Animated Solution for Physics - Thermodynamics: A diatomic ideal gas is compressed adiabatically to $\frac{1}{32}$ of its initial volume. If the initial temperature of the gas is $T_i$ (in kelvin) and the final temperature is $a{T}_i$, the value of $a$ is

Imagine you have a cylinder filled with a diatomic ideal gas, like oxygen or nitrogen. The gas is initially at a temperature $T_i$ and occupies a volume $V_i$. Now, a piston rapidly compresses this gas to a mere $\frac{1}{32}$ of its original volume. Because this compression happens so quickly, there is no time for heat to escape to the surroundings. This makes the process adiabatic.

We are told that the final temperature becomes $a{T}_i$, and our mission is to find the value of this multiplier, $a$.

For an adiabatic process involving an ideal gas, the relationship between temperature and volume is governed by the equation:

This means that the product of temperature and volume raised to the power of $\gamma -1$ remains the same throughout the process. We can write this for our initial and final states as:

Before we can use this equation, we need to figure out the value of $\gamma$ (the adiabatic index) for our gas. We know the gas is diatomic. At normal temperatures, a diatomic molecule has $5$ degrees of freedom (3 translational and 2 rotational). The formula for $\gamma$ is:

Substituting $f=5$, we get:

Therefore, the exponent in our equation, $\gamma -1$, is simply $1.4-1=0.4$.

Now, let's plug everything we know into our master equation. We substitute $\gamma -1=0.4$, $T_f=a{T}_i$, and $V_f=\frac{V_i}{32}$:

Look closely at this equation. The terms $T_i$ and $V_i^{0.4}$ appear on both sides. This is the beauty of such physics problems—the initial unknown variables often cancel out! Let's divide both sides by $T_i{V}_i^{0.4}$:

To isolate $a$, we rearrange the equation:

All that's left is a bit of arithmetic. Let's convert the decimal exponent into a fraction to make it easier to handle:

So, we need to evaluate $(32{)}^{\frac{2}{5}}$. We know that $32$ is $2$ raised to the power of $5$ ($2^5=32$). Let's substitute this in:

When you raise a power to a power, you multiply the exponents. The $5$ in the numerator and the $5$ in the denominator cancel each other out perfectly:

And there we have it! The value of $a$ is $4$. This means the temperature of the gas quadrupled during the compression. This dramatic rise in temperature is exactly how a diesel engine ignites its fuel without a spark plug—pure adiabatic compression!