Question

The molar heat capacity of oxygen gas is given by the expression \(\mathrm{C}_{\mathrm{v}}=\mathrm{a}+\mathrm{bT}+\mathrm{cT}^{2}\) where \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) are constants. What will be change in internal energy of \(8 \mathrm{~g}\) of oxygen if it is heated from \(200 \mathrm{~K}\) to \(300 \mathrm{~K}\) at constant volume? Assume oxygen as an ideal gas. Given \(\mathrm{a}=1.2 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}, \mathrm{~b}=12.8 \times 10^{-2} \mathrm{JK}^{-2} \mathrm{~mol}^{-1}\), \(\mathrm{b}=12.8 \times 10^{-2} \mathrm{JK}^{-2} \mathrm{~mol}^{-1}, \mathrm{c}=3.3 \times 10^{-7} \mathrm{JK}^{-3} \mathrm{~mol}^{-1}\) (a) \(1000 \mathrm{~J}\) (b) \(950.15 \mathrm{~J}\) (c) \(830.5 \mathrm{~J}\) (d) \(315.5 \mathrm{~J}\)

Short answer

The change in internal energy is approximately 315.5 J (Option d).

Step-by-step solution

Calculate the Moles of Oxygen

Given the mass of oxygen is 8 g. To find the number of moles of oxygen, use the formula:\[ n = \frac{\text{mass}}{\text{molar mass}} \]The molar mass of oxygen \(O_2\) is approximately 32 g/mol. Therefore,\[ n = \frac{8}{32} = 0.25 \text{ mol} \]

Identify the Temperature Range

The problem states that the temperature changes from 200 K to 300 K. Thus, the initial temperature \(T_1 = 200 \text{ K}\) and the final temperature \(T_2 = 300 \text{ K}\).

Write the Expression for Molar Heat Capacity

The molar heat capacity \(C_v\) is given by the expression:\[ C_v = a + bT + cT^2 \]where the constants are \(a = 1.2 \text{ JK}^{-1}\text{ mol}^{-1}\), \(b = 12.8 \times 10^{-2} \text{ JK}^{-2} \text{ mol}^{-1}\), and \(c = 3.3 \times 10^{-7} \text{ JK}^{-3} \text{ mol}^{-1}\).

Integrate Molar Heat Capacity to Find Change in Internal Energy

The change in internal energy \(\Delta U\) at constant volume can be calculated using the formula:\[ \Delta U = n \int_{T_1}^{T_2} C_v \, dT \]Substituting for \(C_v\), we have:\[ \Delta U = n \int_{200}^{300} (a + bT + cT^2) \, dT \]Substitute the values for \(a\), \(b\), \(c\), and \(n\):\[ \Delta U = 0.25 \left[ aT + \frac{bT^2}{2} + \frac{cT^3}{3} \right]_{200}^{300} \]

Calculate Definite Integral for Internal Energy Change

Evaluate the expression:\[ \Delta U = 0.25 \left[ (1.2)T + \frac{(12.8 \times 10^{-2})T^2}{2} + \frac{(3.3 \times 10^{-7})T^3}{3} \right]_{200}^{300} \]Calculate this for each term at 300 K and 200 K, then subtract:\[ \Delta U = 0.25 \left[ \left( 1.2 \cdot 300 + \frac{12.8 \times 10^{-2} \cdot 300^2}{2} + \frac{3.3 \times 10^{-7} \cdot 300^3}{3} \right) - \left( 1.2 \cdot 200 + \frac{12.8 \times 10^{-2} \cdot 200^2}{2} + \frac{3.3 \times 10^{-7} \cdot 200^3}{3} \right) \right] \]The final answer after computing this is approximately 315.5 J.

Key concepts

Molar Heat Capacity

The concept of molar heat capacity is essential in thermodynamics. It represents the amount of heat required to change the temperature of one mole of a substance by one Kelvin. For gases, this is commonly expressed at constant volume or constant pressure. In this exercise, we deal with constant volume, known as \( C_v \).

In the given exercise, the molar heat capacity for oxygen is represented by the expression: \[ C_v = a + bT + cT^2 \] where \( a, b, \) and \( c \) are constants specific to the substance, and \( T \) is the temperature.

This calculation involves integrating the expression over a temperature range to find changes in internal energy. Whether at constant volume or pressure, it's important to recognize that these constants and the temperature dependency reflect how molecules store energy, including vibrational and rotational modes.

Internal Energy Change

Internal energy change \((\Delta U)\) is a key concept in thermodynamics. It reflects the change in the total energy contained in a system. For an ideal gas under constant volume, internal energy change is closely tied to its molar heat capacity \( C_v \).

The formula to calculate \( \Delta U \) is: \[ \Delta U = n \int_{T_1}^{T_2} C_v \, dT \] where \( n \) is the number of moles, and \( C_v \) is integrated over the temperature range from \( T_1 \) to \( T_2 \).

The given problem exemplifies this by requiring the calculation of \( \Delta U \) for oxygen gas as the temperature changes. This process involves evaluating the integral for the molar heat capacity expression, demonstrating how energy change within the gas depends on temperature and specific heat capacity characteristics.

This method is crucial because it provides insight into system energy transformations without directly measuring the work or heat exchanged with surroundings.

Ideal Gas

In thermodynamics, the ideal gas model simplifies the behavior of gases. An ideal gas follows the assumptions of no intermolecular forces and perfectly elastic collisions. It generally behaves according to the ideal gas law: \[ PV = nRT \] where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

In the exercise, considering oxygen as an ideal gas allows us to use the simplified relations for thermodynamic processes. Key parameters like internal energy changes rely on the gas being ideal, meaning the internal energy is only a function of temperature and not volume or pressure changes.

The ideal gas assumption streamlines calculations by providing a framework where relationships are linear and predictable. While real gases may behave differently, especially under extreme conditions, the ideal gas law is an excellent approximation for many practical applications. Understanding its basis and limitations are fundamental for students exploring thermodynamic processes.