Question

\(0.1\) mole of gas absorbs \(41.75 \mathrm{~J}\) of heat, the rise in temperature occurs \(20^{\circ} \mathrm{C}\), the gas must be: (a) Triatomic (b) Diatomic (c) Monoatomic (d) (b) and (c) both

Short answer

The gas is diatomic.

Step-by-step solution

Understand the Concepts

The heat capacity at constant volume, for a monoatomic gas is \(C_v = \frac{3}{2}R\), for a diatomic gas it is \(C_v = \frac{5}{2}R\), and for a triatomic gas it is \(C_v = 3R\). The relationship between heat absorbed (\(q\)), change in temperature (\(\Delta T\)), number of moles (\(n\)), and heat capacity (\(C_v\)) is given by \(q = n \cdot C_v \cdot \Delta T\).

Plug in Known Values

We know \(n = 0.1\) mol, \(q = 41.75\) J, and \(\Delta T = 20^{\circ}C\). We need to calculate \(C_v\) from \(q = n \cdot C_v \cdot \Delta T\), with which \(C_v = \frac{q}{n \cdot \Delta T}\).

Calculate \(C_v\)

Substitute the known values into the equation: \(C_v = \frac{41.75}{0.1 \times 20}\). Whereby simplifying, \(C_v = \frac{41.75}{2} = 20.875\) J/mol/K.

Compare with Theoretical Values

Find the theoretical values considering \(R = 8.314\) J/mol/K. For monoatomic gases, \(C_v = \frac{3}{2}R = 12.471\) J/mol/K; for diatomic gases, \(C_v = \frac{5}{2}R = 20.785\) J/mol/K; for triatomic gases \(C_v = 3R = 24.942\) J/mol/K.

Determine the Gas

The experimental \(C_v = 20.875\) J/mol/K is closest to the theoretical value of the diatomic gas \(C_v = 20.785\) J/mol/K.

Key concepts

Heat Capacity

In thermodynamics, heat capacity is a crucial concept that quantifies the amount of heat needed to change a substance's temperature. It is often represented by the symbol \(C\), and is measured in units of joules per mole per Kelvin (J/mol/K).

For individual gases, heat capacity can be specified at constant volume (\(C_v\)) or constant pressure (\(C_p\)). The heat absorbed or released when a gas changes temperature is directly related to its heat capacity, as seen in the formula:

• \( q = n \cdot C_v \cdot \Delta T \)

Where:

• \(q\) is the heat absorbed,
• \(n\) is the number of moles of gas, and
• \(\Delta T\) is the change in temperature.

Understanding heat capacity allows us to predict how substances will respond to heat, which is essential for many scientific and engineering applications.

Diatomic Gas

Diatomic gases are composed of molecules containing two atoms, which can be the same or different elements, such as \(O_2\) (oxygen) or \(CO\) (carbon monoxide). These gases play an essential role in thermodynamics because they possess rotational and vibrational modes of energy in addition to translational energy.

The heat capacity at constant volume for a diatomic gas is expressed as:

• \(C_v = \frac{5}{2}R\)

Where \(R\) is the ideal gas constant (8.314 J/mol/K). This higher heat capacity compared to monatomic gases arises from the additional degrees of freedom (rotational and vibrational) present in diatomic molecules.

Understanding the heat capacity linked to these degrees of freedom helps in studying the energy transfer and thermodynamic properties of diatomic gases.

Monatomic Gas

Monatomic gases consist of single atoms rather than molecules. Examples include noble gases like helium (He) and neon (Ne). Due to their simple atomic structure, these gases only have translational degrees of freedom.

For a monatomic gas, the heat capacity at constant volume is given by:

• \(C_v = \frac{3}{2}R\)

This reflects that monatomic gases lack rotational and vibrational modes of energy, leading to a lower heat capacity compared to polyatomic gases.

Since monatomic gases have simpler behavior, they provide a good model for studying basic thermodynamic systems and help deepen our understanding of more complex gases.

Triatomic Gas

Triatomic gases, as the name suggests, are composed of molecules with three atoms. An example of a triatomic gas is carbon dioxide (\(CO_2\)). Such gases exhibit even more complex behavior than diatomic gases owing to the additional degrees of freedom available to them.

The heat capacity at constant volume for a triatomic gas is:

• \(C_v = 3R\)

This higher heat capacity is due to the inclusion of more vibrational and rotational modes, which enables the storage of more energy.

These gases help us explore more elaborate thermodynamic principles and are often studied in the context of atmospheric science, chemistry, and engineering applications. Understanding their unique heat capacities enhances our knowledge of how different gas molecules absorb and transfer heat.