Answer
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Hint: For calculating the RMS, there are some factors to be calculated like the kinetic energy of the molecules, density of the molecule, molecular mass, etc. The ideal gas equation i.e., PV=RT is also used.
Complete step by step answer:
The molecules are moving in a different direction with different velocity colliding with one another as well as with the walls of the container. As a result, their individual velocity and hence the kinetic energies keep on changing even at the same temperature. However, it is found that at a particular temperature, the average kinetic energy of the gas remains constant.At a particular temperature, if\[{{n}_{1}}\] molecules have velocity\[{{v}_{1}}\] , \[{{n}_{2}}\] molecules have velocity\[{{v}_{2}}\], \[{{n}_{3}}\]molecules have velocity\[{{v}_{3}}\] and so on, then the total kinetic energy (\[{{E}_{K}}\]) of the gas at this temperature will be:\[{{E}_{K}}={{n}_{1}}(\dfrac{1}{2}m{{v}^{2}}_{1})+{{n}_{2}}(\dfrac{1}{2}m{{v}^{2}}_{2})+{{n}_{3}}(\dfrac{1}{2}m{{v}^{2}}_{3})........\]\[=\dfrac{1}{2}m({{n}_{1}}{{v}^{2}}_{1}+{{n}_{2}}{{v}^{2}}_{2}+{{n}_{3}}{{v}^{2}}_{3}+......)\]Where m is the mass is the molecule of the gas.Dividing by total number molecules, average kinetic energy (\[{{E}_{K}}\]) of the gas will be\[\overline{{{E}_{k}}}\]\[=\dfrac{{{E}_{k}}}{n}=\dfrac{1}{2}m\{\dfrac{{{n}_{1}}{{v}^{2}}_{1}+{{n}_{2}}{{v}^{2}}_{2}+{{n}_{3}}{{v}^{2}}_{3}+......}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....}\}\]In this equation, the expression\[\dfrac{{{n}_{1}}{{v}^{2}}_{1}+{{n}_{2}}{{v}^{2}}_{2}+{{n}_{3}}{{v}^{2}}_{3}+......}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....}\] represents the mean of the squares of the velocity of different molecules and hence it is called mean square velocity.Its square root is called root mean square (RMS) velocity and is represented by c. Thus the root means square velocity may be defined as the square root of the mean of the square of the speeds of different molecules of the gas. Mathematically,\[c=\sqrt{\dfrac{{{n}_{1}}{{v}^{2}}_{1}+{{n}_{2}}{{v}^{2}}_{2}+{{n}_{3}}{{v}^{2}}_{3}+......}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....}}\] There are many formulae of calculating RMS velocity:If density of the gas is given: \[c=\sqrt{\dfrac{3P}{d}}\]With ideal gas equation PV=RT, we have: \[c=\sqrt{\dfrac{3PV}{M}}=\sqrt{\dfrac{3RT}{M}}\]Now, calculating the RMS velocity of \[C{{O}_{2}}\] at \[{{27}^{\circ }}C\],Here, we can apply: \[c=\sqrt{\dfrac{3RT}{M}}\]R= gas constant = 8.314 J/K mol.T= Temperature = 27 + 273 = 300 KM= molar mass of \[C{{O}_{2}}\] = 44 gm/mol = 0.044 kg/molSo, substituting all the values, we get:\[\sqrt{\dfrac{3*8.314*300}{0.044}}\] \[=412.3m/s\]Hence, the RMS velocity of \[C{{O}_{2}}\] at \[{{27}^{\circ }}C\] is 412.3m/s.
Note: You can also solve the values in the cgs system by taking the cgs units. Always convert the temperature into Kelvin form otherwise the answer will be wrong. Don't get confused between RMS, average, and most probable speed they all have a different formula.
Q1 Single Correct Medium
An ant is moving on a plane horizontal surface. The number of degrees of freedom of the ant will be
Asked in: Physics - Kinetic Theory
This example problem demonstrates how to calculate the root mean square (RMS) velocity of particles in an ideal gas. This value is the square root of the average velocity-squared of molecules in a gas. While the value is an approximation, especially for real gases, it offers useful information when studying kinetic theory.
What is the average velocity or root mean square velocity of a molecule in a sample of oxygen at 0 degrees Celsius?
Gases consist of atoms or molecules that move at different speeds in random directions. The root mean square velocity (RMS velocity) is a way to find a single velocity value for the particles. The average velocity of gas particles is found using the root mean square velocity formula:
μrms = (3RT/M)½
μrms = root mean square velocity in m/sec
R = ideal gas constant = 8.3145 (kg·m2/sec2)/K·mol
T = absolute temperature in Kelvin
M = mass of a mole of the gas in kilograms.
Really, the RMS calculation gives you root mean square speed, not velocity. This is because velocity is a vector quantity that has magnitude and direction. The RMS calculation only gives the magnitude or speed. The temperature must be converted to Kelvin and the molar mass must be found in kg to complete this problem.
Find the absolute temperature using the Celsius to Kelvin conversion formula:
- T = °C + 273
- T = 0 + 273
- T = 273 K
Find molar mass in kg:
From the periodic table, the molar mass of oxygen = 16 g/mol.
Oxygen gas (O2) is comprised of two oxygen atoms bonded together. Therefore:
- molar mass of O2 = 2 x 16
- molar mass of O2 = 32 g/mol
- Convert this to kg/mol:
- molar mass of O2 = 32 g/mol x 1 kg/1000 g
- molar mass of O2 = 3.2 x 10-2 kg/mol
Find μrms:
- μrms = (3RT/M)½
- μrms = [3(8.3145 (kg·m2/sec2)/K·mol)(273 K)/3.2 x 10-2 kg/mol]½
- μrms = (2.128 x 105 m2/sec2)½
- μrms = 461 m/sec
The average velocity or root mean square velocity of a molecule in a sample of oxygen at 0 degrees Celcius is 461 m/sec.