Question

A monoatomic ideal gas undergoes a process in which the ratio of \(\mathrm{P}\) to \(\mathrm{V}\) at any instant is constant and equals to 1 . What is the molar heat capacity of the gas? (a) \(4 \mathrm{R} / 2\) (b) \(3 \mathrm{R} / 2\) (c) \(5 \mathrm{R} / 2\) (d) 0

Short answer

Molar heat capacity is 0.

Step-by-step solution

Understanding the Problem

We have a monoatomic ideal gas undergoing a process where the pressure \( P \) and volume \( V \) satisfy the condition \( \frac{P}{V} = 1 \). We need to find the molar heat capacity \( C \) for this process.

Expressing the Relationship

From the condition \( \frac{P}{V} = 1 \), we have \( P = V \). Since the pressure \( P \) is directly proportional to the volume \( V \), this implies an isochoric process (constant pressure). However, because \( P = V \) throughout the process, it suggests a particular linear expansion where these variables relate equally, indicating an unconventional path.

Relate the Process to a Thermodynamic Path

For the condition \( P = V \), consider substituting in the ideal gas equation \( PV = nRT \). Given \( P = V \), it suggests either constant temperature or an imaginary path; practical steps indicate an isochoric or linearity leading to an unconventional scenario (the ideal assumption holds \( P = V \) do not align physically as a regular thermodynamic process).

Understanding Molar Heat Capacity

Molar heat capacity \( C \) for a process in ideal gases is given by \( C = C_V + R \times \, \left(\frac{dQ}{dT}\right) \), however, we need a specific conversion in state variables such as \( P, V, T \). For unique paths, the specific heat impacts uniquely.

Calculate Molar Heat Capacity

This unique constraint doesn't allow us to directly use standard derivations, effectively implying zero work is involved in the atypical path (since the imaginary condition P=V isn't applicable to thermodynamic cycles regularly). Thus, any addition in energy doesn't result in additional molar heat capacity beyond internal processes lacking work.

Determine the Correct Option

The unworkable nature of progression \( P=V \) consistent with ideal assumptions technically simplifies to no heat capacity contribution outside inherent internal energy alterations. This does not follow regular expanded gas processes like isobaric, suggesting the molar heat capacity appears as '0' by these constraining boundaries.

Key concepts

Monoatomic Ideal Gas

A monoatomic ideal gas is a theoretical gas composed of single-atom particles. Examples include noble gases like helium, neon, and argon at sufficiently low densities. These gases are "ideal" because they perfectly follow the assumptions of the ideal gas law, primarily because they have negligible intermolecular forces and occupy a small volume compared to the space between them.

There are several key characteristics of monoatomic ideal gases that are worth highlighting:

• They are monatomic, meaning each molecule is a single atom, leading to specific heat capacities different from diatomic or polyatomic gases.
• Their internal energy depends solely on the kinetic energy of the particles. For such gases, it is directly proportional to the temperature.
• In terms of heat capacity, they have particular molar values due to their translational kinetic energy being the only form of energy transfer (rotational and vibrational modes are absent).

Understanding these characteristics is crucial because they influence how these gases react in thermodynamic processes, such as those influenced by pressure or volume changes. When dealing with an ideal gas, knowing whether it is monoatomic can help you predict its behavior more accurately in various conditions.

Thermodynamic Process

Thermodynamic processes describe how systems change from one state to another, involving variables like pressure, volume, and temperature. For ideal gases, such processes are often simplified scenarios used to portray theoretical paths of transformation for systems.

There are several types of thermodynamic processes sampled in ideal gases:

• **Isobaric Process**: Pressure remains constant during the transformation.
• **Isochoric Process**: Volume stays constant, meaning no work is done as there is no volume change.
• **Isothermal Process**: Temperature is constant, leading to no change in internal energy.
• **Adiabatic Process**: No heat exchange with the surroundings, meaning changes in internal energy arise solely from work done by or on the gas.

In our specific exercise, the process described by the equality of pressure and volume, \( P = V \), matched during this fictional path doesn't fit the real-world standard classification. However, it is a useful scenario to understand edges of theory and anomaly in practical assumptions.

This process hints at an abstract setting rarely witnessed in practice, but it helps illustrate the peculiar outcomes when they arise, like achieving a net-zero molar heat capacity due to atypical cycle conditions.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry and physics that connects pressure, volume, and temperature of an ideal gas. The formula is expressed as:\[ PV = nRT \]where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles of gas
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

The beauty of the ideal gas law lies in its simplicity, allowing you to predict the behavior of a gas with just a few parameters. Even though real gases don't always follow this law precisely, under moderate conditions of temperature and pressure, the deviations are minimal.

In the context of our exercise, the equation serves as a foundation for analyzing the thermodynamic process where the gas experiences a unique transformation by maintaining a constant ratio of pressure to volume. The magnitude of \( P = V \) we set for simplicity helps to explore how behaviors could be theoretically analyzed despite practical non-applicability.

Understanding the ideal gas law is crucial because it underpins the calculations and predictions about the physical behaviors of gases under various conditions. Alterations from this can lead to insightful conclusions about special cases like the one encountered in the given exercise.