Question

The pressure of \(6.0 \mathrm{~L}\) of an ideal gas in a flexible container is decreased to one-third of its original pressure, and its absolute temperature is decreased by one-half. What is the final volume of the gas?

Short answer

The final volume of the gas is 18 L.

Step-by-step solution

Understanding the Problem

We need to find the final volume of an ideal gas after its pressure is decreased to one-third and its temperature is decreased by half. We are given the initial volume of the gas as \(6.0 \mathrm{~L}\).

Using the Ideal Gas Law

The ideal gas law is given by \(PV = nRT\). For the sake of our comparison, we will consider changes in pressure \(P\), volume \(V\), and temperature \(T\). We assume \(n\) (number of moles) and \(R\) (ideal gas constant) remain constant.

Setting Up Relationships

We know from the problem that the final pressure \(P_f\) is one-third of the initial pressure \(P_i\), and final temperature \(T_f\) is one-half of the initial temperature \(T_i\), so we have: \(P_f = \frac{1}{3}P_i\) and \(T_f = \frac{1}{2}T_i\).

Applying the Combined Gas Law

Using the combined gas law \(\frac{P_iV_i}{T_i} = \frac{P_fV_f}{T_f}\), we substitute \(V_i = 6.0 \mathrm{~L}\), \(P_f = \frac{1}{3}P_i\), and \(T_f = \frac{1}{2}T_i\) to get:\[ \frac{P_i \cdot 6.0}{T_i} = \frac{\left(\frac{1}{3}P_i\right) \cdot V_f}{\left(\frac{1}{2}T_i\right)} \]

Simplifying the Equation

Simplify the expression:\[ 12P_i = \frac{1}{3}P_i \cdot V_f \cdot 2 \]\[ 12P_i = \frac{2}{3}P_i \cdot V_f \]Cancelling \(P_i\) from both sides (assuming \(P_i eq 0\)), we get:\[ 12 = \frac{2}{3}V_f \]

Solving for Final Volume

Multiply both sides by 3:\[ 36 = 2V_f \]Divide by 2 to isolate \(V_f\):\[ V_f = 18 \mathrm{~L} \]

Key concepts

pressure

Pressure in the context of gases is an expression of the force exerted by gas particles against the walls of its container. It is influenced by the number of particles and their kinetic energy. The more particles present, and the faster they move, the greater the pressure. In ideal gases, pressure is related to other properties such as volume and temperature. The ideal gas law formula \(PV = nRT\) helps illustrate this relationship. Here, \(P\) stands for pressure, highlighting how changes in pressure can affect, and be affected by, the other variables.
In our problem, the pressure of the gas is reduced to one-third, meaning the force exerted by the gas particles is lessened. The result of this decrease can impact the gas's volume, as seen in the step-by-step solution. This relationship underscores why pressure is a fundamental concept when studying gases.

temperature

Temperature measures the average kinetic energy of particles in a substance. In gases, temperature is crucial because it affects how fast the particles move. The higher the temperature, the faster the particles move, leading to higher pressure if volume remains constant, as given by the ideal gas law.
In this exercise, the gas's absolute temperature decreases by half. This reduction means that the particles are moving slower, therefore exerting less force on the container walls. This drop not only affects pressure but also influences the final volume of the gas. This showcases the interconnectedness of the gas properties, illustrating why changes in temperature cannot be ignored when solving gas problems.

volume

Volume refers to the space that a gas occupies. In a flexible container, as described in this problem, the volume has the potential to change as the pressure or temperature changes. According to the ideal gas law and combined gas law, volume is directly proportional to the number of moles and temperature, and inversely proportional to pressure. This relationship is captured in the formula \( \frac{P_iV_i}{T_i} = \frac{P_fV_f}{T_f} \), as used in the solution.
After reducing both pressure and temperature, the initial conditions change, directly impacting the final volume. As calculated, these conditions led to a final volume of 18 L, highlighting the dynamic relationship and balancing act of the gas's properties.