Square mean speed

The importance of the root mean square speed stems from the fact that is proportional to the average kinetic energy of the molecules, [Pg.284]

At this stage, we relate (vj.) to the root mean square speed, frms = (v2)111, the square root of the average of the squares of the molecular speeds (this quantity is explained in more detail in the text following this derivation). First, we note that the speed, v, of a single molecule is related to the velocity parallel to the x, y, and 2 directions ... 

We can now do something remarkable we can use the ideal gas law to calculate the root mean square speed of the molecules of a gas. We know that PV = nRT for an ideal gas therefore, we can set the right-hand side of Eq. 19 equal to nRT and rearrange the resulting expression ( nMv2ms = nRT) into... 

FIGURE 4 J5 The mot mean square speeds of five gases at 25°C, in meters per second. The gases are some of the components of air hydrogen is included to show that the root mean square speed of light molecules is much greater than that of heavy molecules. 

This important result is used to find the root mean square speeds of the gas-phase molecules at any temperature (Fig. 4.25). We can rewrite this equation to emphasize that, for a gas, the temperature is a measure of mean molecular speed. From... 

That is, the temperature is proportional to the mean square speed of the molecules. 

EXAMPLE 4.9 Sample exercise Calculating the root mean square speed of gas molecules... 

The kinetic model of gases is consistent with the ideal gas law and provides an expression for the root mean square speed of the molecules vnns = (3RT/M)l/2. The molar kinetic energy of a gas is proportional to the temperature. 

The molecules of all gases have a wide range of speeds. As the temperature increases, the root mean square speed and the range of speeds both increase. 

Calculate the root mean square speeds of (a) methane,... 

The root mean square speed of gaseous methane molecules, CH4, at a certain temperature was found to be 550. nvs What is the root mean square speed of krypton atoms at the same temperature ... 

In an experiment on gases, you are studying a L.00-L sample of hydrogen gas at 20°C and 2.40 atm. You heat the gas until the root mean square speed of the molecules of the sample has been doubled. What will be the final pressure of the gas ... 

A bottle contains 1.0 mol He(g) and a second bottle contains 1.0 mol Ar(g) at the same temperature. At that temperature, the root mean square speed of He is 1477 m-s 1 and that of Ar is 467 nvs-1. What is the ratio of the number of He atoms in the first bottle to the number of Ar atoms in the second bottle having these speeds Assume that both gases behave ideally. 

Determine the ratio of the number of molecules in a gas having a speed ten times as great as the root mean square speed to the number having a speed equal to the root mean square speed. Is this ratio independent of temperature Why ... 

How does the root mean square speed of gas molecules vary with temperature Illustrate this relationship by plotting the root mean square speed of N, as a function of temperature from T = 100 I< to T = 300 K. ... 

The root mean square speed rrrm of gas molecules was derived in Section 4.10. Using the Maxwell distribution of speeds, we can also calculate the mean speed and most probable (mp) speed of a collection of molecules. The equations used to calculate these two quantities are i/mean = (8RT/-nM),a and... 

This speed is caiied the root-mean-square speed, because it is found by taking the square root of u According to Equation, the average speed of gas moiecuies is directiy proportionai to the square root of the temperature and is inverseiy proportionai to the square root of the moiar mass. 

C05-0075. Determine the root-mean-square speed of SFg molecules under the conditions of Problem 5.31. C05-0076. Determine the root-mean-square speed of H2 molecules under the conditions of Problem 5.32. C05-0077. If a gas line springs a leak, which will diffuse faster through the atmosphere and why, CH4 or... 

Consider, as an example, the calculation of the mean-square speed of an ensemble of molecules which obey the Maxwell-Boltzmann distribution law. This quantity is given by... 

A The gas with the smaller molar mass, NH3at 17.0 g/mol, has the greater root-mean-square speed... 

We equate the two expressions for root mean square speed, cancel the common factors, and solve for the temperature of Ne. Note that the units of molar masses do not have to be in kg/mol in this calculation they simply must be expressed in the same units. 

Table 1.4 The average speeds of gas molecules at 273.15 K, given in order of increasing molecular mass. The speeds c are in fact root-mean-square speeds, obtained by squaring each velocity, taking their mean and then taking the square root of the sum...

Before we leave the Kinetic Molecular Theory (KMT) and start examining the gas law relationships, let s quantify a couple of the postulates of the KMT. Postulate 3 qualitatively describes the motion of the gas particles. The average velocity of the gas particles is called the root mean square speed and is given the symbol rms. This is a special type of average speed. 

The average kinetic energy, e, is related to the root-mean-square (rms) speed u through the equation e = /i mi/ Because the MM of CH4 (16) is slightly less than that of NH3 (17), the root-mean-square speed of CH4 is slightly higher than that of NH3. Root-mean-square speed is inversely proportional to the square root of the molar mass of the gas. 1 point for correct answer and explanation. 

Urrre= root-mean-square speed KE= kinetic energy r = rate of effusion M= molar mass jc= osmotic pressure /= van t Hoff factor... 

---