Animated Solution for Physics - Thermodynamics: One mole of a monoatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100 K and the universal gas constant $R=8.0{\text{ J mol}}^{-1}{\text{ K}}^{-1}$, the decrease in its internal energy in joule, is ......... .

The beauty of thermodynamics lies in its ability to connect abstract mathematical equations with tangible physical realities. In this problem, we are exploring one of the most elegant processes in physics: the adiabatic expansion.

Imagine a perfectly insulated cylinder containing a monoatomic ideal gas. The gas is allowed to expand, pushing a piston outwards. Because the cylinder is insulated, no heat can enter or leave the system ($Q=0$). Yet, the gas is doing work on the piston. Where does the energy for this work come from? It must come from the gas's own internal energy! As a result, the gas cools down. Let's dive into the math to see exactly how much energy is spent.

Adiabatic Expansion

To find the change in internal energy, our first mission is to determine the final temperature of the gas. For an adiabatic process, the pressure, volume, and temperature are locked in a strict mathematical dance. Since we are dealing with temperature and volume, we use the relation:

This means that the initial state and the final state are connected by:

But what is $\gamma$ (gamma)? Gamma is the ratio of specific heats ($C_p/C_V$). For a monoatomic gas (like Helium or Argon), the atoms only have 3 translational degrees of freedom. This gives us a specific $\gamma$ value:

Therefore, the exponent in our equation becomes:

Now, let's substitute the known values into our master equation. We know the initial temperature $T_1=100\text{ K}$, and the final volume is eight times the initial volume ($V_2=8V_1$).

Notice how beautifully the initial volume $V_1$ is poised to cancel out. Let's rearrange the equation to isolate the final temperature $T_2$:

To solve this without a calculator, we can take the cube root first, and then square the result. The cube root of $\frac{1}{8}$ is $\frac{1}{2}$, and squaring that gives $\frac{1}{4}$.

The gas has cooled down from $100\text{ K}$ to a freezing $25\text{ K}$! This massive temperature drop is the direct result of the gas expending its internal energy to expand.

Change in Internal Energy

Now for the final piece of the puzzle. The change in internal energy ($\Delta U$) for any ideal gas depends only on the change in temperature, and is given by the formula:

For a monoatomic gas, the molar heat capacity at constant volume ($C_V$) is $\frac{3}{2}R$. Let's set up our final calculation with $n=1\text{ mole}$ and $R=8.0{\text{ J mol}}^{-1}{\text{K}}^{-1}$:

The negative sign is crucial here. It mathematically confirms our physical intuition: the internal energy of the system has decreased.

Since the question specifically asks for the decrease in internal energy, we take the absolute value. The internal energy decreased by exactly $900\text{ J}$.

According to the First Law of Thermodynamics ($Q=\Delta U+W$), since $Q=0$, the work done by the gas is $W=-\Delta U=900\text{ J}$. The gas performed $900\text{ J}$ of mechanical work, and it paid the exact price of $900\text{ J}$ from its own thermal bank account!