Animated Solution for Chemistry - Chemical Thermodynamics: For a diatomic ideal gas in a closed system, which of the following plots does not correctly describe the relation between various thermodynamic quantities?

Thermodynamics is not just a collection of abstract equations; it is the study of how energy flows and transforms in the physical world. When we look at an ideal gas, its behavior is governed by beautifully simple laws. One of the most fascinating aspects of an ideal gas is how it stores heat energy, which brings us to the concept of molar heat capacities: $C_p$ (at constant pressure) and $C_V$ (at constant volume).

To understand heat capacities, we must first understand the 'degrees of freedom' ($f$). Imagine a diatomic molecule like oxygen ($O_2$) or nitrogen ($N_2$). It's like a tiny dumbbell flying through space. The ways this dumbbell can move and store energy are its degrees of freedom.

For any ideal gas, the molar heat capacities are directly tied to these degrees of freedom through the relations:

Here is the crucial takeaway: The degrees of freedom ($f$) depend exclusively on temperature. They do not care about how much pressure the gas is under or what volume it occupies. At low temperatures, the molecule only translates ($f=3$). As it gets warmer, it starts rotating ($f=5$). At extremely high temperatures, the bond between the atoms starts vibrating like a spring ($f=7$).

The Four Plots

Armed with this knowledge, let's play detective and examine the four plots provided in the question.

Plot (d): $C_V$ vs $V$ This plot shows $C_V$ as a flat, horizontal line as volume increases. Since we know $C_V$ depends only on temperature and is completely independent of volume, this plot is absolutely correct.

Plot (b): $C_V$ vs $T$ This plot shows $C_V$ increasing in discrete steps as temperature rises. This perfectly illustrates the quantization of energy! As the gas heats up, it reaches specific thermal thresholds where rotational and then vibrational modes 'unlock'. Each time a new mode unlocks, $C_V$ jumps to a higher constant value. This plot is a beautiful representation of quantum mechanics in thermodynamics and is entirely correct.

Plot (c): $U$ vs $T$ Internal energy ($U$) is the total energy stored in the gas. The relationship between internal energy and temperature is given by $dU=n{C}_{V}dT$. This means the slope of the $U$ vs $T$ graph is exactly $n{C}_V$. Since we just saw in Plot (b) that $C_V$ increases at higher temperatures, the slope of the $U$ vs $T$ graph must also increase. The curve correctly bends upwards, reflecting this increasing slope.

Plot (a): $C_p$ vs $p$ Finally, we look at Plot (a). It depicts $C_p$ increasing linearly as pressure ($p$) increases. But wait! We established right at the beginning that $C_p$ depends only on temperature. Changing the pressure of an ideal gas does not change its heat capacity. Therefore, the graph of $C_p$ versus $p$ should be a perfectly flat, horizontal line.

Plot (a) violates the fundamental principles of ideal gas thermodynamics by suggesting that heat capacity is a function of pressure. Therefore, it is the incorrect plot, making it the right answer to our question. Always remember to critically analyze the axes of any graph you encounter in physics and chemistry!