# 2.2 Pressure, temperature, and rms speed  (Page 4/18)

 Page 4 / 18
${v}_{\text{rms}}=\sqrt{\frac{3\phantom{\rule{0.2em}{0ex}}RT}{M}}.$

We digress for a moment to answer a question that may have occurred to you: When we apply the model to atoms instead of theoretical point particles, does rotational kinetic energy change our results? To answer this question, we have to appeal to quantum mechanics. In quantum mechanics, rotational kinetic energy cannot take on just any value; it’s limited to a discrete set of values, and the smallest value is inversely proportional to the rotational inertia. The rotational inertia of an atom is tiny because almost all of its mass is in the nucleus, which typically has a radius less than ${10}^{-14}\phantom{\rule{0.2em}{0ex}}\text{m}$ . Thus the minimum rotational energy of an atom is much more than $\frac{1}{2}\phantom{\rule{0.2em}{0ex}}{k}_{\text{B}}T$ for any attainable temperature, and the energy available is not enough to make an atom rotate. We will return to this point when discussing diatomic and polyatomic gases in the next section.

## Calculating kinetic energy and speed of a gas molecule

(a) What is the average kinetic energy of a gas molecule at $20.0\phantom{\rule{0.2em}{0ex}}\text{ºC}$ (room temperature)? (b) Find the rms speed of a nitrogen molecule $\left({\text{N}}_{2}\right)$ at this temperature.

## Strategy

(a) The known in the equation for the average kinetic energy is the temperature:

$\stackrel{\text{–}}{K}=\frac{1}{2}\phantom{\rule{0.2em}{0ex}}m\stackrel{\text{–}}{{v}^{2}}=\frac{3}{2}{k}_{\text{B}}T.$

Before substituting values into this equation, we must convert the given temperature into kelvin: $T=\left(20.0+273\right)\phantom{\rule{0.2em}{0ex}}\text{K}=293\phantom{\rule{0.2em}{0ex}}\text{K}\text{.}$ We can find the rms speed of a nitrogen molecule by using the equation

${v}_{\text{rms}}=\sqrt{\stackrel{\text{–}}{{v}^{2}}}=\sqrt{\frac{3{k}_{\text{B}}T}{m}},$

but we must first find the mass of a nitrogen molecule. Obtaining the molar mass of nitrogen ${\text{N}}_{2}$ from the periodic table, we find

$m=\frac{M}{{N}_{A}}=\frac{2\phantom{\rule{0.2em}{0ex}}\left(14.0067\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{kg/mol}\right)}{6.02\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{23}\phantom{\rule{0.2em}{0ex}}{\text{mol}}^{\text{-1}}}=4.65\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-26}\phantom{\rule{0.2em}{0ex}}\text{kg}.$

## Solution

1. The temperature alone is sufficient for us to find the average translational kinetic energy. Substituting the temperature into the translational kinetic energy equation gives
$\stackrel{\text{–}}{K}=\frac{3}{2}\phantom{\rule{0.2em}{0ex}}{k}_{\text{B}}T=\frac{3}{2}\left(1.38\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-23}\phantom{\rule{0.2em}{0ex}}\text{J/K}\right)\left(293\phantom{\rule{0.2em}{0ex}}\text{K}\right)=6.07\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-21}\phantom{\rule{0.2em}{0ex}}\text{J}.$
2. Substituting this mass and the value for ${k}_{\text{B}}$ into the equation for ${v}_{\text{rms}}$ yields
${v}_{\text{rms}}=\sqrt{\frac{3{k}_{\text{B}}T}{m}}=\sqrt{\frac{3\left(1.38\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-23}\phantom{\rule{0.2em}{0ex}}\text{J/K}\right)\left(293\phantom{\rule{0.2em}{0ex}}\text{K}\right)}{4.65\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-26}\phantom{\rule{0.2em}{0ex}}\text{kg}}}=511\phantom{\rule{0.2em}{0ex}}\text{m/s}\text{.}$

