A gas in a container consists of molecules of mass m. The gas has a well defined temperature T.

A gas in a container consists of molecules of mass m. The...

Key Concepts

Maxwell-Boltzmann Distribution

This statistical distribution describes the spread of molecular speeds in an ideal gas at thermal equilibrium. It shows how the probability of finding a molecule at a specific speed depends on the temperature of the gas and the mass of its molecules, and it provides the framework for determining characteristic speeds such as the most probable speed and the average speed.

Most Probable Speed

The most probable speed is the speed at which the probability density function of the Maxwell-Boltzmann distribution reaches its maximum. It represents the speed that the largest number of molecules have in an ideal gas, reflecting the peak of the speed distribution due to the interplay of the number of accessible states and the damping effect of higher kinetic energies.

Average Speed

The average speed is the arithmetic mean of the speeds of all molecules in the gas, calculated by integrating the product of speed and its probability density over all speeds. It takes into account the entire distribution of speeds, balancing the contributions from both slower and faster molecules, and is generally higher than the most probable speed.

Average Velocity

Average velocity is a vector quantity defined as the sum of all molecular velocity vectors divided by the total number of molecules. In an ideal gas at equilibrium, the distribution of molecular velocities is isotropic (directionally symmetric), causing the vector contributions to cancel out, resulting in an average velocity of zero.

Transcript

00:01 So we have here two identical containers which have which has two types of gases a and b and it says the mass of a molecule of gas of type a ma is 3 .34 into 10 to the power minus 27 kg and it also says that the mass of molecule of gas of gas of gas of type b that is mb is equal to 5 .34 into 3 to the power minus 26 kizu all right so both cancers are under the same pressure that is pressure 8 a is equal to pressure at b and same temperature that is t a is equal to t b and the temperature is 10 degrees celsius so that we convert this into kelvin that is equal to 283 so, now the question asks us for a, which molecules are we have greater to processing kinetic energy per molecule and rms is spitz.

01:24 So processor of kinetic energy e is equal to 3 by 2 of bosom's constant k and absolute temperature t.

01:35 So in this case, in this case, temperature of a is equal to temperature of a is equal to temperature temperature of p and bolzman's constant b is constant for both cases right so this implies translational kinetic energy for molecule for a is equal to that for b now the second part of a asks us about the rm s speed now v rms is equal to root under 3 kt by mass of molecule small n right so as we can see the temperature is the same so only it depends upon is the mass now since m of a is less than m f b we can say that b rms for b is less than b rm s for a right so which has the question is asking which has the greater kinetic energy so that is a so our answer is a has greater kinetic energy so greater rms speed so the next question this is now we want to raise the temperature of only one of the containers so that both have the same rms speed right so both have the same rms speed which means that root under 3k temperature of a upon mass of a should be equal to 3k temperature of b upon mass of b right now if we want this since m a is less than mb for this condition we require t a should be less than t b right so we can say that temperature of b should increase so our answer is b okay now the next question at what temperature will you accomplish your goal so for what temperature will we have just to determine the value of t b so from this from this equation here the value of t b can be calculated as t v is equal m b by m a into t a so m b is 5 .34 into 0 .34 into 10 to the power minus 26 upon m a is 3 .34 into 10 to the power minus 26 upon temperature of a that is equal to 283 so this gives me t b should be equal to 4 5 to 4 4 to 4 cckelvin if you want this in celsius this comes out to be 4251 celsius right so this gives me the temperature of t b when the both a and b will have the same rms speed.

06:24 So let's do question number t.

06:27 Now as we have accomplished our goal that we have made aramasease speed same for both, which will have a greater kinetic energy, translation candy energy for molecule.

06:37 Now again, we have to compare e is equal to 3x2 kt...

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