Animated Solution for Physics - Thermodynamics: In an adiabatic process, the density of a diatomic gas becomes 32 times of its initial value. The final pressure of the gas is found to be $n$ times the initial pressure. The value of $n$ is

Imagine a sturdy, perfectly insulated cylinder filled with a diatomic gas—perhaps oxygen or nitrogen, the very air we breathe. We are about to subject this gas to an adiabatic compression.

What does "adiabatic" mean in the physical world? It means the process happens so fast, or the insulation is so perfect, that absolutely no heat can escape or enter the system. The gas is completely isolated from its thermal surroundings.

The problem throws a fascinating constraint at us: the density of the gas becomes $32$ times its initial value.

Let's pause and think about what density really is. Density, denoted by $\rho$, is simply the mass of the gas divided by its volume:

Since our cylinder is sealed, the mass $m$ of the gas is strictly conserved. It cannot change. Therefore, the density is inversely proportional to the volume. If the gas molecules are suddenly packed $32$ times more densely, it physically means the space they occupy has been violently shrunk.

Mathematically, we can write:

The volume has been crushed to $\frac{1}{32}$ of its original size!

Now that we understand the geometric reality of the compression, we need a tool to connect this volume change to the pressure change.

For an isothermal process, we would use Boyle's Law ($pV=\text{constant}$). But our process is adiabatic. The compression does work on the gas, and with nowhere for the heat to go, the internal energy skyrockets. This means the pressure will increase much more aggressively than it would in an isothermal scenario.

The governing law for an adiabatic process is Poisson's equation:

Here, $\gamma$ (gamma) is the ratio of specific heats ($C_p/C_v$). This is where the specific nature of the gas becomes critical. The problem explicitly states we are dealing with a diatomic gas.

For a diatomic gas, the molecules have $5$ degrees of freedom (3 translational and 2 rotational) at room temperature. This gives us a specific heat ratio of:

This is a high-yield fact for JEE. Never confuse it with the monoatomic value of $\frac{5}{3}$!

We are now armed with all the necessary pieces. Let's set up our initial and final states using Poisson's equation:

We want to find how many times the final pressure is compared to the initial pressure. Let's isolate the pressure ratio:

Substitute the volume ratio we discovered earlier ($\frac{V_i}{V_f}=32$) and our diatomic gamma ($\gamma =\frac{7}{5}$):

At first glance, calculating a fractional exponent might look intimidating. But look closely at the number $32$. It is a beautiful, highly composite number in the world of physics and computer science. It is exactly $2$ raised to the power of $5$.

Let's rewrite the equation:

By the laws of exponents, when you raise a power to a power, you multiply the exponents. The $5$ in the numerator and the $5$ in the denominator cancel out perfectly in a moment of mathematical elegance:

And what is $2^7$?

Therefore, the final pressure is $128$ times the initial pressure.

The value of $n$ is $128$.

This problem is a beautiful symphony of physical intuition (density to volume) and mathematical execution (adiabatic equation and exponent rules). Always remember to read the gas type carefully, and trust that the numbers in JEE problems are often designed to cancel out beautifully!