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

The molar heat capacity is 0 (Option d).

Step-by-step solution

Understand the Process

Since the process is described where the ratio of pressure, \(P\), to volume, \(V\), is constant and equal to 1, we interpret this as the equation \(P = V\). In other terms, \(P/V = 1\).

Apply the Ideal Gas Law

Recall the ideal gas law \(PV = nRT\). Given \(P = V\), substitute this into the equation, resulting in \(V^2 = nRT\). This represents an equation where temperature \(T\) is a function of \(V^2\) as \(T = \frac{V^2}{nR}\).

Identify the Type of Process

In this situation, since \(n, R\) are constants while \(PV = nRT\) evolves as \(V^2\), it implies a non-standard process where neither temperature nor energy remains constant. Typically, work done and energy exchange must be considered integrally.

Calculate the Molar Heat Capacity

To find the molar heat capacity \(C\), use the relationship where \(C = \frac{dQ}{dT}\). However, since \(PV = nRT\) and \(P = V\), derivatives and integrals of the state functions provide that no heat \(Q\) flows in or out as it performs work without temperature change stabilizing differently, thus the molar heat capacity required is \(0\) since \(dQ = 0\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that describes the behavior of ideal gases. It is written as \(PV = nRT\). This equation relates the four key variables associated with a gas:

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

Each variable is crucial in understanding how gases behave under different conditions.
The Ideal Gas Law assumes that gas particles move randomly and do not interact with one another.
This simplification helps in predicting the behavior of gases when other factors remain constant. It's essential in calculations involving changes in temperature, volume, and pressure.

Monoatomic Ideal Gas

A monoatomic ideal gas consists of single atoms, like helium or argon.
These gases follow the Ideal Gas Law but with additional specific details due to their atomic nature.
Monoatomic gases have unique molar heat capacities.
For such gases, when they undergo any process, the heat capacity at constant volume \(C_v\) is given by \(C_v = \frac{3}{2}R\). This is because the internal energy changes only due to translational motion. When considering processes involving a monoatomic ideal gas, its behavior is predictable because there are no complexities added by molecular vibrations or rotations.

• They exhibit simple kinetic energy variations when temperature changes.
• These gases are a core concept in thermodynamics due to their simple structure.

Non-standard Thermodynamic Process

A non-standard thermodynamic process does not adhere to the usual identifiable paths like isothermal, isobaric, or adiabatic processes. In the described scenario, the gas undergoes a process where the ratio \(P/V = 1\) remains constant. This is unusual because:

• The typical laws governing temperature or pressure do not apply clearly.
• Instead, work done and energy changes become integral without straightforward heat flow.

In this specific process, the lack of heat transfer, as indicated by \(dQ = 0\), implies that the energy used for work does not alter the heat content of the system. Thus, the molar heat capacity \(C\), defined as \(C = \frac{dQ}{dT}\), results in zero since there is no heat exchange.
This highlights how unique thermodynamic processes often ignore traditional paths, showing how flexible gas behaviors can be under unconventional conditions.