Animated Solution for Physics - Thermodynamics: 2 kg of a monoatomic gas is at a pressure of $4\times 10^4{\text{ N/m}}^2$. The density of the gas is $8{\text{ kg/m}}^3$. What is the order of energy of the gas due to its thermal motion ?

Imagine you are looking at a sealed container filled with a gas. The molecules inside are constantly zipping around, colliding with each other and the walls of the container. This chaotic, microscopic dance is what we call thermal motion, and the energy associated with it is the internal energy of the gas.

In this problem, we are given a $2\text{ kg}$ sample of a monoatomic gas. We know its pressure is $4\times 10^4{\text{ N/m}}^2$ and its density is $8{\text{ kg/m}}^3$. Our goal is to find the order of magnitude of its thermal energy.

Finding the Hidden Volume

Before we can calculate the energy, we need to know how much space this gas occupies. We aren't given the volume directly, but we have the mass and the density.

Remember the fundamental relationship:

By substituting the given values, we can easily find the volume:

So, our gas occupies exactly $0.25\text{ cubic meters}$.

The Master Equation for Internal Energy

Now, how do we connect pressure and volume to thermal energy? The internal energy $U$ of an ideal gas is directly proportional to the product of its pressure and volume. The exact relationship depends on the degrees of freedom ($f$) of the gas molecules:

Since our gas is monoatomic (like Helium or Neon), its molecules are single atoms. They can only move in three independent directions (up-down, left-right, forward-backward). Therefore, they have exactly $3$ translational degrees of freedom, meaning $f=3$.

Our master equation becomes:

The Final Calculation

We have all the pieces of the puzzle. Let's substitute our pressure and volume into the equation:

Notice the beautiful mathematical elegance here: the $4$ in the pressure perfectly cancels out the $4$ in the denominator of the volume! We are left with a very clean calculation:

Determining the Order of Magnitude

The question specifically asks for the order of energy. In physics, the order of magnitude is the power of $10$ that best approximates the number.

To find the order of magnitude of a number written in scientific notation $A\times 10^n$, we look at the coefficient $A$. If $A<\sqrt{10}$ (which is approximately $3.16$), the order of magnitude is simply $10^n$. If $A\ge 3.16$, the order of magnitude rounds up to $10^{n+1}$.

Since $1.5$ is strictly less than $3.16$, our order of magnitude remains exactly as the power of ten in our result:

Order of Magnitude = $10^4\text{ J}$

And there we have it! By systematically breaking down the given parameters and applying the kinetic theory of gases, we've successfully decoded the thermal energy of the system.