 |
| The Ideal Gas Equation |
|
| Another simple equation of state is the one for an ideal gas. The figure below shows an experimental setup to study the behavior of a gas. The cylinder has a movable piston to vary the volume, heating can vary the temperature, and we can pump the desired amount of any gas into the cylinder. We then measure the pressure, volume, temperature and amount of gas. Note that pressure refers both to the force per unit area exerted by the cylinder of the gas and to the force per unit area exerted by the gas on the cylinder. By Newton's third law, these must be equal. |
| |
| It is usually easy to describe the amount of gas in terms of the number of moles n, rather than the mass. The molar mass M of a compound (some times called molecular weight) is the mass per mole, and the total mass mtot of a given quantity of that compound is the number of moles (n) multiplied by mass per mole (M): |
| |
| mtot = nM (total mass = number of moles into molar mass) ---(2) |
| |
| We are calling the total mass mtot, because later on in the chapter, we use m for the mass of one molecule. |
| |
| Measurements of the behavior of various gases lead to several conclusions. First, the volume V is proportional to the number of moles n. If we double the number of moles, keeping pressure and temperature constant, the volume doubles. |
| |
| Secondly, the volume varies inversely with the absolute pressure p. If we double the pressure while holding the temperature T and number of moles n constant, the gas gets compressed to one half of its initial volume. In other words, pV = constant, when n and T are constants. |
| |
| Thirdly, the pressure is proportional to the absolute temperature. If we double the absolute temperature, keeping the volume and number of moles constant, the pressure doubles. In other words, P = (constant) T, n and V are constant, these three relationships can be combined neatly into a single equation, called the ideal-gas equation: |
| |
| PV=nRT (ideal gas equation) ------(3) |
| |
| where R is proportionality constant. An ideal gas is one for which Equation (3) holds good at all pressures and temperatures. This is an idealized model. It works best at very low pressures and high temperatures, when the gas molecules are far apart and in rapid motion. It is reasonably good (within a few per cent) at moderate pressures (such as a few atmospheres) and at temperatures well above the point at which the gas liquefies. |
| |
| We expect the ideal gas equation to have different values for the constant R for different gases, but R has the same value for all gases, at least at sufficiently high temperature and low pressure. It is called the gas constant (or ideal gas constant). The numerical value of R depends on the units of P, V, and T in SI units, in which the unit of P is Pa (1 Pa = 1 N/m2) and the unit of V is m3. The numerical value of R is |
| |
| R = 8.314510(70) J/mol/K |
| |
| Note that the units of pressure times volume are the same as units of work or energy (for example, N/m2 times m3) that's why R has units of energy per mole per unit of absolute temperature. In chemical calculations, volumes are often expressed in litres (L) and pressures in atmospheres (atm) in this system, to four significant digits, |
| |
 |
| |
| We can express the ideal gas equation, equation (3), in terms of the mass mtot of gas, using mtot =nM from equation (2): |
| |
 -----(4) |
| |
 |
| |
