Question

Calculate the change in internal energy of 1.00 mole of a diatomic ideal gas that starts at room temperature \((293 \mathrm{~K})\) when its temperature is increased by \(2.00 \mathrm{~K}\).

Short answer

Answer: The change in internal energy of the diatomic ideal gas when its temperature is increased by 2.00 K is 41.570 J.

Step-by-step solution

1. Understand the internal energy of a diatomic ideal gas

The internal energy of an ideal gas only depends on its temperature (T) and the number of moles (n). For a diatomic ideal gas, we also need to know the degrees of freedom, which is equal to 5 for a diatomic ideal gas.

2. Formula for change in internal energy

The formula for the change in internal energy (ΔU) is: ΔU = n * C_v * ΔT, where n is the number of moles, C_v is the molar heat capacity (at constant volume), and ΔT is the change in temperature. For a diatomic ideal gas, the molar heat capacity at constant volume, C_v, is given by: C_v = (5/2) * R, where R is the universal gas constant (8.314 J/mol·K).

3. Calculate C_v for a diatomic ideal gas

Using the above formula for C_v, we can calculate the molar heat capacity at constant volume for a diatomic ideal gas: C_v = (5/2) * 8.314 \mathrm{~J/mol·K} = 20.785 \mathrm{~J/mol·K}

4. Calculate the change in temperature

The temperature change (ΔT) is given in the problem: ΔT = +2.00 K.

5. Calculate the change in internal energy (ΔU)

Now, we can use the formula for the change in internal energy (ΔU) and plug in the values: ΔU = n * C_v * ΔT = 1.00 \mathrm{~mol} * 20.785 \mathrm{~J/mol·K} * 2.00 \mathrm{~K} ΔU = 41.570 \mathrm{~J} The change in internal energy of the diatomic ideal gas when its temperature is increased by 2.00 K is 41.570 J.

Key concepts

Molar Heat Capacity

Understanding the molar heat capacity is essential when studying thermodynamics and the behavior of gases. The molar heat capacity at constant volume, denoted as \(C_v\), refers to the amount of heat required to raise the temperature of one mole of a substance by one degree Kelvin (K) while keeping the volume constant.

For diatomic ideal gases, such as oxygen or nitrogen, the molar heat capacity at constant volume is particularly important. These gases have more degrees of freedom compared to monatomic gases, such as helium or neon, due to their ability to rotate and vibrate. In the case of diatomic gases, there are 5 degrees of freedom: 3 translational (movement in three-dimensional space) and 2 rotational. As a result, the molar heat capacity of diatomic ideal gases is \(C_v = \frac{5}{2}R\), where \(R\) is the universal gas constant with a value of 8.314 J/mol·K.

This value comes from the kinetic theory of gases and the equipartition theorem, which states that each degree of freedom contributes \(\frac{1}{2}R\) to the molar heat capacity. Hence, for a diatomic gas with 5 degrees of freedom, the molar heat capacity is \(\frac{5}{2}R\). This concept is pivotal for calculating changes in the internal energy of a gas due to temperature variations.

Degrees of Freedom

The term 'degrees of freedom' in thermodynamics refers to the number of independent ways in which a molecule can store energy. This typically includes translational, rotational, and vibrational motion. In general, monatomic gases have 3 degrees of freedom, all translational, since they can move in three dimensional space - up and down, left and right, forward and backward.

However, diatomic and polyatomic gases have additional degrees of freedom related to rotational and possibly vibrational movements. Diatomic molecules, like hydrogen or oxygen, have 3 translational and 2 rotational degrees of freedom, totaling 5. These rotational degrees of freedom come into play because these molecules can rotate around two axes perpendicular to the bond connecting the two atoms.

Understanding Internal Energy

These degrees of freedom directly correspond to the internal energy of a gas. According to the equipartition theorem, each degree of freedom contributes \(\frac{1}{2}kT\) to the internal energy per molecule, where \(k\) is the Boltzmann constant and \(T\) is the temperature. Therefore, understanding the degrees of freedom helps us predict how energy is distributed in a gas and how it changes with conditions such as temperature changes.

Temperature Change in Ideal Gas

When studying ideal gases, one key aspect is analyzing how their internal energy changes with temperature. Temperature change in an ideal gas is directly related to changes in its kinetic energy because an ideal gas's internal energy is entirely kinetic, disregarding potential energy as gas particles are considered to have no interactions.

The internal energy change \(\Delta U\) of an ideal gas due to a temperature change can be calculated using the equation \(\Delta U = nC_v\Delta T\), where \(n\) represents the number of moles, \(C_v\) is the molar heat capacity at constant volume, and \(\Delta T\) is the change in temperature.

For instance, if the temperature of a diatomic ideal gas increases, its particles move faster, which leads to an increase in internal energy. This relationship shows how sensitive gases are to temperature changes and underscores why controlling temperature is crucial in various scientific and industrial processes that involve gases. The calculation of internal energy change offers valuable insights into the behavior of gases and reflects the fundamental principles governing thermodynamics.