Temperature pressure, volume, mass relationship

The properties of gases that are most easily observed are the relationships among pressure, volume, temperature, and mass. If you have ever inflated a balloon, baked a cake, or slept on an air mattress, you have observed how these properties are related. Because the laws of gases were developed from the study of their properties and behavior, it is now possible to predict the physical behavior of gases by the application of these laws. 

Next, we study the relationship among pressure, volume, temperature, and amount of a gas in terms of various gas laws. We wiU see that these laws can be summarized by the ideal gas equation, which can be used to calculate the density or molar mass of a gas. (5.3 and 5.4)... 

The Joule-Thomson effect occurs without the transfer of heat. Temperature is affected by the relationship between volume, mass, pressure, and temperature. Rapid expansion of a gas from high to low pressure results in a temperature drop. This principle was employed by Dutch physicist Heike Kamerlingh Onnes to liquefy helium in 1908 and is useful in home refrigerators and air conditioners. 

Phase behaviour describes the phase or phases in which a mass of fluid exists at given conditions of pressure, volume (the inverse of the density) and temperature (PVT). The simplest way to start to understand this relationship is by considering a single component, say water, and looking at just two of the variables, say pressure and temperature. 

The relationship between pressure p, volume V, mass m, and temperature T is given by the equation of state ... 

Because moles are the currency of chemistry, all stoichiometric computations require amounts in moles. In the real world, we measure mass, volume, temperature, and pressure. With the ideal gas equation, our catalog of relationships for mole conversion is complete. Table lists three equations, each of which applies to a particular category of chemical substances. 

There are four intrinsic, measurable properties of a gas (or, for that matter, any substance) its pressure P, temperature T, volume (in the case of a gas, the container volume) V, and mass m, or mole number n. The gas density d is a derived quantity, which is m/V. Before the relationships between these properties for a gas are discussed, the units in which they are usually reported will be outlined. 

Of the three quantities (temperature, energy, and entropy) that appear in the laws of thermodynamics, it seems on the surface that only energy has a clear definition, which arises from mechanics. In our study of thermodynamics a number of additional quantities will be introduced. Some of these quantities (for example, pressure, volume, and mass) may be defined from anon-statistical (non-thermodynamic) perspective. Others (for example Gibbs free energy and chemical potential) will require invoking a statistical view of matter, in terms of atoms and molecules, to define them. Our goals here are to see clearly how all of these quantities are defined thermodynamically and to make use of relationships between these quantities in understanding how biochemical systems behave. 

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 number of moles is a fourth variable that can be added to pressure, volume, and temperature as a way to describe a gas sample. Recall that as the other gas laws were presented, care was taken to state that the relationships hold true for a fixed mass or a given amount of a gas sample. Changing the number of gas particles present will affect at least one of the other three variables. 

Figure 12-5 An experiment showing that the volume of an ideal gas increases as the temperature is increased at constant pressure, (a) A mercury plug of constant weight, plus atmospheric pressure, maintains a constant pressure on the trapped air. (b) Some representative volume-temperature data at constant pressure. The relationship becomes clear when t (°C) is converted to T (K) by adding 273°C. (c) A graph in which volume is plotted versus temperature on two different scales. Lines A, B, and C represent the same mass of the same ideal gas at different pressures. Line A represents the data tabulated in part (b). Graph D shows the behavior of a gas that condenses to form a liquid (in this case, at 50°C) as it is cooled.

This simply shows that there is a physical relationship between different quantities that one can measure in a gas system, so that gas pressure can be expressed as a function of gas volume, temperature and number of moles, n. In general, some relationships come from the specific properties of a material and some follow from physical laws that are independent of the material (such as the laws of thermodynamics). There are two different kinds of thermodynamic variables intensive variables (those that do not depend on the size and amount of the system, like temperature, pressure, density, electrostatic potential, electric field, magnetic field and molar properties) and extensive variables (those that scale linearly with the size and amount of the system, like mass, volume, number of molecules, internal energy, enthalpy and entropy). Extensive variables are additive whereas intensive variables are not. 

The volumetric equation of state may also provide another relationship between the temperature, pressure, mass, and volume when the information about the final state of the system is presented in terms of total volume, rather than volume per unit mass or molar volume (see Illustration 3.4-5). 

Equation of state (EOS) n. For an ideal gas, if the pressure and temperature are constant, the volume of of the gas depends on the mass, or amount of gas. Then, a single property called the gas density (ratio of mass/volume). If the mass and temperature are held constant, the product of pressure and volume are observed to be nearly constant for a real gas. The product of pressure and volume is exactly for an ideal gas. This relationship between pressure and volume is called Boyle s Law. Finally, if the mass and pressure are held constant, the volume is directly proportional to the temperature for an ideal gas. This relationship is called Charles and Gay-Lussac s law. The gas laws of Boyle and Charles and Gay-Lussac can be combined into a single equation of state PV = nRT, where P is pressure, V volume, Tabsolute temperature, n number of moles and R is the universal gas constant. Ane-rodynamicists us a different form of the equation of state that is specialized of air. Regarding polymers and monomers, equation of state is an equation giving the specific volume (v) of a polymer from the known temperature and pressure and, sometimes, from its morphological form. An early example is the modified Van der Waals form, successfully tested on amorphous and molten polymers. The equation is ... 

