Animated Solution for Chemistry - Chemical Thermodynamics: For one mole of an ideal gas, which of these statements must be true? (A) $U$ and $H$ each depends only on temperature. (B) Compressibility factor $Z$ is not equal to $1$. (C) $C_{p,m}-C_{V,m}=R$ (D) $dU=C_{V}dT$ for any process.

Unpacking Thermodynamic Truths

When we dive into the world of chemical thermodynamics, the ideal gas serves as our perfect theoretical playground. It strips away the messy complexities of real-world molecular interactions, allowing us to see the pure, elegant mathematical relationships between temperature, pressure, volume, and energy.

In this problem, we are tasked with evaluating four fundamental thermodynamic statements for exactly one mole of an ideal gas. Let's break them down one by one and uncover the physics behind the math.

Let's start with Statement (A), which claims that both internal energy ($U$) and enthalpy ($H$) depend only on temperature.

For an ideal gas, the core assumption is that there are absolutely no intermolecular forces of attraction or repulsion. Because these forces are non-existent, there is no potential energy stored between the molecules. Therefore, the internal energy $U$ is entirely composed of the kinetic energy of the molecules. Since kinetic energy is directly proportional to the absolute temperature, it follows that $U$ is strictly a function of temperature: $U=f(T)$.

Now, what about enthalpy? By definition, enthalpy is given by the equation:

For one mole of an ideal gas, the ideal gas law tells us that $pV=RT$. Substituting this into our enthalpy equation gives:

Since we already established that $U$ depends only on temperature, and $RT$ obviously depends only on temperature, their sum $H$ must also depend exclusively on temperature. Thus, Statement (A) is absolutely true.

Next, we examine Statement (B), which states that the compressibility factor $Z$ is not equal to $1$.

The compressibility factor $Z$ is a dimensionless number used to quantify how much a real gas deviates from ideal behavior. It is defined mathematically as:

However, we are explicitly dealing with an ideal gas! For an ideal gas, the equation $pV=nRT$ is perfectly obeyed under all conditions. If we substitute $pV$ with $nRT$ in the numerator, the ratio simplifies perfectly:

Because $Z$ is exactly $1$ for an ideal gas, Statement (B) is false.

Moving on to Statement (C), we encounter the famous Mayer's relation: $C_{p,m}-C_{V,m}=R$.

For $n$ moles of an ideal gas, the difference between the heat capacity at constant pressure ($C_p$) and the heat capacity at constant volume ($C_V$) is equal to $nR$. Since our problem specifies exactly one mole ($n=1$), we use the molar heat capacities, denoted by the subscript 'm'.

Substituting $n=1$ into the general relation yields exactly $C_{p,m}-C_{V,m}=R$. This is a foundational derivation in thermodynamics, making Statement (C) perfectly true.

Finally, let's look at Statement (D), which claims that $dU=C_{V}dT$ for any process. This is where many students fall into a classic trap.

By definition, the heat capacity at constant volume is the rate of change of internal energy with respect to temperature:

For one mole ($n=1$), this rearranges to $dU=C_{V}dT$.

For a real gas, this equation is strictly valid only if the process occurs at a constant volume (isochoric). However, for an ideal gas, we must remember our conclusion from Statement (A): internal energy $U$ depends solely on temperature. It does not care about changes in volume or pressure.

Because $U$ is a state function dependent only on $T$, a specific change in temperature $dT$ will always produce the exact same change in internal energy $dU$, regardless of the path taken. Whether the process is isobaric, isothermal, or adiabatic, the relationship holds. Therefore, Statement (D) is true.

After a careful conceptual analysis, we have determined that statements (A), (C), and (D) are true, while statement (B) is false. Grouping the true statements together leads us directly to our final correct choice.