Problem 44 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

When an ideal diatomic gas is a heated at a constant pressure, the fraction of heat energy supplied which increase the internal energy of the gas is (a) \(\frac{2}{5}\) (b) \(\frac{3}{5}\) (c) \(\frac{5}{7}\) (d) \(\frac{3}{7}\)

Step-by-Step Solution

Answer

\frac{5}{7}

Step 1: Understand the problem

We are asked to find the fraction of heat energy supplied to an ideal diatomic gas that goes into increasing its internal energy while it is heated at constant pressure. Recall that the heat capacity ratio for a diatomic gas is important for this calculation.

Step 2: Recall the heat capacity ratios

For a diatomic gas at constant pressure, the heat capacity (\(C_p\text{ - molar heat capacity at constant pressure}\) ), is \(C_p = \frac{7}{2} R\text{ (where R is the ideal gas constant). }\) At constant volume, the molar heat capacity (\(C_v\text{ - molar heat capacity at constant volume}\) ) is \(C_v = \frac{5}{2} R.\text{ The difference between }C_p\text{ and }C_v\text{ will give the work done on the gas.}\underline{\phantom{xxx}}\)

Step 3: Calculate the fraction of energy that increases internal energy

Since the total heat supplied (\(Q\text{ - total heat supplied}\) ) at constant pressure goes into doing work (\(W\text{ - work done by the gas}\) ) and increasing internal energy (\( \bigtriangleup U \text{ - change in internal energy}\text{), }\) we have} \(Q = \bigtriangleup U + W\text{. }\) The work done by the gas at constant pressure is equal to the pressure multiplied by the change in volume, which is also expressed by the relation \(W = nR\bigtriangleup T\text{, where }n\text{ is the number of moles and }\bigtriangleup T\text{ is the change in temperature. Since we are dealing with heat capacities, this translates to }W = C_p - C_v\text{. For a diatomic gas, this is }W = \frac{7}{2} R - \frac{5}{2} R = R\text{. Therefore, the fraction of the total heat which increases internal energy is }\) \( \frac{\bigtriangleup U}{Q} = \frac{Q - W}{Q} = \frac{Q - R}{Q} = \frac{\frac{5}{2} R}{\frac{7}{2} R} = \frac{5}{7}\text{. }\)

Key Concepts

Internal Energy of Gas

Understanding the internal energy of a gas is essential while studying thermodynamics and how heat energy interacts with gases. The internal energy, denoted as \( U \), is the total energy contained within a gas. This energy comes from the kinetic energy that the molecules possess due to their random motion.

When a diatomic gas is heated at constant pressure, part of the heat energy supplied is used to raise its internal energy. The rise in internal energy, represented as \( \Delta U \), correlates to an increase in the temperature of the gas. For an ideal gas, this increase is directly proportional to the change in temperature and the number of moles of the gas.

Furthermore, the internal energy for diatomic gases, which have rotational and vibrational modes, is greater than for monatomic gases due to these additional degrees of freedom. Consequently, understanding this concept helps clarify how heat transfer results in temperature changes and work performed by the gas.

Molar Heat Capacity

The molar heat capacity is a measure of how much heat energy is required to raise the temperature of one mole of a substance by one degree Celsius. For gases, it's crucial to differentiate between heat capacities at constant volume (\(C_v\)) and at constant pressure (\(C_p\)).

In the context of a diatomic gas, when heat is supplied at constant pressure, the molar heat capacity, \((C_p\)), is higher than at constant volume, \((C_v\)), since it also includes the energy that does work due to the expansion of the gas. The relationship for a diatomic gas at constant volume is \((C_v = \frac{5}{2} R\)) and at constant pressure is \((C_p = \frac{7}{2} R\)).

Understanding these values is key when dissecting the ways in which heat energy is partitioned between raising internal energy and performing work on the surroundings, which is a fundamental aspect of the first law of thermodynamics.

Ideal Gas Constant

The ideal gas constant, \( R \) is a vital constant in thermodynamics linking the physical properties of an ideal gas to the amount of substance present. Its value comes into play when describing relationships described in the universal gas law, which combines Boyle’s, Charles’s, and Avogadro’s laws into a single equation.

The numerical value of \( R \) is approximately \( 8.314 \, J/(mol \cdot K) \), and it appears in various equations, representing the energy per mole per Kelvin unit. In the context of heating a diatomic gas, \( R \) helps us understand the amount of work done when the gas expands under constant pressure and the division of energy contributions in increasing the gas's internal energy.

Knowing the ideal gas constant allows us to calculate how much of the heat supplied to a diatomic gas is used for raising its internal energy and how much for doing work, as seen in the relationship between \( C_p \) and \( C_v \) for a diatomic gas.