What is the average mechanical energy of a mole of an ideal monatomic gas at a temperature of 333 kelvin?
So in this question, we are asked to work out the average mechanical energy of one mole, a mole, of an ideal monatomic gas at a temperature of 333 kelvin. I feel like I’ve underlined almost everything in this question but that’s because all of it is important. So how do we go about doing this? Well, first of all, let’s look at the mechanical energy there.
The mechanical energy of an object is defined as the sum of the kinetic energy and potential energy used to do work. Now, in this case, we’ve got a mole of an ideal monatomic gas. The kinetic energy and the potential energy stored in this gas could be used to do some work. Therefore, what we’re looking to find is the kinetic energy plus the potential energy of the gas. However, we know that we’ve got an ideal gas.
In an ideal gas, the particles do not interact with each other apart from the course when they collide with each other. And that is the only interaction between ideal gas particles. In other words, when they’re not colliding with each other, they do not attract each other or repel each other. So they have no potential energy. This makes life a lot easier for us. We just need to find now the kinetic energy of the gas.
Another useful piece of information in the question is the fact that this gas is monatonic. In other words, the particles making up the gas are made up of single atoms. Single particles can only have translational kinetic energy. They can move side to side or up and down or in and out through the page, whereas something like say for example a diatomic molecule which is a molecule consisting of two atoms could of course have the translational kinetic energy as before up and down, side to side, in and out the page. But it could have other forms of kinetic energy stored in it as well, for example, rotational or vibrational when the bond vibrates.
Now, you might be thinking that the monatomic particle could also have rotational kinetic energy. Well, not true because this is an ideal gas. And in ideal gases, the particles are point particles they’re about the smallest they can get. And so they cannot have rotational kinetic energy. Anyway, we can recall that the average kinetic energy of a particle in a monatomic gas is given by three by two 𝐾𝑇. Now, this is the kinetic energy of a monatomic particle — so a particle made of just one atom in an ideal gas.
The 𝐾 is the Boltzmann constant and the 𝑇 is the temperature. And this is the average kinetic energy of one particle. However, in our gas, we’ve got an entire mole of that gas. And the mole is 6.02 times 10 to the power of 23 particles. So the total average kinetic energy of the gas which we’ll call 𝐸 sub total is given by 6.02 times 10 to the power of 23 times three by two 𝐾𝑇. In other words, it’s the number of particles that we have multiplied by the average kinetic energy of one particle.
And of course, we can substitute the Boltzmann constant in as well as the temperature of 333 kelvin. And we can evaluate this to give us 4150.845 joules. However, since the value we’ve been given in the question — the temperature of 333 kelvin — is to three significant figures, we need to give our answer to three significant figures as well.
And so our final answer is that the average mechanical energy of a mole of an ideal monatomic gas at a temperature of 333 kelvin is 4150 joules to three significant figures.