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Kinetic Theory of Gases Part II The ideal Gas Pressure

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If an ideal gas rests in a closed container, then the gas will exert pressure on the container wall because the gas particles are continously moving and colliding the container wall. vxvx l vyvy vzvz X Z Y Suppose the length of container side is l, the mass of particle is m, the velocity of particle in the direction of x-axis, y-axis, and z-axis are v x, v y, and v z. Respevtively, the pressed area is A, the pressure force is F and the pressure against the wall is p Obserb the direction of the particle motion in the direction of x-axis : A

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A B m m vxvx -v x l The time required by the particle to collide the wall B and return to its original position (A) is determining by the following equation : Every time the particle collides the wall, then it will give its momentum that is equal to 2 mv x to the right wall. The change of momentum on the particle is p = I = F. t, so : I = -mv x – mv x F. t = -2mv x Because F = p.A, where p = pressure, then :

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Because of the velocity of the gas particle to all direction is assumed to the same v x = v y = v z, then : However, because we observe in x- axis direction, we use : So the equation pV = mv x 2 becoms : For N particles of ideal gas, then we obtain the following equation : Where : N : number of gas particles m : mass of gas particles v : velocity of gas particles V : volume of container S.A

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Average Kinetic energy : If this equation is subtitude into the previous equation, that is : Then will be obtained the equation as follows : This equation express the relationship between gas pressure and average kinetic energy of gas particles. The greater the average kinetic energy of gas particle, the greater the pressure on the gas Presure and Kinetic Energy

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If the gas is heated, then the temperature of the gas will increase, This increase of temperature causes the speed of gas particle motion to increase, so that the average kinetic energy of the gas particle will also increase, The amount of average kinetic energy of the gas particle can be obtained from the following equation ; Enter the value of p into the equation : So that obtained the following equation : This equation expresses that the average kinetic energy of gas particle is only influenced by its absolute temperature It should be note that the equation above only holds for monoatomic gases Absolute Temperature and Average Kinetic Energy

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The velocity of gas particles in container is not all the same, Because of that, the concept of effective velocity of gas particles is needed. Suppose a closed container contains are a number of N gas particles, each particle moves at a certain velocity. N 1 particle moves at velocity of v 1, N 2 particle moves velocity of v 2 and so on, Then the average (mean) velocity square v 2, can be expressed as follow : The effective velocity of gas is defined as the root mean square of gas velocity, and it is determined as follow : The Velocity of Gas

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Hence, the average kinetic energy of gas particle can be expressed using the effective velocity of gas particle as follows : By subtituting that value into the Then obtained the following equation : Another expression for effective velocity : p = gas pressure (Pa) = density of gas (kg/3)

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Student Activity #1 A closed vessel contains 20 L oxygen gas. If the gas is at temperature of 27 o C and atmospheric pressure of 1 atm (1atm = 10 5 Pa) determine the number of moles of the oxygen gas in the vessel. Back

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Student Activity #2 How many molecules are there in 6 grams of hydrogen gas ?

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Student Activity #3 A container of hydrogen of volume 0.1 m 3 and temperature 25 o C contains 3.20 x 10 23 molecules. What is the pressure of the container ?

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Student Activity #4 One mole of gas occupies 100 dm 3 volume, its temperature at the moment is 127 o C Determine the pressure of the gas

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Student Activity #5 The kinetic energy of 2 moles of monoatomic gas in a 10 liter tube is 2.3 x 10 -23 joule. What is the pressure of the gas in the tube ?

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Student Activity #6 Determine the average kinetic energy of 5 moles of neon gas which its volume is 25 liters and its pressure of 100 kPa

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Student Activity #7 If the kinetic energy of gas molecules become twice of the kinetic energy of the gas molecules at 127 o C, what is the gas temperature now ?

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Student Activity #8 The speed of 20 particles are distributed as follows : Determine the rms speed ! Speed m/s1.02.03.04.05.06.0 No. of molecules 1 3 4 5 2 5

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Student Activity #9 What is the temperature at which the rms speed of oxygen molecules is twice as great as their rms speed at 300 o K

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Student Activity #10 Determine the effective velocity of gas particles at normal state, if the gas density is 10 kg/m 3 and its pressure is 3 x 10 5 N/m 2

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Student Activity #11 A tank of 2.4 m 3 in volume is filled with 2 kg gas. The pressure tank is 1.3 atm. What is the effective velocity of the gas molecules ?

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Student Activity #12 The pressure of a gas in a closed tube decreases 81% from the initial pressure. Does this change the speed of the gas particles ?

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Internal Energy In a closed container, an ideal gas only has kinetic energy. The total kinetic energy of the gas particles in the container is called the internal energy. The equation of the internal energy above is valid for monoatomic gases, for example : He, Ne and Ar For diatomic gasses such H 2, O 2, and N 2. at low temperature + 300 K holds : At medium temperature + 500 K holds : At high temperature + 1000K :

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Student Activity A container with temperature of 67 o C and pressure of 1.2 x 10 5 Pa contains 2 g helium gas with molecule mass being of 4g/mole. Calculate the internal energy of the gas

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Student Activity Determine the internal energy of one mole of gas at temperature of -37 o C ( k =1.38 x 10 -23 J/K, No = 6.02 x 10 23 molecules/moles)

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Student Activity What is the internal energy of 2 grams of neon gas (Ne) at temperature of 27 o C (known that neon gas has M = 10 g/mole)

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Thanks for your attention

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