## Significance

Note that the average kinetic energy of the molecule is independent of the type of molecule. The average translational kinetic energy depends only on absolute temperature. The kinetic energy is very small compared to macroscopic energies, so that we do not feel when an air molecule is hitting our skin. On the other hand, it is much greater than the typical difference in gravitational potential energy when a molecule moves from, say, the top to the bottom of a room, so our neglect of gravitation is justified in typical real-world situations. The rms speed of the nitrogen molecule is surprisingly large. These large molecular velocities do not yield macroscopic movement of air, since the molecules move in all directions with equal likelihood. The mean free path (the distance a molecule moves on average between collisions, discussed a bit later in this section) of molecules in air is very small, so the molecules move rapidly but do not get very far in a second. The high value for rms speed is reflected in the speed of sound, which is about 340 m/s at room temperature. The higher the rms speed of air molecules, the faster sound vibrations can be transferred through the air. The speed of sound increases with temperature and is greater in gases with small molecular masses, such as helium (see [link] ).

two point charges +30c and +10c are separated by a distance of 80cm,compute the electric intensity and force on a +5×10^-6c charge place midway between the charges
what is the difference between temperature and heat
Heat is the condition or quality of being hot While Temperature is ameasure of cold or heat, often measurable with a thermometer
Abdul
Temperature is the one of heat indicators of materials that can be measured with thermometers, and Heat is the quantity of calor content in material that can be measured with calorimetry.
Gamma
2. A brass rod of length 50cm and diameter 3mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250°c( degree Celsius) if the original length are 40°c(degree Celsius) is there at thermal stress developed at the junction? The end of the rod are free to expand (coefficient of linear expansion of brass = 2.0×10^-5, steel=1.2×10^-5k^1)
A charge insulator can be discharged by passing it just above a flame. Explain.
of the three vectors in the equation F=qv×b which pairs are always at right angles?
what is an ideal gas?
What is meant by zero Kelvin ?
Justine
Why does water cool when put in the pot ?
Justine
when we pour the water in a vessel(pot) the hot body(water) loses its heat to the surrounding in order to maintain thermal equilibrium.Thus,water cools.
rupendra
when we drop water in the pot, the pot body loses heat to surrounded in order to maintain thermal equilibrium thus,water cool.
Srabon
types of thermometer?
thermometer, Radiation thermometer and vapour pressure thermometer.liquid thermometer use thermometric liquid like mercury ,alcohol etc.
rupendra
liqid thermometer ,gas thermometer, resitance thermometer,thermo electric thermometer , radiation thermometer andvapour pressure thermometer
rupendra
calculate the quantity of heat required to rise the temperature of 1gmail of ice _10 to 110
A 40cm tall glass is filled with water to a depth of 30cm. A.what is the gauge pressure at the bottom of the glass? B.what is the absolute pressure at the bottom of the glass?
A glass bottle full of mercury has mass 50g when heated through 35degree, 2.43g of mercury was expelled. Calculate the mass of the mercury remaining in the bottle
Two electric point charges Q=2micro coulomb are fixed in space a distance 2.0cm apart. calculate the electric potential at the point p located a distance d/2 above the central point between two charges
what is wave
A wave is a periodic disturbance which travel with a finite velocity and remains unchanged in type as it travels.
Justine
What's a wave motion?
Justine
What is charge bodies
which have free elections
Usman
Show that if a vector is gradient of a scaler function then its line around a closed path is zero
Pak
Charge bodies are those which have free electons
Pak
the melting point of gold is 1064degree cencius and is boiling point is 2660 degree cenciu
is Thomas's young experiment interference experiment or diffraction experiment or both
An aqueous solution is prepared by diluting 3.30 mL acetone (d = 0.789 g/mL) with water to a final volume of 75.0 mL. The density of the solution is 0.993 g/mL. What is the molarity, molality and mole fraction of acetone in this solution?
eugene
A 4.0kg mess kit sliding on a fractionless surface explodes into two 2.0 kg parts.3.0 m/s due to north and 0.5 m/s 30 degree north of east. what is the speed of the mess kit
Shahid