of program is not all that common and is hard to find available commercially. There is a tool used by process engineers that contains the elements called for in the preceding paragraph. This tool is called a process simulator. Generally these tools or computer codes are expensive. These programs are used to design process streams and have capability beyond what is needed for working just with compressors. There is one process simulator on the market at a lower cost that includes all the items described above. It can be purchased without all the process components such as packed towers and other items related to the chemistry of process design. This process simulator is Design II by WinSim, Inc

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