Question

Suppose \(15.0 \mathrm{~L}\) of an ideal monatomic gas at a pressure of \(1.50 \cdot 10^{5} \mathrm{kPa}\) is expanded adiabatically (no heat transfer) until the volume is doubled. a) What is the pressure of the gas at the new volume? b) If the initial temperature of the gas was \(300 . \mathrm{K},\) what is its final temperature after the expansion?

Short answer

(Use the initial pressure of \(1.50 \times 10^{5} \mathrm{kPa}\), initial volume of \(15.0 \mathrm{L}\), and initial temperature of \(300 \mathrm{K}\).) Answer: The final pressure and temperature of the gas after the adiabatic expansion are approximately \(4.47 \times 10^4 \mathrm{kPa}\) and \(178.2 \mathrm{K}\), respectively.

Step-by-step solution

Identify the adiabatic equation for ideal gases

The adiabatic equation for ideal gases is as follows: \(P_i V_i^{\gamma} = P_f V_f^{\gamma}\) Where: \(P_i\) = initial pressure \(V_i\) = initial volume \(\gamma\) = adiabatic index (For a monatomic gas, \(\gamma = \dfrac{5}{3}\)) \(P_f\) = final pressure, and \(V_f\) = final volume

Plug in the given values and solve for the final pressure

Using the given initial values and knowing that the volume doubles, we will plug them into the equation and solve for \(P_f\): \(P_i = 1.50 \times 10^{5} \mathrm{kPa}\) \(V_i = 15.0 \mathrm{L}\) \(V_f = 2 \times V_i = 30.0 \mathrm{L}\) \(\gamma = \dfrac{5}{3}\) Inserting these values into the adiabatic equation: \((1.50 \times 10^{5} \mathrm{kPa})(15.0 \mathrm{L})^{\frac{5}{3}} = P_f (30.0 \mathrm{L})^{\frac{5}{3}}\) Now we can solve for \(P_f\): \(P_f = \frac{(1.50 \times 10^{5} \mathrm{kPa})(15.0 \mathrm{L})^{\frac{5}{3}}}{(30.0 \mathrm{L})^{\frac{5}{3}}} \approx 4.47 \times 10^4 \mathrm{kPa}\) So, the pressure of the gas after expansion is approximately \(4.47 \times 10^4 \mathrm{kPa}\).

Use the ideal gas law to find the final temperature

The ideal gas law states: \(PV = nRT\) Where: P = pressure V = volume n = number of moles R = ideal gas constant, and T = temperature In this problem, we have to find the final temperature, \(T_f\), after the adiabatic expansion. However, we can eliminate the number of moles, 'n', by dividing the initial state equation by the final state equation: \(\frac{P_i V_i}{T_i} = \frac{P_f V_f}{T_f}\) Now we can solve for \(T_f\) by inserting the values of \(P_i, V_i, T_i, P_f\), and \(V_f\): \(T_f = \frac{P_f V_f}{P_i V_i} T_i\) Using the given values, we find the final temperature: \(T_f = \frac{(4.47 \times 10^4 \mathrm{kPa})(30.0 \mathrm{L})}{(1.50 \times 10^5 \mathrm{kPa})(15.0 \mathrm{L})} (300 \mathrm{K}) \approx 178.2 \mathrm{K}\) So, the final temperature after adiabatic expansion is approximately \(178.2 \mathrm{K}\). In conclusion, a) The pressure of the gas at the new volume is approximately \(4.47 \times 10^4 \mathrm{kPa}\). b) The final temperature of the gas after the expansion is approximately \(178.2 \mathrm{K}\).

Key concepts

Ideal Monatomic Gas

An ideal monatomic gas is a theoretical model that assumes a perfectly ideal gas composed of singular atoms, not bound to each other. This model is crucial in physics and chemistry for simplifying the study of gases and their behavior under various conditions.

In the context of our adiabatic process example, the gas consists of individual, non-interacting atoms that exhibit ideal behavior. Unlike diatomic or polyatomic gases, monatomic gases have simpler degrees of freedom, typically three, correlating to their motion in three-dimensional space (x, y, and z axes). These degrees of freedom directly impact heat capacity and other thermodynamic properties.

Importantly, ideal monatomic gases adhere to the adiabatic equation, which illustrates how they behave when they expand or compress without exchanging heat with the environment. The adiabatic process depends on the specific heat capacities at constant volume and pressure, which are related to the degrees of freedom of the gas particles.

Ideal Gas Law

The ideal gas law is a fundamental equation in thermodynamics, expressing the relationship between the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas. It is often represented as:
\[ PV = nRT \]
where R is the universal gas constant. This equation is particularly powerful because it assumes a simplified model where gas particles are in constant, random, and elastic collision, and the size of the gas particles and intermolecular forces are negligible.

For practical applications, such as the exercise we're discussing, the ideal gas law allows us to determine a missing variable if the others are known. In adiabatic processes, where the number of moles remains constant, the law can be adapted to relate changes in pressure, volume, and temperature, which is useful when comparing the initial and final states of a gas.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. One of the key principles in thermodynamics is the conservation of energy, which is essential when examining the adiabatic process in the given exercise.

An adiabatic process is a type of thermodynamic process that occurs without transferring heat between a system and its surroundings. This is a very idealized concept, as it's impossible to have a perfectly adiabatic system due to inevitable thermal losses. However, this concept is critical in understanding processes in isolated systems.

The exercise's adiabatic expansion of an ideal monatomic gas illustrates one of the many applications of thermodynamics in real-world phenomena. The details calculated - a change in pressure and temperature after the expansion - are based on thermodynamic principles, showing the interconnectedness of pressure, volume, and temperature in adiabatic systems.