Animated Solution for Physics - Thermodynamics: In a process, temperature and volume of one mole of an ideal monoatomic gas are varied according to the relation $VT=k$, where $k$ is a constant. In this process, the temperature of the gas is increased by $\Delta T$. The amount of heat absorbed by gas is (where, $R$ is gas constant)

Thermodynamics is fundamentally the study of energy in transit. When a gas undergoes a transformation, it exchanges energy with its surroundings in the form of heat and work. In this thrilling problem, we are presented with a rather unusual constraint: a monoatomic gas undergoes a process where the product of its volume and temperature remains constant, mathematically expressed as $VT=k$. Our mission is to determine the total heat absorbed by the gas when its temperature is increased by $\Delta T$.

Let's embark on this thermodynamic journey step-by-step, unraveling the physics hidden within the equations.

From $T$ to $p$

The relation $VT=k$ is intriguing, but it is not immediately useful for calculating thermodynamic work. The standard formulas for work done by a gas are almost exclusively defined in terms of pressure ($p$) and volume ($V$). Therefore, our first strategic move must be to eliminate temperature ($T$) from the given relation.

We invoke the universal Ideal Gas Equation:

Isolating temperature, we get:

Now, we substitute this expression for $T$ back into our original constraint $VT=k$:

By rearranging the terms and moving all constants to the right side, we arrive at a beautiful and highly revealing equation:

Since $n$, $R$, and $k$ are all constants, their product is also a constant. Thus, the process equation simplifies to:

What does $p{V}^2=\text{constant}$ actually mean? This is the signature of a polytropic process, which generally takes the form $p{V}^x=\text{constant}$. By direct comparison, we can immediately identify our polytropic index:

Before we rush into calculations, let's pause and appreciate the physical reality of this process. We are given that $VT=\text{constant}$. This implies that volume and temperature are inversely proportional. The problem states that the temperature of the gas is increased ($\Delta T>0$). For the product $VT$ to remain constant while $T$ increases, the volume $V$ must strictly decrease.

A decrease in volume means the gas is undergoing compression. When a gas is compressed, the surroundings are doing work on the gas, which means the work done by the gas will be negative. Keep this physical intuition in mind; it will serve as a powerful reality check for our upcoming calculations.

Armed with the knowledge that this is a polytropic process with $x=2$, we can deploy the standard formula for work done in a polytropic process:

We are given exactly one mole of gas ($n=1$), and we have determined $x=2$. Substituting these values into our formula yields:

As predicted by our physical intuition, the work done is indeed negative! The gas is being squeezed, absorbing mechanical energy from its surroundings.

Next, we must evaluate the change in the internal energy of the gas. For any ideal gas, the change in internal energy depends solely on the change in temperature and is given by:

We are dealing with a monoatomic gas (like Helium or Argon). Monoatomic gases have exactly 3 translational degrees of freedom, which means their molar heat capacity at constant volume ($C_v$) is:

Substituting $n=1$ and our $C_v$ value, the change in internal energy becomes:

We now possess both pieces of the puzzle: the work done and the change in internal energy. It is time to synthesize them using the granddaddy of all thermodynamic principles—the First Law of Thermodynamics. This law states that the total heat absorbed by a system ($\Delta Q$) is equal to the sum of its change in internal energy and the work it does:

Let's plug in our calculated values:

Factoring out $R\Delta T$, we get:

This is our final, elegant answer. The gas absorbs heat equal to half of $R\Delta T$ during this specific polytropic compression.

Molar Heat Capacity Shortcut

While the First Law approach is robust and builds great conceptual understanding, competitive exams demand speed. There is a powerful shortcut you can use for any polytropic process.

Instead of calculating work and internal energy separately, you can directly calculate the molar heat capacity ($C$) for the specific polytropic process using the formula:

Let's test it with our values ($C_v=\frac{3}{2}R$ and $x=2$):

Once you have the molar heat capacity for the process, finding the heat absorbed is a single, effortless step:

Boom! We arrive at the exact same answer in a fraction of the time. Mastering both the fundamental First Law approach and the rapid molar heat capacity shortcut will make you an unstoppable force in thermodynamics!