(a) What is the total random translational kinetic energy of...

(a) What is the total random translational kinetic energy of 5.00 L of hydrogen gas (molar mass 2.016 $\mathrm{g} / \mathrm{mol} )$ with pressure $1.01 \times 10^{5} \mathrm{Pa}$ and temperature 300 $\mathrm{K} ?$ (Hint: Use the procedure of Problem 18.71 as a guide.) (b) If the tank containing the gas is placed on a swift jet moving at $300.0 \mathrm{m} / \mathrm{s},$ by what percentage is the total kinetic energy of the gas increased? (c) since the kinetic energy of the gas molecules is greater when it is on the jet, does this mean that its temperature has gone up? Explain.

Key Concepts

Temperature and Its Independence from Bulk Motion

Temperature is defined as a measure of the average kinetic energy associated with the random motion of particles in a system, not including any kinetic energy due to bulk motion. Thus, even if the entire system is moving at a high speed, the temperature remains a characteristic of the random internal motions of its molecules and does not change with the addition of uniform translational motion.

Bulk (Translational) Kinetic Energy from Motion

Bulk kinetic energy refers to the kinetic energy that a collection of particles has due to the collective, coherent motion of their center-of-mass, such as when a gas filled container is moving relative to an observer. Unlike random thermal motion, this energy is associated with the entire system moving at a constant velocity and is calculated using classical mechanics (1/2 m v^2).

Translational Kinetic Energy of Gas Molecules

This concept explains that the random translational kinetic energy of gas molecules is directly related to the temperature of the gas. For an ideal gas, the average kinetic energy per molecule is given by (3/2)k_BT, where k_B is the Boltzmann constant. The total translational kinetic energy of the gas can therefore be expressed as (3/2)nRT, linking the microscopic motion of individual molecules to the macroscopic temperature.

Ideal Gas Law

The ideal gas law (pV = nRT) is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It provides the basis for understanding how the macroscopic properties of a gas are interconnected, which is crucial for calculating quantities like the number of moles in a given volume at specified pressure and temperature.

Transcript

00:01 So the question is asking us for the total translational kinetic energy.

00:11 And then it's asking us this tank is then being placed on a jet.

00:17 How does the translational, how does the total kinetic energy change when it's being placed on a jet? and then if it's being placed on a jet, does the temperature change? so let's first start off by writing down our givens.

00:32 So we have t equals 300 kelvin, and then we have the volume being 5 liters.

00:38 We're going to automatically convert this to 0 .005 meters cubed.

00:44 Again, 1 ,000 liters equals one cubic meter.

00:49 And then the molar mass of the air inside would be 0 .002 ,016.

00:58 So there's only hydrogen in this tank.

01:04 The pressure is atmospheric pressure, so 1 .01 times 10 to the 5th pascal's.

01:13 And then it's asking us for the total translational kinetic energy.

01:18 So for part a, let's say pv equals n or t.

01:22 And then we can say that the kinetic energy, the average kinetic energy of the particles, rather it's not the average by the bed.

01:32 My apologies, the total translational kinetic energy would be equal to 3nrt over 2.

01:46 And from the ideal gas law, we know that nrt equals pv.

01:50 So this will become 3 pv over 2.

01:54 And at this point, we can then substitute.

01:58 So k would be equal to 3, and then p would be 1 .01 times 10, the fifth volume would be 0 .005 and then this would be divided by 2.

02:16 So the total translational kinetic energy would be equal to 757 .5 joules.

02:27 Okay? at this point they're saying okay this tank is being placed on a jet liner and this jet is going 300 meters per second.

02:39 So at this point, how does the change in, what is the change in kinetic energy? so the change in kinetic energy would be delta k, and delta k is going to equal half mv squared.

03:00 And we know v, so we're mainly trying to look for m.

03:04 So let's use the ideal gas law, pv equals nrt.

03:08 We know that the number of moles equals the mass of the gas divided by the molar mass of the gas.

03:15 So we can then rewrite this and say pv equals m r t over capital m.

03:21 And then we're going to isolate m.

03:25 So then we'll have m equals pvm over rt...

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