91Ó°ÊÓ

Thermodynamics is the branch of physics that deals with heat, work, and the forms of energy transformation. It plays an essential role in understanding how systems exchange energy and perform work. For instance, when we expand a gas at constant pressure, we use thermodynamic principles to compute work done by the gas and the changes in its internal energy. In the context of the ideal gas laws, thermodynamics helps us comprehend the behavior of gases under varying conditions of temperature, pressure, and volume. Recalling the first law of thermodynamics, which states that energy cannot be created or destroyed but only transferred or transformed, we can say that the change in internal energy of an ideal gas is the sum of the work done by the gas and the heat added to it.

When an ideal gas expands or contracts, it performs work. The work done by a gas is a key concept in thermodynamics, and it is especially straightforward to calculate when the process occurs at a constant pressure. The formula for work done by the gas during expansion or compression at constant pressure is given by:

\ W = P \times ( V_f - V_i ) \

Here, \(P\) is the constant pressure, \(V_f\) is the final volume, and \(V_i\) is the initial volume. This relationship shows that work is the product of the pressure and the change in volume. In our example, we found that the work done by the gas during expansion from 0.200 \(\text{m}^3\) to 0.600 \(\text{m}^3\) at a pressure of 6.00 \( \times 10^{6} \) \(\text{Pa}\) is 2.40 \( \times 10^{6} \) \(\text{J}\).

Internal energy is a measure of the total energy contained within a system, stemming from both kinetic and potential energies of its molecules. For an ideal gas, only the kinetic energy matters, as there are no intermolecular forces. The change in internal energy \(\Delta U\) of an ideal gas can be evaluated using the specific heat at constant volume \(C_V\):

\$ \Delta U = n C_V (T_f - T_i) \

\(n\) is the number of moles, \(C_V\) is the specific heat capacity at constant volume, and \((T_f - T_i)\) is the change in temperature. For a diatomic gas, \(C_V\) is \$ \frac{5}{2}R \ $$ where \(R\) is the universal gas constant (8.314 \(\text{J} \cdot \text{K}^{-1} \cdot \text{mol}\text{}^{-1}\)). In the given problem, we determined that the change in internal energy, \(\Delta U\), is roughly 6.0 \(\times 10^{4} \) \(\text{J}\).

Thermal expansion of gases is the change in volume of a gas with a change in temperature at constant pressure. According to Charles’s Law, at constant pressure, the volume of an ideal gas is directly proportional to its absolute temperature. This can be mathematically represented as:

\ \ \$ \frac{V_i}{T_i} = \$ \frac{V_f}{T_f} \

This relationship allows us to determine the final temperature of the gas after expansion. When the initial and final volumes (\(V_i\) and \(V_f\)) and the initial temperature (\(T_i\)) are known, the final temperature (\(T_f\)) can be found by rearranging the equation:

\ T_f = T_i \$ \frac{V_f}{V_i} \$

In our exercise, the gas expanded from 0.200 \(\text{m}^3\) to 0.600 \(\text{m}^3\). Hence, the final temperature was found to be 900.0 K, tripling from the initial 300.0 K.