Explain why the heat capacities of methane and ethane differ from the values expected for an ideal monatomic gas and from each other. The values of \(C_{P}\) are \(35.31 \mathrm{~J} \cdot \mathrm{K}^{-1} \cdot \mathrm{mol}^{-1}\) for \(\mathrm{CH}_{4}\) and \(52.63 \mathrm{~J} \cdot \mathrm{K}^{-1} \cdot \mathrm{mol}^{-1}\) for \(\mathrm{C}_{2} \mathrm{H}_{6}\).

Short Answer

Expert verified
The heat capacities of methane and ethane are higher than the value expected for an ideal monatomic gas (\(\frac{5}{2}R\)) due to additional degrees of freedom in molecular motions. Methane has a lower heat capacity than ethane because it is a smaller molecule with fewer vibrational modes.

Step by step solution

01

Understanding Heat Capacities

Heat capacity at constant pressure (\(C_P\)) is a measure of the amount of heat required to raise the temperature of one mole of a substance by one Kelvin. Ideal monatomic gases typically have a \(C_P\) of \(\frac{5}{2}R\), where \(R\) is the gas constant (\(8.314 \text{ J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}\))
02

Reasons for Difference in Heat Capacities

Methane (\(CH_4\)) and ethane (\(C_2H_6\)) are not monatomic gases; they are molecules with multiple atoms bonded together. The presence of multiple atoms allows for additional degrees of freedom such as vibrations and rotations, which means these molecules can store more heat energy. Thus, their \(C_P\) values are higher than for a monatomic ideal gas.
03

Explanation of the Differences between Methane and Ethane

Methane, being a smaller molecule with fewer atoms, has fewer vibrational modes than ethane, which results in a lower heat capacity compared to ethane. Ethane (\(C_2H_6\)), with more atoms, has more possible vibrational modes and therefore a higher heat capacity.
04

Comparison to Ideal Gas Heat Capacity

For a monatomic ideal gas, \(C_P\) would be \(\frac{5}{2}R \approx 20.79 \text{ J} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}\). Both methane and ethane have higher \(C_P\) values than this due to their molecular complexity and additional degrees of freedom.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermodynamics
Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy. At its core, it studies how energy is converted from one form to another and how it affects matter. Key concepts in thermodynamics include temperature, heat, internal energy, and the laws of thermodynamics, which govern the principles of energy conservation and entropy.

When discussing heat capacity, we're exploring the thermodynamic property that indicates the amount of heat needed to change a substance's temperature by a given amount. This is essential in understanding how different substances react to heat and how they store energy. In the context of gases, heat capacity is particularly important because it helps predict how much energy is needed to heat up or cool down a specific amount of gas at constant volume or pressure.
Ideal Monatomic Gas
An ideal monatomic gas is a simplified model used in thermodynamics to describe the behavior of gases. It consists of single atoms, with no intermolecular forces except during elastic collisions. This model assumes that the atoms occupy no volume and that there are no interactions between them apart from collisions. The internal energy of an ideal monatomic gas thus arises solely from the kinetic energy of its atoms.

The concept of an ideal monatomic gas lays the foundation for understanding how real gases behave under various conditions. This simplistic model allows scientists and students alike to comprehend gas behaviors without the complicating factors found in polyatomic gases, such as intermolecular forces and additional degrees of freedom from molecular bonds.
Degrees of Freedom
Degrees of freedom in physics and chemistry refer to the number of independent ways by which a system can possess energy. For instance, in the framework of molecular motion, there are three types of degrees of freedom: translational, rotational, and vibrational.

A monatomic ideal gas has only translational degrees of freedom because the atoms can only move in three-dimensional space. In contrast, molecules with two or more atoms, like methane (CH₄) and ethane (C₂H₆), have additional rotational degrees of freedom because the molecules can rotate around different axes. If the energy levels are high enough, they can also have vibrational degrees of freedom, where atoms within the molecules move relative to each other.

The number of available degrees of freedom directly influences the heat capacity because more degrees of freedom means more ways to store thermal energy. Simple monatomic gases have the lowest heat capacity since they can only store kinetic energy in their translational motion. As molecules become more complex, the number of degrees of freedom and therefore the heat capacity increases, as evidenced by methane and ethane's heat capacities exceeding that of an ideal monatomic gas.

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Most popular questions from this chapter

A gas sample in a piston assembly expands, doing \(536 \mathrm{~kJ}\) of work on its surroundings at the same time that \(214 \mathrm{~kJ}\) of heat is added to the gas. (a) What is the change in internal energy of the gas during this process? (b) Will the pressure of the gas be higher or lower when these changes are completed?

Calculate the standard enthalpy of formation of \(\mathrm{PCl}_{5}(\mathrm{~s})\) from the standard enthalpy of formation of \(\mathrm{PCl}_{3}(\mathrm{l})\) (see Appendix \(2 \mathrm{~A}\) ) and \(\mathrm{PCl}_{3}(\mathrm{l})+\mathrm{Cl}_{2}(\mathrm{~g}) \longrightarrow \mathrm{PCl}_{5}(\mathrm{~s}), \Delta H^{\circ}=-124 \mathrm{~kJ}\).

Each of the four cylinders of a new type of combustion engine has a displacement of \(3.60 \mathrm{~L}\). (The volume of the cylinder expands \(3.60 \mathrm{~L}\) each time the fuel is ignited.) (a) If each piston in the four cylinders is displaced against a pressure of \(1.80 \mathrm{kbar}\) and each cylinder is ignited once per second, how much work can the engine do in \(1.00\) minute? (b) Is the work positive or negative with respect to the engine and its contents?

Which molecular substance do you expect to have the higher molar heat capacity, \(\mathrm{NO}\) or \(\mathrm{NO}_{2}\) ? Why?

Rank the following compounds in order of increasing enthalpy of vaporization: \(\mathrm{CH}_{4}, \mathrm{H}_{2} \mathrm{O}, \mathrm{N}_{2}, \mathrm{NaCl}, \mathrm{C}_{6} \mathrm{H}_{6}\), and \(\mathrm{H}_{2}\). Do not look up the enthalpy of vaporization. Explain your ordering.

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