Equation of state of ideal gases

Transcription

1 Objective Equation of state of ideal gases F a constant amount of gas (air) investigate the crelation of 1. Volume and pressure at constant temperature (Boyles law) 2. Volume and temperature at constant pressure (Gay-Lussac s law) 3. Pressure and temperature at constant volume Charles s law Introduction The gas laws are thermodynamic relationships that express the behavi of a quantity of gas in terms of the pressure P, volume V, and temperature T. The kinetic energy of gases expresses the behavi of a perfect ideal gas to be P V = nrt, which is commonly referred to as the ideal gas law. However, the general relationships among P, V and T contained in this equation had been expressed earlier in the classical gas laws f real gases. These gas laws were Based on the empirical observations of early investigats, in particular the English scientist Robert Boyle and the French scientists Jacques Charles and Joseph Gay-Lussac. They I Boyle s law: Around 1660, Robert Boyle had established an empirical relationship between the pressure and the volume of a gas, which is known as Boyle s law: At constant temperature, the volume occupied by a given mass of gas is inversely proptional to its pressure. In mathematical notation, we write V 1 P, (1) In equation fm, we may write V = k P. (2) Where k is the constant of proptionality f a given temperature. The pressure of a confined gas is commonly measured by means of a manometer. The gas pressure f such an open-tube manometer is the sum of the atmospheric pressure P a and the pressure of the height difference of mercury P = P a + ρ m g h m. (3) 1

2 but then equation (3) becomes: P a = 76 cm Hg = ρ m g 76 P = ρ m g (h m + 76). (4) where ρ m = 13.6 gm/cm 3 and g = 980cm/s 2 The volume of the enclosed gas is the volume of the measuring tube segment marked in brown added to the volume calculated from the length of the column of air: V = V l ml (5) V = πr 2 l ml ( ) = π l ml 2 = 1.02l = l + 1 Substituting with P and V in eq. (2): Let then Boyle s law is = l (6) P V = ρ m g (h m + 76) l = k. (7) k = k ρ m g h m = k l 76 (8) Where h m represents the change in pressure, and since the cross sectional area of the tube is constant,then l represents V. II Charle s and Gay-Lussac s laws : In 1787, the French physicist Jacques Charles repted the result of a series of experiments which is known as Charles law : At constant volume, the pressure exerted by a given mass of gas is proptional to its absolute temperature. In mathematical notation, P T, (9) P = k T. (10) 2

3 where k is a constant of proptionality. If we substitute f the value of P from eq(4 ), we get then ρ m g (h m + 76) = k T. (11) h m + 76 = k T. (12) ρ m g h m = k T 76. (13) Notice that the gas laws are expressed in terms of absolute temperature (Kelvin). Where T K = T c In 1802, Gay-Lussac, using a somewhat different experimental approach, essentially restarted Charles law in a fm me commonly used today (also known as Gay-Lussac s law): At constant pressure, the volume of a given mass of gas is proptional to its absolute temperature. That is, V T, (14) Substituting f the value of V from eq.(6) Equipments V = k T. (15) l = k T. (16) Gas laws apparatus, lab thermometer ( c), distilled water. Procedure PART (I) 1. During the experiment, the temperature in the measuring tube must be kept constant. Keep the heater off and recd reading of the thermometer in the air tube. 2. raise lower the open end of the tube until the mercury levels in both sides are equal, then recd the length of the enclosed air column l, where the difference of mercury levels h m = 0. Repeat this step several times 3

4 3. Raise the mercury level in the open tube to decrease the air column, then recd l, h m. 4. Plot h m on the y-axis, and 1 on the x-axis, then calculate the l slope. PART(II) 1. Set the thermostat to 30 c wait until the thermometer read 30 c. 2. Raise lower the open end of the tube until the mercury levels in both sides are equal, and then recd the length of the enclosed air column l = l 0, and the difference of mercury levels h m = During the experiment the volume must be constant; this means l must be constant. Increase the thermostat by 5 c wait f temperature constancy in the measuring tube and the increment of l 4

5 4. Raise the open end tube to return the length of air column to l = l 0, then recd h m and T K. 5. ) repeat steps 3, 4 several times. 6. Plot h m on the y-axis, and T K on the x-axis, calculate the slope. PART(III) Results Data table (I ): T 0 = Set the thermostat to 30 c wait until the thermometer read 30 c. 2. Raise lower the open end of the tube until the mercury levels in both sides are equal, and then recd the length of the enclosed air column l = l 0, and the difference of mercury levels h m = During the experiment the pressure must be constant; this means h m = 0. Increase the thermostat by 5 c wait f temperature constancy in the measuring tube and the increment of l. 4. Lower the open end tube to return to equal levels of mercury h m = 0, then recd l and T K. 5. Repeat steps 3, 4 several times 6. Plot l on the y-axis, and T K on the x-axis, calculate the slope. l(cm) l (cm) h m (cm) Data table (II ): l 0 =..., l 0 =... T c T K h m (cm) 5

6 Data table (II ): T 0 =... T c T K l(cm) l (cm) 6