Ideal Gas Law Derivation Statistical Mechanics Pdf

Ideal gas law derivation statistical mechanics pdf

Ideal gas law derivation statistical mechanics pdf

The ideal gas law , also called the general gas equation , is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations.

It is the same for all gases. The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms.

Ideal gas law derivation statistical mechanics pdf

The temperature used in the equation of state is an absolute temperature: the appropriate SI unit is the kelvin. In SI units , p is measured in pascals , V is measured in cubic metres , n is measured in moles , and T in kelvins the Kelvin scale is a shifted Celsius scale, where 0. R has the value 8. How much gas is present could be specified by giving the mass instead of the chemical amount of gas.

Therefore, an alternative form of the ideal gas law may be useful. The chemical amount n in moles is equal to total mass of the gas kg in kilograms divided by the molar mass M in kilograms per mole :. This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas.

Alternatively, the law may be written in terms of the specific volume v , the reciprocal of density, as.

Ideal gas law derivation statistical mechanics pdf

It is common, especially in engineering and meteorological applications, to represent the specific gas constant by the symbol R. In statistical mechanics the following molecular equation is derived from first principles.

When comparing the same substance under two different sets of conditions, the law can be written as. According to assumptions of the kinetic theory of ideal gases, we assume that there are no intermolecular attractions between the molecules of an ideal gas.

In other words, its potential energy is zero.

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Hence, all the energy possessed by the gas is in the kinetic energy of the molecules of the gas. This is the kinetic energy of n moles of a monatomic gas having 3 degrees of freedom ; x , y , z. The table below essentially simplifies the ideal gas equation for a particular processes, thus making this equation easier to solve using numerical methods.

A thermodynamic process is defined as a system that moves from state 1 to state 2, where the state number is denoted by subscript. As shown in the first column of the table, basic thermodynamic processes are defined such that one of the gas properties P , V , T , S , or H is constant throughout the process.

For a given thermodynamics process, in order to specify the extent of a particular process, one of the properties ratios which are listed under the column labeled "known ratio" must be specified either directly or indirectly. Also, the property for which the ratio is known must be distinct from the property held constant in the previous column otherwise the ratio would be unity, and not enough information would be available to simplify the gas law equation.

In the final three columns, the properties p , V , or T at state 2 can be calculated from the properties at state 1 using the equations listed.

In an isentropic process, system entropy S is constant. In an isenthalpic process, system enthalpy H is constant. In the case of free expansion for an ideal gas, there are no molecular interactions, and the temperature remains constant. For real gasses, the molecules do interact via attraction or repulsion depending on temperature and pressure, and heating or cooling does occur.

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This is known as the Joule—Thomson effect. There are in fact many different forms of the equation of state. Since the ideal gas law neglects both molecular size and inter molecular attractions, it is most accurate for monatomic gases at high temperatures and low pressures.

The neglect of molecular size becomes less important for lower densities, i. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy , i. More detailed equations of state , such as the van der Waals equation , account for deviations from ideality caused by molecular size and intermolecular forces.

A residual property is defined as the difference between a real gas property and an ideal gas property, both considered at the same pressure, temperature, and composition.

The empirical laws that led to the derivation of the ideal gas law were discovered with experiments that changed only 2 state variables of the gas and kept every other one constant.

Ideal gas law derivation statistical mechanics pdf

To derive the ideal gas law one does not need to know all 6 formulas, one can just know 3 and with those derive the rest or just one more to be able to get the ideal gas law, which needs 4. Since each formula only holds when only the state variables involved in said formula change while the others remain constant, we cannot simply use algebra and directly combine them all.

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Boyle did his experiments while keeping N and T constant and this must be taken into account. Keeping this in mind, to carry the derivation on correctly, one must imagine the gas being altered by one process at a time. The derivation using 4 formulas can look like this:. Say, starting to change only pressure and volume , according to Boyle's law , then:. Using then Eq. Using then Charles's law to change the volume and temperature of the gas,. If you know or have found with an experiment 3 of the 6 formulas, you can easily derive the rest using the same method explained above; but due to the properties of said equations, namely that they only have 2 variables in them, they can't be any 3 formulas.

For example, if you were to have Eqs. If you were to use the same method used above on 2 of the 3 laws on the vertices of one triangle that has a "O" inside it, you would get the third. The ideal gas law can also be derived from first principles using the kinetic theory of gases , in which several simplifying assumptions are made, chief among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.

The root-mean-square speed can be calculated by. Let F denote the net force on that particle. Then the time-averaged kinetic energy of the particle is:. Summing over a system of N particles yields.

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By Newton's third law and the ideal gas assumption, the net force of the system is the force applied by the walls of the container, and this force is given by the pressure P of the gas. Since the divergence of the position vector q is. For a d -dimensional system, the ideal gas pressure is: [8].

Note that the dimensions of the pressure changes with dimensionality. From Wikipedia, the free encyclopedia.

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