Question

Which of the following gas molecule has the maxi mum specific heat at constant pressure? (a) helium (b) argon (c) nitrogen (d) oxygen

Short answer

Nitrogen and oxygen have the maximum specific heat at constant pressure, being equal.

Step-by-step solution

Understand Specific Heat at Constant Pressure

The specific heat at constant pressure, denoted as \( C_p \), is the amount of heat required to raise the temperature of a unit mass of a gas by one degree Celsius at constant pressure. For ideal gases, \( C_p \) depends on the degree of freedom of the gas molecules.

Determine Degrees of Freedom

Monoatomic gases like helium and argon have 3 translational degrees of freedom. Diatomic gases like nitrogen and oxygen have 5 degrees of freedom (3 translational and 2 rotational) under standard conditions.

Use the Formula for \( C_p \)

The specific heat at constant pressure can be approximated using the formula \( C_p = \frac{f}{2} R + R \), where \( f \) is the degrees of freedom and \( R \) is the universal gas constant. For monoatomic gases, \( C_p = \frac{3}{2}R + R = \frac{5}{2} R \). For diatomic gases, \( C_p = \frac{5}{2}R + R = \frac{7}{2} R \).

Compare \( C_p \) Values

Comparing the calculated \( C_p \) values, we have \( \frac{5}{2} R \) for helium and argon, and \( \frac{7}{2} R \) for nitrogen and oxygen. Diatomic gases have a higher \( C_p \) than monoatomic gases.

Determine Which Gas Has Maximum \( C_p \)

Since both nitrogen and oxygen are diatomic gases with the same formula for \( C_p \), they both possess the maximum specific heat at constant pressure in this list. There is no difference between the \( C_p \) of nitrogen and oxygen under these conditions.

Key concepts

Degrees of Freedom

Degrees of freedom refer to the number of independent ways a molecule in a gas can move or store energy. In simple terms, it represents the different types of motion or vibration a molecule can have. For monatomic gases, such as helium and argon, there are 3 translational degrees of freedom. This is because their motion can be described by their movements in the three-dimensional space: x, y, and z directions.

Diatomic gases, like nitrogen and oxygen, have 5 degrees of freedom under normal conditions. They have 3 translational degrees, similar to monatomic gases, but due to having two atoms bonded, they have 2 additional rotational degrees of freedom. These rotations occur around two perpendicular axes passing through their center of mass, giving them more ways to absorb heat.

Monoatomic Gases

Monoatomic gases consist of single atoms. Examples include noble gases like helium and argon. These types of gases are simpler in structure compared to diatomic or polyatomic gases. Because they are monoatomic, they only translate or move in the three-dimensional space without any internal rotation or vibration.

This simplicity limits their degrees of freedom to just the three translational motions. Monoatomic gases typically have lower heat capacities compared to more complex gases, primarily because they have fewer modes or ways to store energy when heat is added. This is important when understanding why their specific heat at constant pressure, denoted as \( C_p \), is lower than that of diatomic gases.

Diatomic Gases

Diatomic gases contain molecules composed of two atoms. Common examples are nitrogen \((N_2)\) and oxygen \((O_2)\). Due to their two-atom structure, they can move and rotate in space in more ways than monoatomic gases can.

In addition to the three translational degrees of freedom, they have two rotational degrees of freedom. This means that they can rotate about two axes perpendicular to the line joining the two atoms. Under certain conditions, they might also vibrate along the line joining the atoms, although this contribution often comes into effect at higher temperatures.

Heat Capacity

Heat capacity is a measure of the amount of heat energy required to change the temperature of a substance by a certain amount. Specific heat at constant pressure \( (C_p) \) is a type of heat capacity that tells us how much energy is needed to raise the temperature of a given mass of gas by one degree Celsius at constant pressure.

The greater the number of degrees of freedom, the higher the heat capacity. This is because the gas can distribute the absorbed heat into more modes of motion. For example, diatomic gases with 5 degrees of freedom have a higher \( C_p \) than monoatomic gases, which only have 3. Hence, diatomic gases require more energy to increase the temperature by the same amount, as compared to monoatomic gases.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the relationship between the pressure \((P)\), volume \((V)\), temperature \((T)\), and amount \((n)\) of an ideal gas. It is often expressed as \( PV = nRT \), where \( R \) is the universal gas constant.

This law is an approximation that helps understand the behavior of gases under various conditions by assuming no interaction between gas particles and that they occupy no volume. Although real gases may deviate from these assumptions at high pressures and low temperatures, the Ideal Gas Law provides a useful approximation for explaining the basic properties and behavior of gases, and it often underlies how we think about concepts such as degrees of freedom and specific heat.