Thermodynamic properties are characteristics of a system (e.g., pressure, temperature, density, specific volume, enthalpy, entropy, etc.). Because properties depend only on the state of a system, they are said to be path independent (unlike heat and work). Extensive properties are mass dependent (e.g., total system energy and system mass), whereas intensive properties are independent of mass (e.g., temperature and pressure). Specific properties are intensive properties that represent extensive properties divided by the system mass, for example, specific enthalpy is enthalpy per unit mass, h = H/m. In order to apply thermodynamic balance equations, it is necessary to develop thermodynamic property relationships. Properties of certain idealized substances (incompressible liquids and ideal gases with constant specific heats) can be calculated with simple equations of state however, in general, properties require the use of tabulated data or computer solutions of generalized equations of state. 

The graph shows gas volume versus temperature fora given mass of gas at 1.00 atm pressure. This linear relationship is independent of amount or type of gas. Note that all lines extrapolate to -273°C at zero volume. 

For a semi-batch operation for the first stages, optimal variations of pressure and temperature can be calculated based on the above relationships plus the assumption of phase equilibrium, or on a simple relationship for the mass transfer of each volatile component Y (Eq. (55), with the mass transfer rates per unit volume Ji of component Y , mass transfer coefficient of component i kfi, interface area per unit volume a , and equilibrium concentration [Yj at the interface). 

This equation is sometimes referred to as the combined gas law. In Example 6-6, both volume and mass are constant, and this establishes the simple relationship between gas pressure and temperature known as Amontons s law The pressure of a fixed amount of gas confined to a fixed volume is directly proportional to the Kelvin temperature. 

It is thus seen that heat capacity at constant volume is the rate of change of internal energy with temperature, while heat capacity at constant pressure is the rate of change of enthalpy with temperature. Like internal energy, enthalpy and heat capacity are also extensive properties. The heat capacity values of substances are usually expressed per unit mass or mole. For instance, the specific heat which is the heat capacity per gram of the substance or the molar heat, which is the heat capacity per mole of the substance, are generally considered. The heat capacity of a substance increases with increase in temperature. This variation is usually represented by an empirical relationship such as... 

Generally, the higher the pressure, the higher is the solubility of a gas in a liquid. This relationship is expressed quantitatively by Henry s Law which states that the mass of gas (m) dissolved by a given volume of solvent at a constant temperature is proportional to the gas pressure (p) with which it is in equilibrium ... 

The kinetic molecular theory (KMT see Sidebar 2.7) of Bernoulli, Maxwell, and others provides deep insight into the molecular origin of thermodynamic gas properties. From the KMT viewpoint, pressure P arises merely from the innumerable molecular collisions with the walls of a container, whereas temperature T is proportional to the average kinetic energy of random molecular motions in the container of volume V. KMT starts from an ultrasimplified picture of each molecule as a mathematical point particle (i.e., with no volume ) with mass m and average velocity v, but no potential energy of interaction with other particles. From this purely kinetic picture of chaotic molecular motions and wall collisions, one deduces that the PVT relationships must be those of an ideal gas, (2.2). Hence, the inaccuracies of the ideal gas approximation can be attributed to the unrealistically oversimplified noninteracting point mass picture of molecules that underlies the KMT description. 

The head flow meter actually measures volume flow rate rather than mass flow rate. Mass flow rate is easily calculated or computed from volumetric flow rate by knowing or sensing temperature and/or pressure. Temperature and pressure affect the density of the fluid and, therefore, the mass of fluid flowing past a certain point. If the volumetric flow rate signal is compensated for changes in temperature and/or pressure, a true mass flow rate signal can be obtained. In Thermodynamics it is described that temperature and density are inversely proportional, while pressure and density are directly proportional. To show the relationship between temperature or pressure, the mass flow rate equation is often written as either Equation 4-1 or 4-2. 

Here we will use a simplified example to illustrate some basic aspects of the mass transport process for carbonates that avoids most of the more complex relationships. In this example, the calcium and carbonate ion concentrations are set equal, and values of the activity coefficients, temperature, and pressure are held constant. The carbonate ion concentration is considered to be independent of the carbonic acid system. The resulting simple (and approximate) relation between the change in saturation state of a solution and volume of calcite that can be dissolved or precipitated (Vc) is given by equation 7.4, where v is the molar volume of calcite. 

Although Charles discovered that the volume of a fixed amount of gas at constant pressure was proportional to its temperature, he never published this finding. In 1802, Joseph Louis Gay-Lussac (1778-1850), a French scientist, made reference to Charles work in a published paper. The relationship between temperature and volume has since become known as Charles law. Charles law states that the volume of a fixed mass of gas is proportional to its temperature when the pressure is kept constant. 

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