Animated Solution for Physics - Thermodynamics: The work of $146\text{ kJ}$ is performed in order to compress one kilo mole of a gas adiabatically and in this process, the temperature of the gas increases by $7^{\circ }\text{C}$. The gas is ($R=8.3{\text{ J mol}}^{-1}{\text{ K}}^{-1}$)

Imagine you are holding a bicycle pump, and you block the nozzle with your thumb. When you push the handle down quickly, you are doing work on the air inside, compressing it. Because you do it so fast, the heat doesn't have time to escape. This is the essence of an adiabatic compression, and it's exactly what is happening in this problem!

We are given a scenario where $146\text{ kJ}$ of work is performed to compress $1\text{ kilo mole}$ of a gas adiabatically. The temperature of the gas increases by $7^{\circ }\text{C}$. Our goal is to identify the nature of the gas (monatomic, diatomic, etc.).

First, let's list our knowns carefully: - Work done on the gas, $W_{on}=146\text{ kJ}=146\times 10^3\text{ J}$. - Number of moles, $n=1\text{ kilo mole}=10^3\text{ moles}$. - Change in temperature, $\Delta T=7^{\circ }\text{C}$. (Remember, a change of $7^{\circ }\text{C}$ is exactly the same as a change of $7\text{ K}$). - Universal gas constant, $R=8.3{\text{ J mol}}^{-1}{\text{ K}}^{-1}$.

First Law of Thermodynamics

The First Law of Thermodynamics states that the heat added to a system equals the change in its internal energy plus the work done by the system:

Since the process is adiabatic, there is no heat exchange with the surroundings, meaning $Q=0$. Therefore, the equation simplifies beautifully to:

Wait, the problem says work is performed in order to compress the gas. This means work is done on the gas. The work done by the gas is negative. So, $-W_{by}$ is simply the work done on the gas ($W_{on}$).

This makes perfect physical sense: all the mechanical energy you put into compressing the gas goes directly into increasing its internal energy, which is why it heats up!

Now, we need to connect this internal energy to the identity of the gas. The change in internal energy for any ideal gas is given by:

We also know that the molar heat capacity at constant volume, $C_v$, is intimately tied to the gas's degree of freedom ($f$):

Substituting this into our internal energy equation gives us our master working formula:

Let's plug in our known values and solve for $f$:

Notice how the $10^3$ gracefully cancels out from both sides. This is why keeping track of units (like kilo moles and kilo joules) is so crucial!

Since the degree of freedom must be an integer, we can safely round this to $f=5$.

A degree of freedom of 5 corresponds to a gas with 3 translational and 2 rotational modes of motion. This is the classic signature of a diatomic gas at room temperature (like $O_2$ or $N_2$).

Therefore, the gas being compressed is diatomic!